[SOLVED] Why do we need quantum fields in QED?

[Eugene Stefanovich]

This question was raised a few times in recent discussions,
but I don't think we got to the bottom of it.
The commonly accepted wisdom is that quantum fields are
the most fundamental ingredients of the theory. I would like
to challenge this wisdom. Let me give you my take on this, and
see what you think. Here is my logic:

We are looking for a relativistic quantum description of interacting
electrons, positrons, and photons. I would like to distinguish
between fundamental and technical ingredients in this description.
Fundamental objects MUST be present. You simply cannot formulate a
theory without them. Technical objects are optional. They
may be helpful for simplifying some derivations/calculations,
but, in principle, a theory can be formulated without them.

First, I would like to list theoretical objects that I consider
fundamental:

1. Since this is a quantum theory we must have a Hilbert space of
states. Relativity implies that particles can be created and
annihilated. Therefore, we build the Hilbert space as a Fock space:
a direct sum of n-particle spaces where n varies from 0 to
infinity.

2. Any state of the system is represented by a state vector
|Psi> in the Fock space.

3. We must define Hermitian operators of observables
(position, momentum, spin, etc.). The operators corresponding
to individual particles come together with the Fock space
construction in 1. The "total" operators for the entire system
(total momentum, mass, energy, etc.) are defined as functions
of the generators of the Poincare group representation in the
Fock space. This representation allows us to translate the
description of the system between different inertial frames in
accordance with the principle of relativity.

4. The eigenvectors of the observables provide for us
convenient bases in the Fock space. So, we can always
expand the state vector |Psi> in one of those bases and
find the probabilities for measuring different values of
observables.

5. The most important generator of the Poincare group
is the Hamiltonian H. If we know H, then for any |Psi>
we can find its time evolution, we can calculate the
scattering matrix, or the spectrum of bound states,
in other words, all we want to know about the physical system.
Once ingredients 1. - 5. are present, the theory is complete.
Any physical property can be calculated by a routine application
of known formulas.

The next question is how do we get H? In principle, we could choose
some basis in the Fock space, write H as a matrix in this basis and
simply fit the matrix elements to experimental observations.
This is not impossible, but nobody is doing this for obvious
reasons. The point of mentioning this approach is to emphasise that
finding the correct expression for H is not a fundamental
problem, but rather a technical one. So, now I switch to technical
ingredients that are introduced in QED with the sole purpose of
assisting the derivation of H.

a) it is convenient to introduce particle creation and annihilation
operators in the Fock space. They do not have any physical meaning
by themselves, but they allow to simplify notation for operators of
observables, including H

b) it is convenient to form certain linear combinations
of creation and annihilation operators, called "quantum fields".
Weinberg did a good job in explaining how the Hamiltonian H
can be constructed from these fields, so that the full set
of 10 generators satisfies the commutation relations of the Poincare
group.

After the Hamiltonian is constructed as above, we don't need
the temporary
crutches a) and b) anymore. For example, we can just write the
Hamiltonian as a matrix in our favorite basis, and do all calculations
in this matrix form. In other words, creation/annihilation operators
and quantum fields are nice technical additions to the theory, but
they do not belong to the fundamental ingredients. We can do without
them.

Eugene.

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[markwh04@yahoo.com]

Eugene Stefanovich wrote:
> We are looking for a relativistic quantum description [...]

That's why. It falls out, almost automatically, from the Wigner
classification of the irreducible representations of the Poincare'
group -- which, in turn, gives you all the fundamental systems that
satisfy the principle of relativity (i.e., whose state spaces are
invariant subspaces under the action of the Poincare' group).

> 1. Since this is a quantum theory we must have a Hilbert space of
> states.

Hilbert spaces have nothing, per se, to do with quantum theory.
Classical physics can also be represented by Hilbert spaces; as well as
hybrid classico-quantum theories comprising quantum theories with
superselection. The only distinctive feature of a (pure) quantum
theory is the absence of superselection.

> Relativity implies that particles can be created and
> annihilated.

Relativity doesn't even imply a particle picture in the first place.
And for General Relativitic quantum theories it implies the very
opposite -- no particle picture! Only in the special cases of
stationary spacetimes or asymptotically flat spacetimes can you build
up a consistent particle interpretation.

Furthermore ...

> 2. Any state of the system is represented by a state vector
> |Psi> in the Fock space.

... no-go theorems, such as Haag's theorem absolutely proscribe the use
of Fock spaces in any theory involving interacting fields. That is: if
there is interaction then (by Haag's theorem), the Fock space
representation can NOT be correct!

Your approach is way to shallow. These issues have been well studied
in much greater depth over the past 50 years; the main consensus is
that the foundation of any approach lies with the underlying algebra of
observables -- whether it be a classical, quantum theory or any
combination of the two.

The existence and form of the corresponding state spaces emerges as a
THEOREM from the definition of the algebra. One can, in general,
define a concept of "state" for any of a large variety of algebras;
more (in fact) that those which are used in classical or quantum
theory.

There are such things as pure states and mixed states -- the latter
which lay completely beyond your ken in the above. There are an
incredibly large number of representations of the algebra of
observables associated with the free fields, besides the Fock space
representation -- something which goes way beyond and way deeper that
what you're contemplating.

All of these issues have been studied and delineated in exhaustive
depth actually since the 1930's and von Neumann; and particularly since
the 1960's and Haag's reintroduction of the algebraic approach into
quantum field theory (following the no-go result of Haag's theorem; and
the discovery of the difficulties behind the particle concept in a
general relativistic quantum field theory).

[Igor Khavkine]

Eugene Stefanovich wrote:

Regarding your original post. Your argument falls through when
you declare that particles are "fundamental" in the way you define that
term. One can take fields as a starting point and never mention
particles at all, they will fall out automatically. Just as fields will
fall out automatically when you start with particles. Both formulations
are equally "fundamental" or equally not so, however you want to
consider them.

> I am trying to avoid discussion of gravity and curved spacetimes.
> There are enough troubles in understanding simple electro-magnetic
> interactions. Can we stick to "flat spacetimes" as you call them.

First, I don't see much trouble understanding simple electro-magnetic
interactions. Moreover, if you wish to talk of fundamentals, you must
take the most general situation possible, which includes curved
backgrounds. If you stick to flat space-time, you are stuck in a
stale-mate since the particle and field formulations are equaivalent.
If you refuse to consider curved backgrounds, you are simply refusing
to acknoledge a failing of the particle approach, since it's been known
for decades that the field approach comes out a clear winner there.

> The statement of [Haag's] the theorem is that interacting quantum field
> (i.e., the one whose time evolution is described by the full interacting
> Hamiltonian) cannot have usual tensor transformations with respect to
> Lorentz boosts. If you insist on the field-based description of nature,
> then this theorem is a big obstacle. However, this theorem can be safely
> ignored in the particle-based description. I do not assign any physical
> meaning to free fields and to interacting fields. So, I do not care
> whether interacting field is "Lorentz invariant" or not.

As soon as you have a Fock space, you have a field theory. This is a
mathematical fact, no matter how you constructed the Fock space.
Therefore, any consequence of Haag's theorem will apply to such a
theory equally well. And, unlike you claim, Haag's theorem is not a
major obstacle. The non-trivial representations of the operator algebra
do exist and they are implicitly (perturbatively) constructed during
any QFT calculation involving interactions.

> You are saying that all issues "have been studied and delineated in
> exhaustive depth" What about the time evolution of interacting systems?
> Take the simplest interacting system - two electrons. Suppose that
> I gave you a full description (wave function) of this system at time
> t=0. How would you find the state of the system at a later time t?
> The amazing thing (to me) is that field-based quantum electrodynamics
> doesn't have a clue how to do that. I mean a rigorous approach.
> I don't ask you to solve
> the equations of motion. I even don't ask you to write the full set of
> equations to be solved. I am just asking about a general algorithm how
> to do that. Which steps in what order should be taken? So that, if we
> had a supercomputer we could give it all necessary instructions.

This algorithm is called the closed time path formalism. It exist, it
works, and it has been extensively discussed here in the past.

> Normally, in quantum theory in order to do the time evolution,
> we need a Hamiltonian.
> But there is no explicit expression for a finite interacting Hamiltonian
> in the field-based QED. That's the problem.

Explicit expression? I thought you wanted an algorithm. You'll get an
explicit expression once you apply the algorithm.

Igor

[Igor Khavkine]

Eugene Stefanovich wrote:

Regarding your original post. Your argument falls through when
you declare that particles are "fundamental" in the way you define that
term. One can take fields as a starting point and never mention
particles at all, they will fall out automatically. Just as fields will
fall out automatically when you start with particles. Both formulations
are equally "fundamental" or equally not so, however you want to
consider them.

> I am trying to avoid discussion of gravity and curved spacetimes.
> There are enough troubles in understanding simple electro-magnetic
> interactions. Can we stick to "flat spacetimes" as you call them.

First, I don't see much trouble understanding simple electro-magnetic
interactions. Moreover, if you wish to talk of fundamentals, you must
take the most general situation possible, which includes curved
backgrounds. If you stick to flat space-time, you are stuck in a
stale-mate since the particle and field formulations are equaivalent.
If you refuse to consider curved backgrounds, you are simply refusing
to acknoledge a failing of the particle approach, since it's been known
for decades that the field approach comes out a clear winner there.

> The statement of [Haag's] the theorem is that interacting quantum field
> (i.e., the one whose time evolution is described by the full interacting
> Hamiltonian) cannot have usual tensor transformations with respect to
> Lorentz boosts. If you insist on the field-based description of nature,
> then this theorem is a big obstacle. However, this theorem can be safely
> ignored in the particle-based description. I do not assign any physical
> meaning to free fields and to interacting fields. So, I do not care
> whether interacting field is "Lorentz invariant" or not.

As soon as you have a Fock space, you have a field theory. This is a
mathematical fact, no matter how you constructed the Fock space.
Therefore, any consequence of Haag's theorem will apply to such a
theory equally well. And, unlike you claim, Haag's theorem is not a
major obstacle. The non-trivial representations of the operator algebra
do exist and they are implicitly (perturbatively) constructed during
any QFT calculation involving interactions.

> You are saying that all issues "have been studied and delineated in
> exhaustive depth" What about the time evolution of interacting systems?
> Take the simplest interacting system - two electrons. Suppose that
> I gave you a full description (wave function) of this system at time
> t=0. How would you find the state of the system at a later time t?
> The amazing thing (to me) is that field-based quantum electrodynamics
> doesn't have a clue how to do that. I mean a rigorous approach.
> I don't ask you to solve
> the equations of motion. I even don't ask you to write the full set of
> equations to be solved. I am just asking about a general algorithm how
> to do that. Which steps in what order should be taken? So that, if we
> had a supercomputer we could give it all necessary instructions.

This algorithm is called the closed time path formalism. It exist, it
works, and it has been extensively discussed here in the past.

> Normally, in quantum theory in order to do the time evolution,
> we need a Hamiltonian.
> But there is no explicit expression for a finite interacting Hamiltonian
> in the field-based QED. That's the problem.

Explicit expression? I thought you wanted an algorithm. You'll get an
explicit expression once you apply the algorithm.

Igor

[Igor Khavkine]

Eugene Stefanovich wrote:

Regarding your original post. Your argument falls through when
you declare that particles are "fundamental" in the way you define that
term. One can take fields as a starting point and never mention
particles at all, they will fall out automatically. Just as fields will
fall out automatically when you start with particles. Both formulations
are equally "fundamental" or equally not so, however you want to
consider them.

> I am trying to avoid discussion of gravity and curved spacetimes.
> There are enough troubles in understanding simple electro-magnetic
> interactions. Can we stick to "flat spacetimes" as you call them.

First, I don't see much trouble understanding simple electro-magnetic
interactions. Moreover, if you wish to talk of fundamentals, you must
take the most general situation possible, which includes curved
backgrounds. If you stick to flat space-time, you are stuck in a
stale-mate since the particle and field formulations are equaivalent.
If you refuse to consider curved backgrounds, you are simply refusing
to acknoledge a failing of the particle approach, since it's been known
for decades that the field approach comes out a clear winner there.

> The statement of [Haag's] the theorem is that interacting quantum field
> (i.e., the one whose time evolution is described by the full interacting
> Hamiltonian) cannot have usual tensor transformations with respect to
> Lorentz boosts. If you insist on the field-based description of nature,
> then this theorem is a big obstacle. However, this theorem can be safely
> ignored in the particle-based description. I do not assign any physical
> meaning to free fields and to interacting fields. So, I do not care
> whether interacting field is "Lorentz invariant" or not.

As soon as you have a Fock space, you have a field theory. This is a
mathematical fact, no matter how you constructed the Fock space.
Therefore, any consequence of Haag's theorem will apply to such a
theory equally well. And, unlike you claim, Haag's theorem is not a
major obstacle. The non-trivial representations of the operator algebra
do exist and they are implicitly (perturbatively) constructed during
any QFT calculation involving interactions.

> You are saying that all issues "have been studied and delineated in
> exhaustive depth" What about the time evolution of interacting systems?
> Take the simplest interacting system - two electrons. Suppose that
> I gave you a full description (wave function) of this system at time
> t=0. How would you find the state of the system at a later time t?
> The amazing thing (to me) is that field-based quantum electrodynamics
> doesn't have a clue how to do that. I mean a rigorous approach.
> I don't ask you to solve
> the equations of motion. I even don't ask you to write the full set of
> equations to be solved. I am just asking about a general algorithm how
> to do that. Which steps in what order should be taken? So that, if we
> had a supercomputer we could give it all necessary instructions.

This algorithm is called the closed time path formalism. It exist, it
works, and it has been extensively discussed here in the past.

> Normally, in quantum theory in order to do the time evolution,
> we need a Hamiltonian.
> But there is no explicit expression for a finite interacting Hamiltonian
> in the field-based QED. That's the problem.

Explicit expression? I thought you wanted an algorithm. You'll get an
explicit expression once you apply the algorithm.

Igor

[Charles Francis]

In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
<eugenev@synopsys.com> writes
>
[snip]

What you describe is essentially correct, and is used as the basis of
what JB once called "naive" treatments of QED. I have put a lot of work
into making it rigorous. The fundamental problem is that you can carry
out a discrete construction pretty much exactly as stated, but a
discrete construction does not obey manifest covariance. If you try and
take the limit and move over to a continuum in order to recover
covariance, the theory breaks down in the Landau pole.

I have a paper on discrete qed, but the real issue is not whether it is
a sound mathematical construction, but whether it is valid as a physical
model. This actually depends on one's philosophical stance re the
meaning of covariance and the nature of physical measurement. In order
to establish that a discrete model is physically legitimate I have
written a paper, gr-qc/0508077, currently being refereed which, imv,
pretty much does the job. It also makes some unexpected predictions in
cosmology, so hopefully it is testable.

Regards

--
Charles Francis

[Charles Francis]

In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
<eugenev@synopsys.com> writes
>
[snip]

What you describe is essentially correct, and is used as the basis of
what JB once called "naive" treatments of QED. I have put a lot of work
into making it rigorous. The fundamental problem is that you can carry
out a discrete construction pretty much exactly as stated, but a
discrete construction does not obey manifest covariance. If you try and
take the limit and move over to a continuum in order to recover
covariance, the theory breaks down in the Landau pole.

I have a paper on discrete qed, but the real issue is not whether it is
a sound mathematical construction, but whether it is valid as a physical
model. This actually depends on one's philosophical stance re the
meaning of covariance and the nature of physical measurement. In order
to establish that a discrete model is physically legitimate I have
written a paper, gr-qc/0508077, currently being refereed which, imv,
pretty much does the job. It also makes some unexpected predictions in
cosmology, so hopefully it is testable.

Regards

--
Charles Francis

[Charles Francis]

In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
<eugenev@synopsys.com> writes
>
[snip]

What you describe is essentially correct, and is used as the basis of
what JB once called "naive" treatments of QED. I have put a lot of work
into making it rigorous. The fundamental problem is that you can carry
out a discrete construction pretty much exactly as stated, but a
discrete construction does not obey manifest covariance. If you try and
take the limit and move over to a continuum in order to recover
covariance, the theory breaks down in the Landau pole.

I have a paper on discrete qed, but the real issue is not whether it is
a sound mathematical construction, but whether it is valid as a physical
model. This actually depends on one's philosophical stance re the
meaning of covariance and the nature of physical measurement. In order
to establish that a discrete model is physically legitimate I have
written a paper, gr-qc/0508077, currently being refereed which, imv,
pretty much does the job. It also makes some unexpected predictions in
cosmology, so hopefully it is testable.

Regards

--
Charles Francis

[Charles Francis]

In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
<eugenev@synopsys.com> writes
>
[snip]

What you describe is essentially correct, and is used as the basis of
what JB once called "naive" treatments of QED. I have put a lot of work
into making it rigorous. The fundamental problem is that you can carry
out a discrete construction pretty much exactly as stated, but a
discrete construction does not obey manifest covariance. If you try and
take the limit and move over to a continuum in order to recover
covariance, the theory breaks down in the Landau pole.

I have a paper on discrete qed, but the real issue is not whether it is
a sound mathematical construction, but whether it is valid as a physical
model. This actually depends on one's philosophical stance re the
meaning of covariance and the nature of physical measurement. In order
to establish that a discrete model is physically legitimate I have
written a paper, gr-qc/0508077, currently being refereed which, imv,
pretty much does the job. It also makes some unexpected predictions in
cosmology, so hopefully it is testable.

Regards

--
Charles Francis

[Eugene Stefanovich]

Charles Francis wrote:
> In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
> <eugenev@synopsys.com> writes
>
> [snip]
>
> What you describe is essentially correct, and is used as the basis of
> what JB once called "naive" treatments of QED. I have put a lot of work
> into making it rigorous. The fundamental problem is that you can carry
> out a discrete construction pretty much exactly as stated, but a
> discrete construction does not obey manifest covariance. If you try and
> take the limit and move over to a continuum in order to recover
> covariance, the theory breaks down in the Landau pole.
>
> I have a paper on discrete qed, but the real issue is not whether it is
> a sound mathematical construction, but whether it is valid as a physical
> model. This actually depends on one's philosophical stance re the
> meaning of covariance and the nature of physical measurement. In order
> to establish that a discrete model is physically legitimate I have
> written a paper, gr-qc/0508077, currently being refereed which, imv,
> pretty much does the job. It also makes some unexpected predictions in
> cosmology, so hopefully it is testable.

Thank you for the reference, but I suspect we are talking about
different things. I was referring to the standard QED in which
positions can be measured with unlimited precision and the spectrum of
the position operator is continuous. The theory I had in mind has
full relativistic invariance: the interacting generators of the
Poincare group form a representation of the Poincare Lie algebra
as in S. Weinberg "The quantum theory of fields", vol. 1
eqs. (3.3.11) - (3.3.17).

Eugene.

[Eugene Stefanovich]

Charles Francis wrote:
> In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
> <eugenev@synopsys.com> writes
>
> [snip]
>
> What you describe is essentially correct, and is used as the basis of
> what JB once called "naive" treatments of QED. I have put a lot of work
> into making it rigorous. The fundamental problem is that you can carry
> out a discrete construction pretty much exactly as stated, but a
> discrete construction does not obey manifest covariance. If you try and
> take the limit and move over to a continuum in order to recover
> covariance, the theory breaks down in the Landau pole.
>
> I have a paper on discrete qed, but the real issue is not whether it is
> a sound mathematical construction, but whether it is valid as a physical
> model. This actually depends on one's philosophical stance re the
> meaning of covariance and the nature of physical measurement. In order
> to establish that a discrete model is physically legitimate I have
> written a paper, gr-qc/0508077, currently being refereed which, imv,
> pretty much does the job. It also makes some unexpected predictions in
> cosmology, so hopefully it is testable.

Thank you for the reference, but I suspect we are talking about
different things. I was referring to the standard QED in which
positions can be measured with unlimited precision and the spectrum of
the position operator is continuous. The theory I had in mind has
full relativistic invariance: the interacting generators of the
Poincare group form a representation of the Poincare Lie algebra
as in S. Weinberg "The quantum theory of fields", vol. 1
eqs. (3.3.11) - (3.3.17).

Eugene.

[Eugene Stefanovich]

Charles Francis wrote:
> In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
> <eugenev@synopsys.com> writes
>
> [snip]
>
> What you describe is essentially correct, and is used as the basis of
> what JB once called "naive" treatments of QED. I have put a lot of work
> into making it rigorous. The fundamental problem is that you can carry
> out a discrete construction pretty much exactly as stated, but a
> discrete construction does not obey manifest covariance. If you try and
> take the limit and move over to a continuum in order to recover
> covariance, the theory breaks down in the Landau pole.
>
> I have a paper on discrete qed, but the real issue is not whether it is
> a sound mathematical construction, but whether it is valid as a physical
> model. This actually depends on one's philosophical stance re the
> meaning of covariance and the nature of physical measurement. In order
> to establish that a discrete model is physically legitimate I have
> written a paper, gr-qc/0508077, currently being refereed which, imv,
> pretty much does the job. It also makes some unexpected predictions in
> cosmology, so hopefully it is testable.

Thank you for the reference, but I suspect we are talking about
different things. I was referring to the standard QED in which
positions can be measured with unlimited precision and the spectrum of
the position operator is continuous. The theory I had in mind has
full relativistic invariance: the interacting generators of the
Poincare group form a representation of the Poincare Lie algebra
as in S. Weinberg "The quantum theory of fields", vol. 1
eqs. (3.3.11) - (3.3.17).

Eugene.

[Eugene Stefanovich]

Charles Francis wrote:
> In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
> <eugenev@synopsys.com> writes
>
> [snip]
>
> What you describe is essentially correct, and is used as the basis of
> what JB once called "naive" treatments of QED. I have put a lot of work
> into making it rigorous. The fundamental problem is that you can carry
> out a discrete construction pretty much exactly as stated, but a
> discrete construction does not obey manifest covariance. If you try and
> take the limit and move over to a continuum in order to recover
> covariance, the theory breaks down in the Landau pole.
>
> I have a paper on discrete qed, but the real issue is not whether it is
> a sound mathematical construction, but whether it is valid as a physical
> model. This actually depends on one's philosophical stance re the
> meaning of covariance and the nature of physical measurement. In order
> to establish that a discrete model is physically legitimate I have
> written a paper, gr-qc/0508077, currently being refereed which, imv,
> pretty much does the job. It also makes some unexpected predictions in
> cosmology, so hopefully it is testable.

Thank you for the reference, but I suspect we are talking about
different things. I was referring to the standard QED in which
positions can be measured with unlimited precision and the spectrum of
the position operator is continuous. The theory I had in mind has
full relativistic invariance: the interacting generators of the
Poincare group form a representation of the Poincare Lie algebra
as in S. Weinberg "The quantum theory of fields", vol. 1
eqs. (3.3.11) - (3.3.17).

Eugene.

[Eugene Stefanovich]

Charles Francis wrote:
> In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
> <eugenev@synopsys.com> writes
>
> [snip]
>
> What you describe is essentially correct, and is used as the basis of
> what JB once called "naive" treatments of QED. I have put a lot of work
> into making it rigorous. The fundamental problem is that you can carry
> out a discrete construction pretty much exactly as stated, but a
> discrete construction does not obey manifest covariance. If you try and
> take the limit and move over to a continuum in order to recover
> covariance, the theory breaks down in the Landau pole.
>
> I have a paper on discrete qed, but the real issue is not whether it is
> a sound mathematical construction, but whether it is valid as a physical
> model. This actually depends on one's philosophical stance re the
> meaning of covariance and the nature of physical measurement. In order
> to establish that a discrete model is physically legitimate I have
> written a paper, gr-qc/0508077, currently being refereed which, imv,
> pretty much does the job. It also makes some unexpected predictions in
> cosmology, so hopefully it is testable.

Thank you for the reference, but I suspect we are talking about
different things. I was referring to the standard QED in which
positions can be measured with unlimited precision and the spectrum of
the position operator is continuous. The theory I had in mind has
full relativistic invariance: the interacting generators of the
Poincare group form a representation of the Poincare Lie algebra
as in S. Weinberg "The quantum theory of fields", vol. 1
eqs. (3.3.11) - (3.3.17).

Eugene.

[Charles Francis]

In message <1127528209.775506.155930@o13g2000cwo.googlegroups. com>, Igor
Khavkine <igor.kh@gmail.com> writes
>Eugene Stefanovich wrote:
>
>Regarding your original post. Your argument falls through when
>you declare that particles are "fundamental" in the way you define that
>term. One can take fields as a starting point and never mention
>particles at all, they will fall out automatically. Just as fields will
>fall out automatically when you start with particles. Both formulations
>are equally "fundamental" or equally not so, however you want to
>consider them.

Mathematically this is, of course, true, but for me it doesn't hold
water when one tries to put a physical interpretation on the fundamental
entities. A particle is a simple entity without extent and with minimal
properties. A field takes a different value at each point in spacetime -
in complexity it may be likened to a machine with an infinite number of
moving parts. Of course these days the fashion is to say "oh we mustn't
think about physical interpretation". So I guess I'm just pig headed,
because I think it is the main thing we should think about if we are
going to advance our understanding of nature.

>> I am trying to avoid discussion of gravity and curved spacetimes.
>> There are enough troubles in understanding simple electro-magnetic
>> interactions. Can we stick to "flat spacetimes" as you call them.
>
>First, I don't see much trouble understanding simple electro-magnetic
>interactions. Moreover, if you wish to talk of fundamentals, you must
>take the most general situation possible, which includes curved
>backgrounds. If you stick to flat space-time, you are stuck in a
>stale-mate since the particle and field formulations are equaivalent.
>If you refuse to consider curved backgrounds, you are simply refusing
>to acknoledge a failing of the particle approach, since it's been known
>for decades that the field approach comes out a clear winner there.

Not as far as I know. Wald and Fulling have both written books on
quantum field theory in curved space-time, as I recall they both point
to serious difficulties even in defining fields.

>> The statement of [Haag's] the theorem is that interacting quantum field
>> (i.e., the one whose time evolution is described by the full interacting
>> Hamiltonian) cannot have usual tensor transformations with respect to
>> Lorentz boosts. If you insist on the field-based description of nature,
>> then this theorem is a big obstacle. However, this theorem can be safely
>> ignored in the particle-based description. I do not assign any physical
>> meaning to free fields and to interacting fields. So, I do not care
>> whether interacting field is "Lorentz invariant" or not.
>
>As soon as you have a Fock space, you have a field theory. This is a
>mathematical fact, no matter how you constructed the Fock space.
>Therefore, any consequence of Haag's theorem will apply to such a
>theory equally well. And, unlike you claim, Haag's theorem is not a
>major obstacle. The non-trivial representations of the operator algebra
>do exist and they are implicitly (perturbatively) constructed during
>any QFT calculation involving interactions.

If you have a Fock space you have a field theory of "non-interacting
fields". Haag's theorem is an obstacle to having a theory of
"interacting fields". I inclined to agree with Eugene. "Interacting
fields" are unnecessary and a fiction. They arise by taking a wrong
approach to the subject, via quantisation of classical fields.



Regards

--
Charles Francis

[Charles Francis]

In message <1127528209.775506.155930@o13g2000cwo.googlegroups. com>, Igor
Khavkine <igor.kh@gmail.com> writes
>Eugene Stefanovich wrote:
>
>Regarding your original post. Your argument falls through when
>you declare that particles are "fundamental" in the way you define that
>term. One can take fields as a starting point and never mention
>particles at all, they will fall out automatically. Just as fields will
>fall out automatically when you start with particles. Both formulations
>are equally "fundamental" or equally not so, however you want to
>consider them.

Mathematically this is, of course, true, but for me it doesn't hold
water when one tries to put a physical interpretation on the fundamental
entities. A particle is a simple entity without extent and with minimal
properties. A field takes a different value at each point in spacetime -
in complexity it may be likened to a machine with an infinite number of
moving parts. Of course these days the fashion is to say "oh we mustn't
think about physical interpretation". So I guess I'm just pig headed,
because I think it is the main thing we should think about if we are
going to advance our understanding of nature.

>> I am trying to avoid discussion of gravity and curved spacetimes.
>> There are enough troubles in understanding simple electro-magnetic
>> interactions. Can we stick to "flat spacetimes" as you call them.
>
>First, I don't see much trouble understanding simple electro-magnetic
>interactions. Moreover, if you wish to talk of fundamentals, you must
>take the most general situation possible, which includes curved
>backgrounds. If you stick to flat space-time, you are stuck in a
>stale-mate since the particle and field formulations are equaivalent.
>If you refuse to consider curved backgrounds, you are simply refusing
>to acknoledge a failing of the particle approach, since it's been known
>for decades that the field approach comes out a clear winner there.

Not as far as I know. Wald and Fulling have both written books on
quantum field theory in curved space-time, as I recall they both point
to serious difficulties even in defining fields.

>> The statement of [Haag's] the theorem is that interacting quantum field
>> (i.e., the one whose time evolution is described by the full interacting
>> Hamiltonian) cannot have usual tensor transformations with respect to
>> Lorentz boosts. If you insist on the field-based description of nature,
>> then this theorem is a big obstacle. However, this theorem can be safely
>> ignored in the particle-based description. I do not assign any physical
>> meaning to free fields and to interacting fields. So, I do not care
>> whether interacting field is "Lorentz invariant" or not.
>
>As soon as you have a Fock space, you have a field theory. This is a
>mathematical fact, no matter how you constructed the Fock space.
>Therefore, any consequence of Haag's theorem will apply to such a
>theory equally well. And, unlike you claim, Haag's theorem is not a
>major obstacle. The non-trivial representations of the operator algebra
>do exist and they are implicitly (perturbatively) constructed during
>any QFT calculation involving interactions.

If you have a Fock space you have a field theory of "non-interacting
fields". Haag's theorem is an obstacle to having a theory of
"interacting fields". I inclined to agree with Eugene. "Interacting
fields" are unnecessary and a fiction. They arise by taking a wrong
approach to the subject, via quantisation of classical fields.



Regards

--
Charles Francis

[Charles Francis]

In message <1127528209.775506.155930@o13g2000cwo.googlegroups. com>, Igor
Khavkine <igor.kh@gmail.com> writes
>Eugene Stefanovich wrote:
>
>Regarding your original post. Your argument falls through when
>you declare that particles are "fundamental" in the way you define that
>term. One can take fields as a starting point and never mention
>particles at all, they will fall out automatically. Just as fields will
>fall out automatically when you start with particles. Both formulations
>are equally "fundamental" or equally not so, however you want to
>consider them.

Mathematically this is, of course, true, but for me it doesn't hold
water when one tries to put a physical interpretation on the fundamental
entities. A particle is a simple entity without extent and with minimal
properties. A field takes a different value at each point in spacetime -
in complexity it may be likened to a machine with an infinite number of
moving parts. Of course these days the fashion is to say "oh we mustn't
think about physical interpretation". So I guess I'm just pig headed,
because I think it is the main thing we should think about if we are
going to advance our understanding of nature.

>> I am trying to avoid discussion of gravity and curved spacetimes.
>> There are enough troubles in understanding simple electro-magnetic
>> interactions. Can we stick to "flat spacetimes" as you call them.
>
>First, I don't see much trouble understanding simple electro-magnetic
>interactions. Moreover, if you wish to talk of fundamentals, you must
>take the most general situation possible, which includes curved
>backgrounds. If you stick to flat space-time, you are stuck in a
>stale-mate since the particle and field formulations are equaivalent.
>If you refuse to consider curved backgrounds, you are simply refusing
>to acknoledge a failing of the particle approach, since it's been known
>for decades that the field approach comes out a clear winner there.

Not as far as I know. Wald and Fulling have both written books on
quantum field theory in curved space-time, as I recall they both point
to serious difficulties even in defining fields.

>> The statement of [Haag's] the theorem is that interacting quantum field
>> (i.e., the one whose time evolution is described by the full interacting
>> Hamiltonian) cannot have usual tensor transformations with respect to
>> Lorentz boosts. If you insist on the field-based description of nature,
>> then this theorem is a big obstacle. However, this theorem can be safely
>> ignored in the particle-based description. I do not assign any physical
>> meaning to free fields and to interacting fields. So, I do not care
>> whether interacting field is "Lorentz invariant" or not.
>
>As soon as you have a Fock space, you have a field theory. This is a
>mathematical fact, no matter how you constructed the Fock space.
>Therefore, any consequence of Haag's theorem will apply to such a
>theory equally well. And, unlike you claim, Haag's theorem is not a
>major obstacle. The non-trivial representations of the operator algebra
>do exist and they are implicitly (perturbatively) constructed during
>any QFT calculation involving interactions.

If you have a Fock space you have a field theory of "non-interacting
fields". Haag's theorem is an obstacle to having a theory of
"interacting fields". I inclined to agree with Eugene. "Interacting
fields" are unnecessary and a fiction. They arise by taking a wrong
approach to the subject, via quantisation of classical fields.



Regards

--
Charles Francis

[Charles Francis]

In message <1127528209.775506.155930@o13g2000cwo.googlegroups. com>, Igor
Khavkine <igor.kh@gmail.com> writes
>Eugene Stefanovich wrote:
>
>Regarding your original post. Your argument falls through when
>you declare that particles are "fundamental" in the way you define that
>term. One can take fields as a starting point and never mention
>particles at all, they will fall out automatically. Just as fields will
>fall out automatically when you start with particles. Both formulations
>are equally "fundamental" or equally not so, however you want to
>consider them.

Mathematically this is, of course, true, but for me it doesn't hold
water when one tries to put a physical interpretation on the fundamental
entities. A particle is a simple entity without extent and with minimal
properties. A field takes a different value at each point in spacetime -
in complexity it may be likened to a machine with an infinite number of
moving parts. Of course these days the fashion is to say "oh we mustn't
think about physical interpretation". So I guess I'm just pig headed,
because I think it is the main thing we should think about if we are
going to advance our understanding of nature.

>> I am trying to avoid discussion of gravity and curved spacetimes.
>> There are enough troubles in understanding simple electro-magnetic
>> interactions. Can we stick to "flat spacetimes" as you call them.
>
>First, I don't see much trouble understanding simple electro-magnetic
>interactions. Moreover, if you wish to talk of fundamentals, you must
>take the most general situation possible, which includes curved
>backgrounds. If you stick to flat space-time, you are stuck in a
>stale-mate since the particle and field formulations are equaivalent.
>If you refuse to consider curved backgrounds, you are simply refusing
>to acknoledge a failing of the particle approach, since it's been known
>for decades that the field approach comes out a clear winner there.

Not as far as I know. Wald and Fulling have both written books on
quantum field theory in curved space-time, as I recall they both point
to serious difficulties even in defining fields.

>> The statement of [Haag's] the theorem is that interacting quantum field
>> (i.e., the one whose time evolution is described by the full interacting
>> Hamiltonian) cannot have usual tensor transformations with respect to
>> Lorentz boosts. If you insist on the field-based description of nature,
>> then this theorem is a big obstacle. However, this theorem can be safely
>> ignored in the particle-based description. I do not assign any physical
>> meaning to free fields and to interacting fields. So, I do not care
>> whether interacting field is "Lorentz invariant" or not.
>
>As soon as you have a Fock space, you have a field theory. This is a
>mathematical fact, no matter how you constructed the Fock space.
>Therefore, any consequence of Haag's theorem will apply to such a
>theory equally well. And, unlike you claim, Haag's theorem is not a
>major obstacle. The non-trivial representations of the operator algebra
>do exist and they are implicitly (perturbatively) constructed during
>any QFT calculation involving interactions.

If you have a Fock space you have a field theory of "non-interacting
fields". Haag's theorem is an obstacle to having a theory of
"interacting fields". I inclined to agree with Eugene. "Interacting
fields" are unnecessary and a fiction. They arise by taking a wrong
approach to the subject, via quantisation of classical fields.



Regards

--
Charles Francis

[Charles Francis]

In message <1127528209.775506.155930@o13g2000cwo.googlegroups. com>, Igor
Khavkine <igor.kh@gmail.com> writes
>Eugene Stefanovich wrote:
>
>Regarding your original post. Your argument falls through when
>you declare that particles are "fundamental" in the way you define that
>term. One can take fields as a starting point and never mention
>particles at all, they will fall out automatically. Just as fields will
>fall out automatically when you start with particles. Both formulations
>are equally "fundamental" or equally not so, however you want to
>consider them.

Mathematically this is, of course, true, but for me it doesn't hold
water when one tries to put a physical interpretation on the fundamental
entities. A particle is a simple entity without extent and with minimal
properties. A field takes a different value at each point in spacetime -
in complexity it may be likened to a machine with an infinite number of
moving parts. Of course these days the fashion is to say "oh we mustn't
think about physical interpretation". So I guess I'm just pig headed,
because I think it is the main thing we should think about if we are
going to advance our understanding of nature.

>> I am trying to avoid discussion of gravity and curved spacetimes.
>> There are enough troubles in understanding simple electro-magnetic
>> interactions. Can we stick to "flat spacetimes" as you call them.
>
>First, I don't see much trouble understanding simple electro-magnetic
>interactions. Moreover, if you wish to talk of fundamentals, you must
>take the most general situation possible, which includes curved
>backgrounds. If you stick to flat space-time, you are stuck in a
>stale-mate since the particle and field formulations are equaivalent.
>If you refuse to consider curved backgrounds, you are simply refusing
>to acknoledge a failing of the particle approach, since it's been known
>for decades that the field approach comes out a clear winner there.

Not as far as I know. Wald and Fulling have both written books on
quantum field theory in curved space-time, as I recall they both point
to serious difficulties even in defining fields.

>> The statement of [Haag's] the theorem is that interacting quantum field
>> (i.e., the one whose time evolution is described by the full interacting
>> Hamiltonian) cannot have usual tensor transformations with respect to
>> Lorentz boosts. If you insist on the field-based description of nature,
>> then this theorem is a big obstacle. However, this theorem can be safely
>> ignored in the particle-based description. I do not assign any physical
>> meaning to free fields and to interacting fields. So, I do not care
>> whether interacting field is "Lorentz invariant" or not.
>
>As soon as you have a Fock space, you have a field theory. This is a
>mathematical fact, no matter how you constructed the Fock space.
>Therefore, any consequence of Haag's theorem will apply to such a
>theory equally well. And, unlike you claim, Haag's theorem is not a
>major obstacle. The non-trivial representations of the operator algebra
>do exist and they are implicitly (perturbatively) constructed during
>any QFT calculation involving interactions.

If you have a Fock space you have a field theory of "non-interacting
fields". Haag's theorem is an obstacle to having a theory of
"interacting fields". I inclined to agree with Eugene. "Interacting
fields" are unnecessary and a fiction. They arise by taking a wrong
approach to the subject, via quantisation of classical fields.



Regards

--
Charles Francis

[Juan R.]

Igor Khavkine wrote:
>
> Regarding your original post. Your argument falls through when
> you declare that particles are "fundamental" in the way you define that
> term. One can take fields as a starting point and never mention
> particles at all, they will fall out automatically. Just as fields will
> fall out automatically when you start with particles. Both formulations
> are equally "fundamental" or equally not so, however you want to
> consider them.

Particles are more fundamental that fields because fields are ALWAYS
-by definition- unobserved, one measures in scattering experiments are
particles, newer fields. As clearly stated by Weinberg in his volume 1,
we know more about particles that about fields.

> > I am trying to avoid discussion of gravity and curved spacetimes.
> > There are enough troubles in understanding simple electro-magnetic
> > interactions. Can we stick to "flat spacetimes" as you call them.
>
> First, I don't see much trouble understanding simple electro-magnetic
> interactions. Moreover, if you wish to talk of fundamentals, you must
> take the most general situation possible, which includes curved
> backgrounds. If you stick to flat space-time, you are stuck in a
> stale-mate since the particle and field formulations are equaivalent.
> If you refuse to consider curved backgrounds, you are simply refusing
> to acknoledge a failing of the particle approach, since it's been known
> for decades that the field approach comes out a clear winner there.

1) curved spacetime is just a view, only that. As was claimed by
Feynmann (in his lectures on gravitation) the curved spacetime
interpretation of GR is unnecesary for physics. In fact, in the torsion
re-geometrization of gravity, spacetime curvature is exactly zero.
Therefore, i think that Stefanovich could generalize his work to
torsion gravity using the flat spacetime he uses for QED.

2) That in curved spacetime there is no posibility for description of
particles is not very convincing just as shows the curved spacetime
generalization of the FFW action (See Hoyle and Narkilar gravitational
theory).

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.

I am not sure i say next, but in string heory one uses the Fock space
for the representation of states of vibrating string and one has not
fields.

> > You are saying that all issues "have been studied and delineated in
> > exhaustive depth" What about the time evolution of interacting systems?
> > Take the simplest interacting system - two electrons. Suppose that
> > I gave you a full description (wave function) of this system at time
> > t=0. How would you find the state of the system at a later time t?
> > The amazing thing (to me) is that field-based quantum electrodynamics
> > doesn't have a clue how to do that. I mean a rigorous approach.
> > I don't ask you to solve
> > the equations of motion. I even don't ask you to write the full set of
> > equations to be solved. I am just asking about a general algorithm how
> > to do that. Which steps in what order should be taken? So that, if we
> > had a supercomputer we could give it all necessary instructions.
>
> This algorithm is called the closed time path formalism. It exist, it
> works, and it has been extensively discussed here in the past.

Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
the FULL two body problem in QFT still. Moreover it is well-known that
path formalism has several flaws. For example, the standard version of
the PF offers incorrect S matrices in 'complex' problems: a typical
example is the nonlinear sigma model.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Igor Khavkine wrote:
>
> Regarding your original post. Your argument falls through when
> you declare that particles are "fundamental" in the way you define that
> term. One can take fields as a starting point and never mention
> particles at all, they will fall out automatically. Just as fields will
> fall out automatically when you start with particles. Both formulations
> are equally "fundamental" or equally not so, however you want to
> consider them.

Particles are more fundamental that fields because fields are ALWAYS
-by definition- unobserved, one measures in scattering experiments are
particles, newer fields. As clearly stated by Weinberg in his volume 1,
we know more about particles that about fields.

> > I am trying to avoid discussion of gravity and curved spacetimes.
> > There are enough troubles in understanding simple electro-magnetic
> > interactions. Can we stick to "flat spacetimes" as you call them.
>
> First, I don't see much trouble understanding simple electro-magnetic
> interactions. Moreover, if you wish to talk of fundamentals, you must
> take the most general situation possible, which includes curved
> backgrounds. If you stick to flat space-time, you are stuck in a
> stale-mate since the particle and field formulations are equaivalent.
> If you refuse to consider curved backgrounds, you are simply refusing
> to acknoledge a failing of the particle approach, since it's been known
> for decades that the field approach comes out a clear winner there.

1) curved spacetime is just a view, only that. As was claimed by
Feynmann (in his lectures on gravitation) the curved spacetime
interpretation of GR is unnecesary for physics. In fact, in the torsion
re-geometrization of gravity, spacetime curvature is exactly zero.
Therefore, i think that Stefanovich could generalize his work to
torsion gravity using the flat spacetime he uses for QED.

2) That in curved spacetime there is no posibility for description of
particles is not very convincing just as shows the curved spacetime
generalization of the FFW action (See Hoyle and Narkilar gravitational
theory).

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.

I am not sure i say next, but in string heory one uses the Fock space
for the representation of states of vibrating string and one has not
fields.

> > You are saying that all issues "have been studied and delineated in
> > exhaustive depth" What about the time evolution of interacting systems?
> > Take the simplest interacting system - two electrons. Suppose that
> > I gave you a full description (wave function) of this system at time
> > t=0. How would you find the state of the system at a later time t?
> > The amazing thing (to me) is that field-based quantum electrodynamics
> > doesn't have a clue how to do that. I mean a rigorous approach.
> > I don't ask you to solve
> > the equations of motion. I even don't ask you to write the full set of
> > equations to be solved. I am just asking about a general algorithm how
> > to do that. Which steps in what order should be taken? So that, if we
> > had a supercomputer we could give it all necessary instructions.
>
> This algorithm is called the closed time path formalism. It exist, it
> works, and it has been extensively discussed here in the past.

Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
the FULL two body problem in QFT still. Moreover it is well-known that
path formalism has several flaws. For example, the standard version of
the PF offers incorrect S matrices in 'complex' problems: a typical
example is the nonlinear sigma model.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Igor Khavkine wrote:
>
> Regarding your original post. Your argument falls through when
> you declare that particles are "fundamental" in the way you define that
> term. One can take fields as a starting point and never mention
> particles at all, they will fall out automatically. Just as fields will
> fall out automatically when you start with particles. Both formulations
> are equally "fundamental" or equally not so, however you want to
> consider them.

Particles are more fundamental that fields because fields are ALWAYS
-by definition- unobserved, one measures in scattering experiments are
particles, newer fields. As clearly stated by Weinberg in his volume 1,
we know more about particles that about fields.

> > I am trying to avoid discussion of gravity and curved spacetimes.
> > There are enough troubles in understanding simple electro-magnetic
> > interactions. Can we stick to "flat spacetimes" as you call them.
>
> First, I don't see much trouble understanding simple electro-magnetic
> interactions. Moreover, if you wish to talk of fundamentals, you must
> take the most general situation possible, which includes curved
> backgrounds. If you stick to flat space-time, you are stuck in a
> stale-mate since the particle and field formulations are equaivalent.
> If you refuse to consider curved backgrounds, you are simply refusing
> to acknoledge a failing of the particle approach, since it's been known
> for decades that the field approach comes out a clear winner there.

1) curved spacetime is just a view, only that. As was claimed by
Feynmann (in his lectures on gravitation) the curved spacetime
interpretation of GR is unnecesary for physics. In fact, in the torsion
re-geometrization of gravity, spacetime curvature is exactly zero.
Therefore, i think that Stefanovich could generalize his work to
torsion gravity using the flat spacetime he uses for QED.

2) That in curved spacetime there is no posibility for description of
particles is not very convincing just as shows the curved spacetime
generalization of the FFW action (See Hoyle and Narkilar gravitational
theory).

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.

I am not sure i say next, but in string heory one uses the Fock space
for the representation of states of vibrating string and one has not
fields.

> > You are saying that all issues "have been studied and delineated in
> > exhaustive depth" What about the time evolution of interacting systems?
> > Take the simplest interacting system - two electrons. Suppose that
> > I gave you a full description (wave function) of this system at time
> > t=0. How would you find the state of the system at a later time t?
> > The amazing thing (to me) is that field-based quantum electrodynamics
> > doesn't have a clue how to do that. I mean a rigorous approach.
> > I don't ask you to solve
> > the equations of motion. I even don't ask you to write the full set of
> > equations to be solved. I am just asking about a general algorithm how
> > to do that. Which steps in what order should be taken? So that, if we
> > had a supercomputer we could give it all necessary instructions.
>
> This algorithm is called the closed time path formalism. It exist, it
> works, and it has been extensively discussed here in the past.

Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
the FULL two body problem in QFT still. Moreover it is well-known that
path formalism has several flaws. For example, the standard version of
the PF offers incorrect S matrices in 'complex' problems: a typical
example is the nonlinear sigma model.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Igor Khavkine wrote:
>
> Regarding your original post. Your argument falls through when
> you declare that particles are "fundamental" in the way you define that
> term. One can take fields as a starting point and never mention
> particles at all, they will fall out automatically. Just as fields will
> fall out automatically when you start with particles. Both formulations
> are equally "fundamental" or equally not so, however you want to
> consider them.

Particles are more fundamental that fields because fields are ALWAYS
-by definition- unobserved, one measures in scattering experiments are
particles, newer fields. As clearly stated by Weinberg in his volume 1,
we know more about particles that about fields.

> > I am trying to avoid discussion of gravity and curved spacetimes.
> > There are enough troubles in understanding simple electro-magnetic
> > interactions. Can we stick to "flat spacetimes" as you call them.
>
> First, I don't see much trouble understanding simple electro-magnetic
> interactions. Moreover, if you wish to talk of fundamentals, you must
> take the most general situation possible, which includes curved
> backgrounds. If you stick to flat space-time, you are stuck in a
> stale-mate since the particle and field formulations are equaivalent.
> If you refuse to consider curved backgrounds, you are simply refusing
> to acknoledge a failing of the particle approach, since it's been known
> for decades that the field approach comes out a clear winner there.

1) curved spacetime is just a view, only that. As was claimed by
Feynmann (in his lectures on gravitation) the curved spacetime
interpretation of GR is unnecesary for physics. In fact, in the torsion
re-geometrization of gravity, spacetime curvature is exactly zero.
Therefore, i think that Stefanovich could generalize his work to
torsion gravity using the flat spacetime he uses for QED.

2) That in curved spacetime there is no posibility for description of
particles is not very convincing just as shows the curved spacetime
generalization of the FFW action (See Hoyle and Narkilar gravitational
theory).

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.

I am not sure i say next, but in string heory one uses the Fock space
for the representation of states of vibrating string and one has not
fields.

> > You are saying that all issues "have been studied and delineated in
> > exhaustive depth" What about the time evolution of interacting systems?
> > Take the simplest interacting system - two electrons. Suppose that
> > I gave you a full description (wave function) of this system at time
> > t=0. How would you find the state of the system at a later time t?
> > The amazing thing (to me) is that field-based quantum electrodynamics
> > doesn't have a clue how to do that. I mean a rigorous approach.
> > I don't ask you to solve
> > the equations of motion. I even don't ask you to write the full set of
> > equations to be solved. I am just asking about a general algorithm how
> > to do that. Which steps in what order should be taken? So that, if we
> > had a supercomputer we could give it all necessary instructions.
>
> This algorithm is called the closed time path formalism. It exist, it
> works, and it has been extensively discussed here in the past.

Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
the FULL two body problem in QFT still. Moreover it is well-known that
path formalism has several flaws. For example, the standard version of
the PF offers incorrect S matrices in 'complex' problems: a typical
example is the nonlinear sigma model.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Igor Khavkine wrote:
>
> Regarding your original post. Your argument falls through when
> you declare that particles are "fundamental" in the way you define that
> term. One can take fields as a starting point and never mention
> particles at all, they will fall out automatically. Just as fields will
> fall out automatically when you start with particles. Both formulations
> are equally "fundamental" or equally not so, however you want to
> consider them.

Particles are more fundamental that fields because fields are ALWAYS
-by definition- unobserved, one measures in scattering experiments are
particles, newer fields. As clearly stated by Weinberg in his volume 1,
we know more about particles that about fields.

> > I am trying to avoid discussion of gravity and curved spacetimes.
> > There are enough troubles in understanding simple electro-magnetic
> > interactions. Can we stick to "flat spacetimes" as you call them.
>
> First, I don't see much trouble understanding simple electro-magnetic
> interactions. Moreover, if you wish to talk of fundamentals, you must
> take the most general situation possible, which includes curved
> backgrounds. If you stick to flat space-time, you are stuck in a
> stale-mate since the particle and field formulations are equaivalent.
> If you refuse to consider curved backgrounds, you are simply refusing
> to acknoledge a failing of the particle approach, since it's been known
> for decades that the field approach comes out a clear winner there.

1) curved spacetime is just a view, only that. As was claimed by
Feynmann (in his lectures on gravitation) the curved spacetime
interpretation of GR is unnecesary for physics. In fact, in the torsion
re-geometrization of gravity, spacetime curvature is exactly zero.
Therefore, i think that Stefanovich could generalize his work to
torsion gravity using the flat spacetime he uses for QED.

2) That in curved spacetime there is no posibility for description of
particles is not very convincing just as shows the curved spacetime
generalization of the FFW action (See Hoyle and Narkilar gravitational
theory).

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.

I am not sure i say next, but in string heory one uses the Fock space
for the representation of states of vibrating string and one has not
fields.

> > You are saying that all issues "have been studied and delineated in
> > exhaustive depth" What about the time evolution of interacting systems?
> > Take the simplest interacting system - two electrons. Suppose that
> > I gave you a full description (wave function) of this system at time
> > t=0. How would you find the state of the system at a later time t?
> > The amazing thing (to me) is that field-based quantum electrodynamics
> > doesn't have a clue how to do that. I mean a rigorous approach.
> > I don't ask you to solve
> > the equations of motion. I even don't ask you to write the full set of
> > equations to be solved. I am just asking about a general algorithm how
> > to do that. Which steps in what order should be taken? So that, if we
> > had a supercomputer we could give it all necessary instructions.
>
> This algorithm is called the closed time path formalism. It exist, it
> works, and it has been extensively discussed here in the past.

Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
the FULL two body problem in QFT still. Moreover it is well-known that
path formalism has several flaws. For example, the standard version of
the PF offers incorrect S matrices in 'complex' problems: a typical
example is the nonlinear sigma model.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Igor Khavkine wrote:
>
> Regarding your original post. Your argument falls through when
> you declare that particles are "fundamental" in the way you define that
> term. One can take fields as a starting point and never mention
> particles at all, they will fall out automatically. Just as fields will
> fall out automatically when you start with particles. Both formulations
> are equally "fundamental" or equally not so, however you want to
> consider them.

Particles are more fundamental that fields because fields are ALWAYS
-by definition- unobserved, one measures in scattering experiments are
particles, newer fields. As clearly stated by Weinberg in his volume 1,
we know more about particles that about fields.

> > I am trying to avoid discussion of gravity and curved spacetimes.
> > There are enough troubles in understanding simple electro-magnetic
> > interactions. Can we stick to "flat spacetimes" as you call them.
>
> First, I don't see much trouble understanding simple electro-magnetic
> interactions. Moreover, if you wish to talk of fundamentals, you must
> take the most general situation possible, which includes curved
> backgrounds. If you stick to flat space-time, you are stuck in a
> stale-mate since the particle and field formulations are equaivalent.
> If you refuse to consider curved backgrounds, you are simply refusing
> to acknoledge a failing of the particle approach, since it's been known
> for decades that the field approach comes out a clear winner there.

1) curved spacetime is just a view, only that. As was claimed by
Feynmann (in his lectures on gravitation) the curved spacetime
interpretation of GR is unnecesary for physics. In fact, in the torsion
re-geometrization of gravity, spacetime curvature is exactly zero.
Therefore, i think that Stefanovich could generalize his work to
torsion gravity using the flat spacetime he uses for QED.

2) That in curved spacetime there is no posibility for description of
particles is not very convincing just as shows the curved spacetime
generalization of the FFW action (See Hoyle and Narkilar gravitational
theory).

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.

I am not sure i say next, but in string heory one uses the Fock space
for the representation of states of vibrating string and one has not
fields.

> > You are saying that all issues "have been studied and delineated in
> > exhaustive depth" What about the time evolution of interacting systems?
> > Take the simplest interacting system - two electrons. Suppose that
> > I gave you a full description (wave function) of this system at time
> > t=0. How would you find the state of the system at a later time t?
> > The amazing thing (to me) is that field-based quantum electrodynamics
> > doesn't have a clue how to do that. I mean a rigorous approach.
> > I don't ask you to solve
> > the equations of motion. I even don't ask you to write the full set of
> > equations to be solved. I am just asking about a general algorithm how
> > to do that. Which steps in what order should be taken? So that, if we
> > had a supercomputer we could give it all necessary instructions.
>
> This algorithm is called the closed time path formalism. It exist, it
> works, and it has been extensively discussed here in the past.

Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
the FULL two body problem in QFT still. Moreover it is well-known that
path formalism has several flaws. For example, the standard version of
the PF offers incorrect S matrices in 'complex' problems: a typical
example is the nonlinear sigma model.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Igor Khavkine wrote:
>
> Regarding your original post. Your argument falls through when
> you declare that particles are "fundamental" in the way you define that
> term. One can take fields as a starting point and never mention
> particles at all, they will fall out automatically. Just as fields will
> fall out automatically when you start with particles. Both formulations
> are equally "fundamental" or equally not so, however you want to
> consider them.

Particles are more fundamental that fields because fields are ALWAYS
-by definition- unobserved, one measures in scattering experiments are
particles, newer fields. As clearly stated by Weinberg in his volume 1,
we know more about particles that about fields.

> > I am trying to avoid discussion of gravity and curved spacetimes.
> > There are enough troubles in understanding simple electro-magnetic
> > interactions. Can we stick to "flat spacetimes" as you call them.
>
> First, I don't see much trouble understanding simple electro-magnetic
> interactions. Moreover, if you wish to talk of fundamentals, you must
> take the most general situation possible, which includes curved
> backgrounds. If you stick to flat space-time, you are stuck in a
> stale-mate since the particle and field formulations are equaivalent.
> If you refuse to consider curved backgrounds, you are simply refusing
> to acknoledge a failing of the particle approach, since it's been known
> for decades that the field approach comes out a clear winner there.

1) curved spacetime is just a view, only that. As was claimed by
Feynmann (in his lectures on gravitation) the curved spacetime
interpretation of GR is unnecesary for physics. In fact, in the torsion
re-geometrization of gravity, spacetime curvature is exactly zero.
Therefore, i think that Stefanovich could generalize his work to
torsion gravity using the flat spacetime he uses for QED.

2) That in curved spacetime there is no posibility for description of
particles is not very convincing just as shows the curved spacetime
generalization of the FFW action (See Hoyle and Narkilar gravitational
theory).

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.

I am not sure i say next, but in string heory one uses the Fock space
for the representation of states of vibrating string and one has not
fields.

> > You are saying that all issues "have been studied and delineated in
> > exhaustive depth" What about the time evolution of interacting systems?
> > Take the simplest interacting system - two electrons. Suppose that
> > I gave you a full description (wave function) of this system at time
> > t=0. How would you find the state of the system at a later time t?
> > The amazing thing (to me) is that field-based quantum electrodynamics
> > doesn't have a clue how to do that. I mean a rigorous approach.
> > I don't ask you to solve
> > the equations of motion. I even don't ask you to write the full set of
> > equations to be solved. I am just asking about a general algorithm how
> > to do that. Which steps in what order should be taken? So that, if we
> > had a supercomputer we could give it all necessary instructions.
>
> This algorithm is called the closed time path formalism. It exist, it
> works, and it has been extensively discussed here in the past.

Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
the FULL two body problem in QFT still. Moreover it is well-known that
path formalism has several flaws. For example, the standard version of
the PF offers incorrect S matrices in 'complex' problems: a typical
example is the nonlinear sigma model.

Juan R.

Center for CANONICAL |SCIENCE)

[Charles Francis]

In message <4339DFFF.1070703@synopsys.com>, Eugene Stefanovich
<eugenev@synopsys.com> writes
>
>
>Charles Francis wrote:
>> In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
>> <eugenev@synopsys.com> writes
>> [snip]
>> What you describe is essentially correct, and is used as the basis
>>of
>> what JB once called "naive" treatments of QED. I have put a lot of work
>> into making it rigorous. The fundamental problem is that you can carry
>> out a discrete construction pretty much exactly as stated, but a
>> discrete construction does not obey manifest covariance. If you try and
>> take the limit and move over to a continuum in order to recover
>> covariance, the theory breaks down in the Landau pole.
>> I have a paper on discrete qed, but the real issue is not whether it
>>is
>> a sound mathematical construction, but whether it is valid as a physical
>> model. This actually depends on one's philosophical stance re the
>> meaning of covariance and the nature of physical measurement. In order
>> to establish that a discrete model is physically legitimate I have
>> written a paper, gr-qc/0508077, currently being refereed which, imv,
>> pretty much does the job. It also makes some unexpected predictions in
>> cosmology, so hopefully it is testable.
>
>Thank you for the reference, but I suspect we are talking about
>different things. I was referring to the standard QED in which
>positions can be measured with unlimited precision and the spectrum of
>the position operator is continuous.

Yes. Then the problem is, as I say, that the theory breaks down in the
Landau pole. It is inconsistent, and therefore it must be wrong.

What I was saying is that in order to create a theory like this using a
continuum one should formulate it using a finite lattice and let lattice
spacing go to zero. The simple answer to your question is that the limit
does not exist.

> The theory I had in mind has
>full relativistic invariance: the interacting generators of the
>Poincare group form a representation of the Poincare Lie algebra
>as in S. Weinberg "The quantum theory of fields", vol. 1
>eqs. (3.3.11) - (3.3.17).
>
>Eugene.
>
>



Regards

--
Charles Francis

[Charles Francis]

In message <4339DFFF.1070703@synopsys.com>, Eugene Stefanovich
<eugenev@synopsys.com> writes
>
>
>Charles Francis wrote:
>> In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
>> <eugenev@synopsys.com> writes
>> [snip]
>> What you describe is essentially correct, and is used as the basis
>>of
>> what JB once called "naive" treatments of QED. I have put a lot of work
>> into making it rigorous. The fundamental problem is that you can carry
>> out a discrete construction pretty much exactly as stated, but a
>> discrete construction does not obey manifest covariance. If you try and
>> take the limit and move over to a continuum in order to recover
>> covariance, the theory breaks down in the Landau pole.
>> I have a paper on discrete qed, but the real issue is not whether it
>>is
>> a sound mathematical construction, but whether it is valid as a physical
>> model. This actually depends on one's philosophical stance re the
>> meaning of covariance and the nature of physical measurement. In order
>> to establish that a discrete model is physically legitimate I have
>> written a paper, gr-qc/0508077, currently being refereed which, imv,
>> pretty much does the job. It also makes some unexpected predictions in
>> cosmology, so hopefully it is testable.
>
>Thank you for the reference, but I suspect we are talking about
>different things. I was referring to the standard QED in which
>positions can be measured with unlimited precision and the spectrum of
>the position operator is continuous.

Yes. Then the problem is, as I say, that the theory breaks down in the
Landau pole. It is inconsistent, and therefore it must be wrong.

What I was saying is that in order to create a theory like this using a
continuum one should formulate it using a finite lattice and let lattice
spacing go to zero. The simple answer to your question is that the limit
does not exist.

> The theory I had in mind has
>full relativistic invariance: the interacting generators of the
>Poincare group form a representation of the Poincare Lie algebra
>as in S. Weinberg "The quantum theory of fields", vol. 1
>eqs. (3.3.11) - (3.3.17).
>
>Eugene.
>
>



Regards

--
Charles Francis

[Charles Francis]

In message <4339DFFF.1070703@synopsys.com>, Eugene Stefanovich
<eugenev@synopsys.com> writes
>
>
>Charles Francis wrote:
>> In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
>> <eugenev@synopsys.com> writes
>> [snip]
>> What you describe is essentially correct, and is used as the basis
>>of
>> what JB once called "naive" treatments of QED. I have put a lot of work
>> into making it rigorous. The fundamental problem is that you can carry
>> out a discrete construction pretty much exactly as stated, but a
>> discrete construction does not obey manifest covariance. If you try and
>> take the limit and move over to a continuum in order to recover
>> covariance, the theory breaks down in the Landau pole.
>> I have a paper on discrete qed, but the real issue is not whether it
>>is
>> a sound mathematical construction, but whether it is valid as a physical
>> model. This actually depends on one's philosophical stance re the
>> meaning of covariance and the nature of physical measurement. In order
>> to establish that a discrete model is physically legitimate I have
>> written a paper, gr-qc/0508077, currently being refereed which, imv,
>> pretty much does the job. It also makes some unexpected predictions in
>> cosmology, so hopefully it is testable.
>
>Thank you for the reference, but I suspect we are talking about
>different things. I was referring to the standard QED in which
>positions can be measured with unlimited precision and the spectrum of
>the position operator is continuous.

Yes. Then the problem is, as I say, that the theory breaks down in the
Landau pole. It is inconsistent, and therefore it must be wrong.

What I was saying is that in order to create a theory like this using a
continuum one should formulate it using a finite lattice and let lattice
spacing go to zero. The simple answer to your question is that the limit
does not exist.

> The theory I had in mind has
>full relativistic invariance: the interacting generators of the
>Poincare group form a representation of the Poincare Lie algebra
>as in S. Weinberg "The quantum theory of fields", vol. 1
>eqs. (3.3.11) - (3.3.17).
>
>Eugene.
>
>



Regards

--
Charles Francis

[Charles Francis]

In message <4339DFFF.1070703@synopsys.com>, Eugene Stefanovich
<eugenev@synopsys.com> writes
>
>
>Charles Francis wrote:
>> In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
>> <eugenev@synopsys.com> writes
>> [snip]
>> What you describe is essentially correct, and is used as the basis
>>of
>> what JB once called "naive" treatments of QED. I have put a lot of work
>> into making it rigorous. The fundamental problem is that you can carry
>> out a discrete construction pretty much exactly as stated, but a
>> discrete construction does not obey manifest covariance. If you try and
>> take the limit and move over to a continuum in order to recover
>> covariance, the theory breaks down in the Landau pole.
>> I have a paper on discrete qed, but the real issue is not whether it
>>is
>> a sound mathematical construction, but whether it is valid as a physical
>> model. This actually depends on one's philosophical stance re the
>> meaning of covariance and the nature of physical measurement. In order
>> to establish that a discrete model is physically legitimate I have
>> written a paper, gr-qc/0508077, currently being refereed which, imv,
>> pretty much does the job. It also makes some unexpected predictions in
>> cosmology, so hopefully it is testable.
>
>Thank you for the reference, but I suspect we are talking about
>different things. I was referring to the standard QED in which
>positions can be measured with unlimited precision and the spectrum of
>the position operator is continuous.

Yes. Then the problem is, as I say, that the theory breaks down in the
Landau pole. It is inconsistent, and therefore it must be wrong.

What I was saying is that in order to create a theory like this using a
continuum one should formulate it using a finite lattice and let lattice
spacing go to zero. The simple answer to your question is that the limit
does not exist.

> The theory I had in mind has
>full relativistic invariance: the interacting generators of the
>Poincare group form a representation of the Poincare Lie algebra
>as in S. Weinberg "The quantum theory of fields", vol. 1
>eqs. (3.3.11) - (3.3.17).
>
>Eugene.
>
>



Regards

--
Charles Francis

[Charles Francis]

In message <4339DFFF.1070703@synopsys.com>, Eugene Stefanovich
<eugenev@synopsys.com> writes
>
>
>Charles Francis wrote:
>> In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
>> <eugenev@synopsys.com> writes
>> [snip]
>> What you describe is essentially correct, and is used as the basis
>>of
>> what JB once called "naive" treatments of QED. I have put a lot of work
>> into making it rigorous. The fundamental problem is that you can carry
>> out a discrete construction pretty much exactly as stated, but a
>> discrete construction does not obey manifest covariance. If you try and
>> take the limit and move over to a continuum in order to recover
>> covariance, the theory breaks down in the Landau pole.
>> I have a paper on discrete qed, but the real issue is not whether it
>>is
>> a sound mathematical construction, but whether it is valid as a physical
>> model. This actually depends on one's philosophical stance re the
>> meaning of covariance and the nature of physical measurement. In order
>> to establish that a discrete model is physically legitimate I have
>> written a paper, gr-qc/0508077, currently being refereed which, imv,
>> pretty much does the job. It also makes some unexpected predictions in
>> cosmology, so hopefully it is testable.
>
>Thank you for the reference, but I suspect we are talking about
>different things. I was referring to the standard QED in which
>positions can be measured with unlimited precision and the spectrum of
>the position operator is continuous.

Yes. Then the problem is, as I say, that the theory breaks down in the
Landau pole. It is inconsistent, and therefore it must be wrong.

What I was saying is that in order to create a theory like this using a
continuum one should formulate it using a finite lattice and let lattice
spacing go to zero. The simple answer to your question is that the limit
does not exist.

> The theory I had in mind has
>full relativistic invariance: the interacting generators of the
>Poincare group form a representation of the Poincare Lie algebra
>as in S. Weinberg "The quantum theory of fields", vol. 1
>eqs. (3.3.11) - (3.3.17).
>
>Eugene.
>
>



Regards

--
Charles Francis

[Charles Francis]

In message <4339DFFF.1070703@synopsys.com>, Eugene Stefanovich
<eugenev@synopsys.com> writes
>
>
>Charles Francis wrote:
>> In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
>> <eugenev@synopsys.com> writes
>> [snip]
>> What you describe is essentially correct, and is used as the basis
>>of
>> what JB once called "naive" treatments of QED. I have put a lot of work
>> into making it rigorous. The fundamental problem is that you can carry
>> out a discrete construction pretty much exactly as stated, but a
>> discrete construction does not obey manifest covariance. If you try and
>> take the limit and move over to a continuum in order to recover
>> covariance, the theory breaks down in the Landau pole.
>> I have a paper on discrete qed, but the real issue is not whether it
>>is
>> a sound mathematical construction, but whether it is valid as a physical
>> model. This actually depends on one's philosophical stance re the
>> meaning of covariance and the nature of physical measurement. In order
>> to establish that a discrete model is physically legitimate I have
>> written a paper, gr-qc/0508077, currently being refereed which, imv,
>> pretty much does the job. It also makes some unexpected predictions in
>> cosmology, so hopefully it is testable.
>
>Thank you for the reference, but I suspect we are talking about
>different things. I was referring to the standard QED in which
>positions can be measured with unlimited precision and the spectrum of
>the position operator is continuous.

Yes. Then the problem is, as I say, that the theory breaks down in the
Landau pole. It is inconsistent, and therefore it must be wrong.

What I was saying is that in order to create a theory like this using a
continuum one should formulate it using a finite lattice and let lattice
spacing go to zero. The simple answer to your question is that the limit
does not exist.

> The theory I had in mind has
>full relativistic invariance: the interacting generators of the
>Poincare group form a representation of the Poincare Lie algebra
>as in S. Weinberg "The quantum theory of fields", vol. 1
>eqs. (3.3.11) - (3.3.17).
>
>Eugene.
>
>



Regards

--
Charles Francis

[Charles Francis]

In message <4339DFFF.1070703@synopsys.com>, Eugene Stefanovich
<eugenev@synopsys.com> writes
>
>
>Charles Francis wrote:
>> In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
>> <eugenev@synopsys.com> writes
>> [snip]
>> What you describe is essentially correct, and is used as the basis
>>of
>> what JB once called "naive" treatments of QED. I have put a lot of work
>> into making it rigorous. The fundamental problem is that you can carry
>> out a discrete construction pretty much exactly as stated, but a
>> discrete construction does not obey manifest covariance. If you try and
>> take the limit and move over to a continuum in order to recover
>> covariance, the theory breaks down in the Landau pole.
>> I have a paper on discrete qed, but the real issue is not whether it
>>is
>> a sound mathematical construction, but whether it is valid as a physical
>> model. This actually depends on one's philosophical stance re the
>> meaning of covariance and the nature of physical measurement. In order
>> to establish that a discrete model is physically legitimate I have
>> written a paper, gr-qc/0508077, currently being refereed which, imv,
>> pretty much does the job. It also makes some unexpected predictions in
>> cosmology, so hopefully it is testable.
>
>Thank you for the reference, but I suspect we are talking about
>different things. I was referring to the standard QED in which
>positions can be measured with unlimited precision and the spectrum of
>the position operator is continuous.

Yes. Then the problem is, as I say, that the theory breaks down in the
Landau pole. It is inconsistent, and therefore it must be wrong.

What I was saying is that in order to create a theory like this using a
continuum one should formulate it using a finite lattice and let lattice
spacing go to zero. The simple answer to your question is that the limit
does not exist.

> The theory I had in mind has
>full relativistic invariance: the interacting generators of the
>Poincare group form a representation of the Poincare Lie algebra
>as in S. Weinberg "The quantum theory of fields", vol. 1
>eqs. (3.3.11) - (3.3.17).
>
>Eugene.
>
>



Regards

--
Charles Francis

[Charles Francis]

In message <4339DFFF.1070703@synopsys.com>, Eugene Stefanovich
<eugenev@synopsys.com> writes
>
>
>Charles Francis wrote:
>> In message <4329CDEA.8050902@synopsys.com>, Eugene Stefanovich
>> <eugenev@synopsys.com> writes
>> [snip]
>> What you describe is essentially correct, and is used as the basis
>>of
>> what JB once called "naive" treatments of QED. I have put a lot of work
>> into making it rigorous. The fundamental problem is that you can carry
>> out a discrete construction pretty much exactly as stated, but a
>> discrete construction does not obey manifest covariance. If you try and
>> take the limit and move over to a continuum in order to recover
>> covariance, the theory breaks down in the Landau pole.
>> I have a paper on discrete qed, but the real issue is not whether it
>>is
>> a sound mathematical construction, but whether it is valid as a physical
>> model. This actually depends on one's philosophical stance re the
>> meaning of covariance and the nature of physical measurement. In order
>> to establish that a discrete model is physically legitimate I have
>> written a paper, gr-qc/0508077, currently being refereed which, imv,
>> pretty much does the job. It also makes some unexpected predictions in
>> cosmology, so hopefully it is testable.
>
>Thank you for the reference, but I suspect we are talking about
>different things. I was referring to the standard QED in which
>positions can be measured with unlimited precision and the spectrum of
>the position operator is continuous.

Yes. Then the problem is, as I say, that the theory breaks down in the
Landau pole. It is inconsistent, and therefore it must be wrong.

What I was saying is that in order to create a theory like this using a
continuum one should formulate it using a finite lattice and let lattice
spacing go to zero. The simple answer to your question is that the limit
does not exist.

> The theory I had in mind has
>full relativistic invariance: the interacting generators of the
>Poincare group form a representation of the Poincare Lie algebra
>as in S. Weinberg "The quantum theory of fields", vol. 1
>eqs. (3.3.11) - (3.3.17).
>
>Eugene.
>
>



Regards

--
Charles Francis

[J. Horta]

On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:

> Igor Khavkine wrote:
>>
>> Regarding your original post. Your argument falls through when
>> you declare that particles are "fundamental" in the way you define that
>> term. One can take fields as a starting point and never mention
>> particles at all, they will fall out automatically. Just as fields will
>> fall out automatically when you start with particles. Both formulations
>> are equally "fundamental" or equally not so, however you want to
>> consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.
>

I very much agree with Igor. What is or is not observable depends on
the situation. At very low energies electric fields may be measured
with a volt meter and magnetic fields with a magnetometer. A radio
picks up em waves. Sure, each one of these may be formulated as
interaction with quantum fields or, as follows from the formalism,
absorption and emission of particles. The physics should be the same
independent of how it is formulated. If you should find, however,
a formulation of the physics using a particle approach which includes
phenomena which a quantum field approach can't then you could well
claim a more fundamental theory.

Paul C.

[J. Horta]

On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:

> Igor Khavkine wrote:
>>
>> Regarding your original post. Your argument falls through when
>> you declare that particles are "fundamental" in the way you define that
>> term. One can take fields as a starting point and never mention
>> particles at all, they will fall out automatically. Just as fields will
>> fall out automatically when you start with particles. Both formulations
>> are equally "fundamental" or equally not so, however you want to
>> consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.
>

I very much agree with Igor. What is or is not observable depends on
the situation. At very low energies electric fields may be measured
with a volt meter and magnetic fields with a magnetometer. A radio
picks up em waves. Sure, each one of these may be formulated as
interaction with quantum fields or, as follows from the formalism,
absorption and emission of particles. The physics should be the same
independent of how it is formulated. If you should find, however,
a formulation of the physics using a particle approach which includes
phenomena which a quantum field approach can't then you could well
claim a more fundamental theory.

Paul C.

[J. Horta]

On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:

> Igor Khavkine wrote:
>>
>> Regarding your original post. Your argument falls through when
>> you declare that particles are "fundamental" in the way you define that
>> term. One can take fields as a starting point and never mention
>> particles at all, they will fall out automatically. Just as fields will
>> fall out automatically when you start with particles. Both formulations
>> are equally "fundamental" or equally not so, however you want to
>> consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.
>

I very much agree with Igor. What is or is not observable depends on
the situation. At very low energies electric fields may be measured
with a volt meter and magnetic fields with a magnetometer. A radio
picks up em waves. Sure, each one of these may be formulated as
interaction with quantum fields or, as follows from the formalism,
absorption and emission of particles. The physics should be the same
independent of how it is formulated. If you should find, however,
a formulation of the physics using a particle approach which includes
phenomena which a quantum field approach can't then you could well
claim a more fundamental theory.

Paul C.

[J. Horta]

On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:

> Igor Khavkine wrote:
>>
>> Regarding your original post. Your argument falls through when
>> you declare that particles are "fundamental" in the way you define that
>> term. One can take fields as a starting point and never mention
>> particles at all, they will fall out automatically. Just as fields will
>> fall out automatically when you start with particles. Both formulations
>> are equally "fundamental" or equally not so, however you want to
>> consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.
>

I very much agree with Igor. What is or is not observable depends on
the situation. At very low energies electric fields may be measured
with a volt meter and magnetic fields with a magnetometer. A radio
picks up em waves. Sure, each one of these may be formulated as
interaction with quantum fields or, as follows from the formalism,
absorption and emission of particles. The physics should be the same
independent of how it is formulated. If you should find, however,
a formulation of the physics using a particle approach which includes
phenomena which a quantum field approach can't then you could well
claim a more fundamental theory.

Paul C.

[J. Horta]

On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:

> Igor Khavkine wrote:
>>
>> Regarding your original post. Your argument falls through when
>> you declare that particles are "fundamental" in the way you define that
>> term. One can take fields as a starting point and never mention
>> particles at all, they will fall out automatically. Just as fields will
>> fall out automatically when you start with particles. Both formulations
>> are equally "fundamental" or equally not so, however you want to
>> consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.
>

I very much agree with Igor. What is or is not observable depends on
the situation. At very low energies electric fields may be measured
with a volt meter and magnetic fields with a magnetometer. A radio
picks up em waves. Sure, each one of these may be formulated as
interaction with quantum fields or, as follows from the formalism,
absorption and emission of particles. The physics should be the same
independent of how it is formulated. If you should find, however,
a formulation of the physics using a particle approach which includes
phenomena which a quantum field approach can't then you could well
claim a more fundamental theory.

Paul C.

[J. Horta]

On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:

> Igor Khavkine wrote:
>>
>> Regarding your original post. Your argument falls through when
>> you declare that particles are "fundamental" in the way you define that
>> term. One can take fields as a starting point and never mention
>> particles at all, they will fall out automatically. Just as fields will
>> fall out automatically when you start with particles. Both formulations
>> are equally "fundamental" or equally not so, however you want to
>> consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.
>

I very much agree with Igor. What is or is not observable depends on
the situation. At very low energies electric fields may be measured
with a volt meter and magnetic fields with a magnetometer. A radio
picks up em waves. Sure, each one of these may be formulated as
interaction with quantum fields or, as follows from the formalism,
absorption and emission of particles. The physics should be the same
independent of how it is formulated. If you should find, however,
a formulation of the physics using a particle approach which includes
phenomena which a quantum field approach can't then you could well
claim a more fundamental theory.

Paul C.

[J. Horta]

On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:

> Igor Khavkine wrote:
>>
>> Regarding your original post. Your argument falls through when
>> you declare that particles are "fundamental" in the way you define that
>> term. One can take fields as a starting point and never mention
>> particles at all, they will fall out automatically. Just as fields will
>> fall out automatically when you start with particles. Both formulations
>> are equally "fundamental" or equally not so, however you want to
>> consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.
>

I very much agree with Igor. What is or is not observable depends on
the situation. At very low energies electric fields may be measured
with a volt meter and magnetic fields with a magnetometer. A radio
picks up em waves. Sure, each one of these may be formulated as
interaction with quantum fields or, as follows from the formalism,
absorption and emission of particles. The physics should be the same
independent of how it is formulated. If you should find, however,
a formulation of the physics using a particle approach which includes
phenomena which a quantum field approach can't then you could well
claim a more fundamental theory.

Paul C.

[J. Horta]

On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:

> Igor Khavkine wrote:
>>
>> Regarding your original post. Your argument falls through when
>> you declare that particles are "fundamental" in the way you define that
>> term. One can take fields as a starting point and never mention
>> particles at all, they will fall out automatically. Just as fields will
>> fall out automatically when you start with particles. Both formulations
>> are equally "fundamental" or equally not so, however you want to
>> consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.
>

I very much agree with Igor. What is or is not observable depends on
the situation. At very low energies electric fields may be measured
with a volt meter and magnetic fields with a magnetometer. A radio
picks up em waves. Sure, each one of these may be formulated as
interaction with quantum fields or, as follows from the formalism,
absorption and emission of particles. The physics should be the same
independent of how it is formulated. If you should find, however,
a formulation of the physics using a particle approach which includes
phenomena which a quantum field approach can't then you could well
claim a more fundamental theory.

Paul C.

[Igor Khavkine]

Charles Francis wrote:
> In message <1127528209.775506.155930@o13g2000cwo.googlegroups. com>,
> Igor Khavkine <igor.kh@gmail.com> writes
> >Eugene Stefanovich wrote:
> >
> >Regarding your original post. Your argument falls through when you
> >declare that particles are "fundamental" in the way you define that
> >term. One can take fields as a starting point and never mention
> >particles at all, they will fall out automatically. Just as fields
> >will fall out automatically when you start with particles. Both
> >formulations are equally "fundamental" or equally not so, however
> >you want to consider them.
>
> Mathematically this is, of course, true, but for me it doesn't hold
> water when one tries to put a physical interpretation on the
> fundamental entities. A particle is a simple entity without extent
> and with minimal properties. A field takes a different value at each
> point in spacetime - in complexity it may be likened to a machine
> with an infinite number of moving parts. Of course these days the
> fashion is to say "oh we mustn't think about physical
> interpretation". So I guess I'm just pig headed, because I think it
> is the main thing we should think about if we are going to advance
> our understanding of nature.

I completely agree that physical interpretation must be considered.
However, since we are dealing with a scientific theory, the
interpretation must be held to as high a standard as any other part of
the theory. Namely, the interpretation must consist of a dictionary to
translate the properties of objects of a theory into measurable and
verifiable quantities, and vice versa. For a successful theory, the
dictionary is required to be as complete as possible going from
experiment to theory, but there is no such requirement going the
opposite way. Hence an incompleteness in this second half of the
dictionary does not have a lot of weight in the discussion. I think the
point that you bring up, a mechanical interpretation of a particle as
opposed to a field, belongs to this second half of the dictionary.

> >> I am trying to avoid discussion of gravity and curved spacetimes.
> >> There are enough troubles in understanding simple electro-magnetic
> >> interactions. Can we stick to "flat spacetimes" as you call them.
> >
> >First, I don't see much trouble understanding simple
> >electro-magnetic interactions. Moreover, if you wish to talk of
> >fundamentals, you must take the most general situation possible,
> >which includes curved backgrounds. If you stick to flat space-time,
> >you are stuck in a stale-mate since the particle and field
> >formulations are equaivalent. If you refuse to consider curved
> >backgrounds, you are simply refusing to acknoledge a failing of the
> >particle approach, since it's been known for decades that the field
> >approach comes out a clear winner there.
>
> Not as far as I know. Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both
> point to serious difficulties even in defining fields.

There is no question that there are difficulties formulating the
corresponding mathematical theory. But let me quote from page 2 of
Wald's book on QFT in Curved Spacetime (1984):

The difficulties involved in explaining the formulation of quantum
field theory in curved spacetime to a reader familiar with standard
treatments of quantum field theory in flat spacetime is somewhat
analogous to the difficulties involved in explaning general
relativity to a reader familiar with special relativity in the manner
in which it normally is formulated---where primary emphasis is placed
upon the existence of global families of inertial observers and the
relationships between these families described by Poincare
transformations. Neither the notion of global inertial observers nor
Poincare transformations generalize in a meaningful way to curved
spacetime. However, when one recognizes that the structure of
spacetime in special relativity is most naturally and simply
described by a flat spacetime metric---and that the existence of
global families of inertial observers may be viewed as a secondary
consequence of the presence of this flat metric---the transition fo
the framework of general relativity is straightforward: One simply
allows the spacetime metric to be curved.

In a similar manner, in quantum field theory in a flat spacetime, the
Poincare group plays a key role in picking out a preferred vacuum
state and defining the notion of a "particle". In the past, much
attention has been devoted to the issue of how to generalize the
notion of "particles" to curved spacetime. One of the key points
which will be emphasized by our presentation here is that this issue
is irrelvant to the formulation of quantum field theory in curved
spacetime---in much the same manner as the issue of how to generalize
the definition of global inertial coordinates to curved spacetime is
irrelevant to the formulation of general relativity. Quantum field
theory is a quantum theory of /fields/, not particles. Although, in
appropriate circumstances a particle interpretation of the theory may
be available, the notion of "particles" plays no fundamental role
either in the formulation or interpretation of the theory.

As you can see, Wald has much less kind words about the formulation of
quantum particles on curved spacetime.

> >As soon as you have a Fock space, you have a field theory. This is a
> >mathematical fact, no matter how you constructed the Fock space.
> >Therefore, any consequence of Haag's theorem will apply to such a
> >theory equally well. And, unlike you claim, Haag's theorem is not a
> >major obstacle. The non-trivial representations of the operator
> >algebra do exist and they are implicitly (perturbatively)
> >constructed during any QFT calculation involving interactions.
>
> If you have a Fock space you have a field theory of "non-interacting
> fields". Haag's theorem is an obstacle to having a theory of
> "interacting fields". I inclined to agree with Eugene. "Interacting
> fields" are unnecessary and a fiction. They arise by taking a wrong
> approach to the subject, via quantisation of classical fields.

A Fock space is merely a Hilbert space with some extra structure
(it's closed under tensor products of states). A Fock space can arise
in two ways. The Hilbert space on which a theory of quantum fields is
formulated can be given a Fock space structure, in which case matrix
elements of the field operators give wave functions for free.
A quantum theory of an indefinite number of identitcal particles can
also be constructed on a Hilbert space with a Fock structure, in which
case single particle wave functions yield quantum field operators,
again for free. The equivalence of these two constructions is the main
theorem of second quantization. In other words, quantum fields are
unavoidable, even if you forget about the quantization of classical
fields.

Now, Haag's theorem says nothing about Fock spaces. It only talks of
Hilbert spaces and quantum fields. It says nothing about the
impossibility of constructing interacting fields. It only says that if
a free theory and an interacting theory are constructed, they cannot be
related by a unitary transformation (where both theories are Poincare
invariant). Compare to ordinary quantum mechanics where knowing the
matrix elements of the x and p operators between the momentum states of
a free particle and between the bound states of the harmonic
oscillator, we can connect one set of matrix elements to the other
through a unitary transformation whose own matrix elements are given by
the Fourier transforms of weighted Hermite polynomials.

Note that the paragraph on second quantization specified nothing of
interaction, while the paragraph on Haag's theorem specified nothing of
Fock spaces. Which means that you can put them together and draw your
own conclusions about the fictitiousness of interacting quantum fields
and the applicability of Haag's theorem to a particle description.

This theorem is usually invoked as an obstruction in the construction
of ineracting fields via the "interaction representation" applied to
free fields. However, due to the renormalization procedure,
regularization breaks at least one of the hypotheses of Haag's theorem.
Renormalization allows us to construct multipoint correlation
functions which are finite in the limit where the regulator is removed.
These correlation functions are in turn sufficient to reconstruct the
quantum fields (see Streater & Wightman), which will necessarily be
inequivalent to those obtained by quantizing a free theory. Of course,
all of this is done order by order in perturbation theory.

Igor

[Igor Khavkine]

Charles Francis wrote:
> In message <1127528209.775506.155930@o13g2000cwo.googlegroups. com>,
> Igor Khavkine <igor.kh@gmail.com> writes
> >Eugene Stefanovich wrote:
> >
> >Regarding your original post. Your argument falls through when you
> >declare that particles are "fundamental" in the way you define that
> >term. One can take fields as a starting point and never mention
> >particles at all, they will fall out automatically. Just as fields
> >will fall out automatically when you start with particles. Both
> >formulations are equally "fundamental" or equally not so, however
> >you want to consider them.
>
> Mathematically this is, of course, true, but for me it doesn't hold
> water when one tries to put a physical interpretation on the
> fundamental entities. A particle is a simple entity without extent
> and with minimal properties. A field takes a different value at each
> point in spacetime - in complexity it may be likened to a machine
> with an infinite number of moving parts. Of course these days the
> fashion is to say "oh we mustn't think about physical
> interpretation". So I guess I'm just pig headed, because I think it
> is the main thing we should think about if we are going to advance
> our understanding of nature.

I completely agree that physical interpretation must be considered.
However, since we are dealing with a scientific theory, the
interpretation must be held to as high a standard as any other part of
the theory. Namely, the interpretation must consist of a dictionary to
translate the properties of objects of a theory into measurable and
verifiable quantities, and vice versa. For a successful theory, the
dictionary is required to be as complete as possible going from
experiment to theory, but there is no such requirement going the
opposite way. Hence an incompleteness in this second half of the
dictionary does not have a lot of weight in the discussion. I think the
point that you bring up, a mechanical interpretation of a particle as
opposed to a field, belongs to this second half of the dictionary.

> >> I am trying to avoid discussion of gravity and curved spacetimes.
> >> There are enough troubles in understanding simple electro-magnetic
> >> interactions. Can we stick to "flat spacetimes" as you call them.
> >
> >First, I don't see much trouble understanding simple
> >electro-magnetic interactions. Moreover, if you wish to talk of
> >fundamentals, you must take the most general situation possible,
> >which includes curved backgrounds. If you stick to flat space-time,
> >you are stuck in a stale-mate since the particle and field
> >formulations are equaivalent. If you refuse to consider curved
> >backgrounds, you are simply refusing to acknoledge a failing of the
> >particle approach, since it's been known for decades that the field
> >approach comes out a clear winner there.
>
> Not as far as I know. Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both
> point to serious difficulties even in defining fields.

There is no question that there are difficulties formulating the
corresponding mathematical theory. But let me quote from page 2 of
Wald's book on QFT in Curved Spacetime (1984):

The difficulties involved in explaining the formulation of quantum
field theory in curved spacetime to a reader familiar with standard
treatments of quantum field theory in flat spacetime is somewhat
analogous to the difficulties involved in explaning general
relativity to a reader familiar with special relativity in the manner
in which it normally is formulated---where primary emphasis is placed
upon the existence of global families of inertial observers and the
relationships between these families described by Poincare
transformations. Neither the notion of global inertial observers nor
Poincare transformations generalize in a meaningful way to curved
spacetime. However, when one recognizes that the structure of
spacetime in special relativity is most naturally and simply
described by a flat spacetime metric---and that the existence of
global families of inertial observers may be viewed as a secondary
consequence of the presence of this flat metric---the transition fo
the framework of general relativity is straightforward: One simply
allows the spacetime metric to be curved.

In a similar manner, in quantum field theory in a flat spacetime, the
Poincare group plays a key role in picking out a preferred vacuum
state and defining the notion of a "particle". In the past, much
attention has been devoted to the issue of how to generalize the
notion of "particles" to curved spacetime. One of the key points
which will be emphasized by our presentation here is that this issue
is irrelvant to the formulation of quantum field theory in curved
spacetime---in much the same manner as the issue of how to generalize
the definition of global inertial coordinates to curved spacetime is
irrelevant to the formulation of general relativity. Quantum field
theory is a quantum theory of /fields/, not particles. Although, in
appropriate circumstances a particle interpretation of the theory may
be available, the notion of "particles" plays no fundamental role
either in the formulation or interpretation of the theory.

As you can see, Wald has much less kind words about the formulation of
quantum particles on curved spacetime.

> >As soon as you have a Fock space, you have a field theory. This is a
> >mathematical fact, no matter how you constructed the Fock space.
> >Therefore, any consequence of Haag's theorem will apply to such a
> >theory equally well. And, unlike you claim, Haag's theorem is not a
> >major obstacle. The non-trivial representations of the operator
> >algebra do exist and they are implicitly (perturbatively)
> >constructed during any QFT calculation involving interactions.
>
> If you have a Fock space you have a field theory of "non-interacting
> fields". Haag's theorem is an obstacle to having a theory of
> "interacting fields". I inclined to agree with Eugene. "Interacting
> fields" are unnecessary and a fiction. They arise by taking a wrong
> approach to the subject, via quantisation of classical fields.

A Fock space is merely a Hilbert space with some extra structure
(it's closed under tensor products of states). A Fock space can arise
in two ways. The Hilbert space on which a theory of quantum fields is
formulated can be given a Fock space structure, in which case matrix
elements of the field operators give wave functions for free.
A quantum theory of an indefinite number of identitcal particles can
also be constructed on a Hilbert space with a Fock structure, in which
case single particle wave functions yield quantum field operators,
again for free. The equivalence of these two constructions is the main
theorem of second quantization. In other words, quantum fields are
unavoidable, even if you forget about the quantization of classical
fields.

Now, Haag's theorem says nothing about Fock spaces. It only talks of
Hilbert spaces and quantum fields. It says nothing about the
impossibility of constructing interacting fields. It only says that if
a free theory and an interacting theory are constructed, they cannot be
related by a unitary transformation (where both theories are Poincare
invariant). Compare to ordinary quantum mechanics where knowing the
matrix elements of the x and p operators between the momentum states of
a free particle and between the bound states of the harmonic
oscillator, we can connect one set of matrix elements to the other
through a unitary transformation whose own matrix elements are given by
the Fourier transforms of weighted Hermite polynomials.

Note that the paragraph on second quantization specified nothing of
interaction, while the paragraph on Haag's theorem specified nothing of
Fock spaces. Which means that you can put them together and draw your
own conclusions about the fictitiousness of interacting quantum fields
and the applicability of Haag's theorem to a particle description.

This theorem is usually invoked as an obstruction in the construction
of ineracting fields via the "interaction representation" applied to
free fields. However, due to the renormalization procedure,
regularization breaks at least one of the hypotheses of Haag's theorem.
Renormalization allows us to construct multipoint correlation
functions which are finite in the limit where the regulator is removed.
These correlation functions are in turn sufficient to reconstruct the
quantum fields (see Streater & Wightman), which will necessarily be
inequivalent to those obtained by quantizing a free theory. Of course,
all of this is done order by order in perturbation theory.

Igor

[Igor Khavkine]

Charles Francis wrote:
> In message <1127528209.775506.155930@o13g2000cwo.googlegroups. com>,
> Igor Khavkine <igor.kh@gmail.com> writes
> >Eugene Stefanovich wrote:
> >
> >Regarding your original post. Your argument falls through when you
> >declare that particles are "fundamental" in the way you define that
> >term. One can take fields as a starting point and never mention
> >particles at all, they will fall out automatically. Just as fields
> >will fall out automatically when you start with particles. Both
> >formulations are equally "fundamental" or equally not so, however
> >you want to consider them.
>
> Mathematically this is, of course, true, but for me it doesn't hold
> water when one tries to put a physical interpretation on the
> fundamental entities. A particle is a simple entity without extent
> and with minimal properties. A field takes a different value at each
> point in spacetime - in complexity it may be likened to a machine
> with an infinite number of moving parts. Of course these days the
> fashion is to say "oh we mustn't think about physical
> interpretation". So I guess I'm just pig headed, because I think it
> is the main thing we should think about if we are going to advance
> our understanding of nature.

I completely agree that physical interpretation must be considered.
However, since we are dealing with a scientific theory, the
interpretation must be held to as high a standard as any other part of
the theory. Namely, the interpretation must consist of a dictionary to
translate the properties of objects of a theory into measurable and
verifiable quantities, and vice versa. For a successful theory, the
dictionary is required to be as complete as possible going from
experiment to theory, but there is no such requirement going the
opposite way. Hence an incompleteness in this second half of the
dictionary does not have a lot of weight in the discussion. I think the
point that you bring up, a mechanical interpretation of a particle as
opposed to a field, belongs to this second half of the dictionary.

> >> I am trying to avoid discussion of gravity and curved spacetimes.
> >> There are enough troubles in understanding simple electro-magnetic
> >> interactions. Can we stick to "flat spacetimes" as you call them.
> >
> >First, I don't see much trouble understanding simple
> >electro-magnetic interactions. Moreover, if you wish to talk of
> >fundamentals, you must take the most general situation possible,
> >which includes curved backgrounds. If you stick to flat space-time,
> >you are stuck in a stale-mate since the particle and field
> >formulations are equaivalent. If you refuse to consider curved
> >backgrounds, you are simply refusing to acknoledge a failing of the
> >particle approach, since it's been known for decades that the field
> >approach comes out a clear winner there.
>
> Not as far as I know. Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both
> point to serious difficulties even in defining fields.

There is no question that there are difficulties formulating the
corresponding mathematical theory. But let me quote from page 2 of
Wald's book on QFT in Curved Spacetime (1984):

The difficulties involved in explaining the formulation of quantum
field theory in curved spacetime to a reader familiar with standard
treatments of quantum field theory in flat spacetime is somewhat
analogous to the difficulties involved in explaning general
relativity to a reader familiar with special relativity in the manner
in which it normally is formulated---where primary emphasis is placed
upon the existence of global families of inertial observers and the
relationships between these families described by Poincare
transformations. Neither the notion of global inertial observers nor
Poincare transformations generalize in a meaningful way to curved
spacetime. However, when one recognizes that the structure of
spacetime in special relativity is most naturally and simply
described by a flat spacetime metric---and that the existence of
global families of inertial observers may be viewed as a secondary
consequence of the presence of this flat metric---the transition fo
the framework of general relativity is straightforward: One simply
allows the spacetime metric to be curved.

In a similar manner, in quantum field theory in a flat spacetime, the
Poincare group plays a key role in picking out a preferred vacuum
state and defining the notion of a "particle". In the past, much
attention has been devoted to the issue of how to generalize the
notion of "particles" to curved spacetime. One of the key points
which will be emphasized by our presentation here is that this issue
is irrelvant to the formulation of quantum field theory in curved
spacetime---in much the same manner as the issue of how to generalize
the definition of global inertial coordinates to curved spacetime is
irrelevant to the formulation of general relativity. Quantum field
theory is a quantum theory of /fields/, not particles. Although, in
appropriate circumstances a particle interpretation of the theory may
be available, the notion of "particles" plays no fundamental role
either in the formulation or interpretation of the theory.

As you can see, Wald has much less kind words about the formulation of
quantum particles on curved spacetime.

> >As soon as you have a Fock space, you have a field theory. This is a
> >mathematical fact, no matter how you constructed the Fock space.
> >Therefore, any consequence of Haag's theorem will apply to such a
> >theory equally well. And, unlike you claim, Haag's theorem is not a
> >major obstacle. The non-trivial representations of the operator
> >algebra do exist and they are implicitly (perturbatively)
> >constructed during any QFT calculation involving interactions.
>
> If you have a Fock space you have a field theory of "non-interacting
> fields". Haag's theorem is an obstacle to having a theory of
> "interacting fields". I inclined to agree with Eugene. "Interacting
> fields" are unnecessary and a fiction. They arise by taking a wrong
> approach to the subject, via quantisation of classical fields.

A Fock space is merely a Hilbert space with some extra structure
(it's closed under tensor products of states). A Fock space can arise
in two ways. The Hilbert space on which a theory of quantum fields is
formulated can be given a Fock space structure, in which case matrix
elements of the field operators give wave functions for free.
A quantum theory of an indefinite number of identitcal particles can
also be constructed on a Hilbert space with a Fock structure, in which
case single particle wave functions yield quantum field operators,
again for free. The equivalence of these two constructions is the main
theorem of second quantization. In other words, quantum fields are
unavoidable, even if you forget about the quantization of classical
fields.

Now, Haag's theorem says nothing about Fock spaces. It only talks of
Hilbert spaces and quantum fields. It says nothing about the
impossibility of constructing interacting fields. It only says that if
a free theory and an interacting theory are constructed, they cannot be
related by a unitary transformation (where both theories are Poincare
invariant). Compare to ordinary quantum mechanics where knowing the
matrix elements of the x and p operators between the momentum states of
a free particle and between the bound states of the harmonic
oscillator, we can connect one set of matrix elements to the other
through a unitary transformation whose own matrix elements are given by
the Fourier transforms of weighted Hermite polynomials.

Note that the paragraph on second quantization specified nothing of
interaction, while the paragraph on Haag's theorem specified nothing of
Fock spaces. Which means that you can put them together and draw your
own conclusions about the fictitiousness of interacting quantum fields
and the applicability of Haag's theorem to a particle description.

This theorem is usually invoked as an obstruction in the construction
of ineracting fields via the "interaction representation" applied to
free fields. However, due to the renormalization procedure,
regularization breaks at least one of the hypotheses of Haag's theorem.
Renormalization allows us to construct multipoint correlation
functions which are finite in the limit where the regulator is removed.
These correlation functions are in turn sufficient to reconstruct the
quantum fields (see Streater & Wightman), which will necessarily be
inequivalent to those obtained by quantizing a free theory. Of course,
all of this is done order by order in perturbation theory.

Igor

[Igor Khavkine]

Charles Francis wrote:
> In message <1127528209.775506.155930@o13g2000cwo.googlegroups. com>,
> Igor Khavkine <igor.kh@gmail.com> writes
> >Eugene Stefanovich wrote:
> >
> >Regarding your original post. Your argument falls through when you
> >declare that particles are "fundamental" in the way you define that
> >term. One can take fields as a starting point and never mention
> >particles at all, they will fall out automatically. Just as fields
> >will fall out automatically when you start with particles. Both
> >formulations are equally "fundamental" or equally not so, however
> >you want to consider them.
>
> Mathematically this is, of course, true, but for me it doesn't hold
> water when one tries to put a physical interpretation on the
> fundamental entities. A particle is a simple entity without extent
> and with minimal properties. A field takes a different value at each
> point in spacetime - in complexity it may be likened to a machine
> with an infinite number of moving parts. Of course these days the
> fashion is to say "oh we mustn't think about physical
> interpretation". So I guess I'm just pig headed, because I think it
> is the main thing we should think about if we are going to advance
> our understanding of nature.

I completely agree that physical interpretation must be considered.
However, since we are dealing with a scientific theory, the
interpretation must be held to as high a standard as any other part of
the theory. Namely, the interpretation must consist of a dictionary to
translate the properties of objects of a theory into measurable and
verifiable quantities, and vice versa. For a successful theory, the
dictionary is required to be as complete as possible going from
experiment to theory, but there is no such requirement going the
opposite way. Hence an incompleteness in this second half of the
dictionary does not have a lot of weight in the discussion. I think the
point that you bring up, a mechanical interpretation of a particle as
opposed to a field, belongs to this second half of the dictionary.

> >> I am trying to avoid discussion of gravity and curved spacetimes.
> >> There are enough troubles in understanding simple electro-magnetic
> >> interactions. Can we stick to "flat spacetimes" as you call them.
> >
> >First, I don't see much trouble understanding simple
> >electro-magnetic interactions. Moreover, if you wish to talk of
> >fundamentals, you must take the most general situation possible,
> >which includes curved backgrounds. If you stick to flat space-time,
> >you are stuck in a stale-mate since the particle and field
> >formulations are equaivalent. If you refuse to consider curved
> >backgrounds, you are simply refusing to acknoledge a failing of the
> >particle approach, since it's been known for decades that the field
> >approach comes out a clear winner there.
>
> Not as far as I know. Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both
> point to serious difficulties even in defining fields.

There is no question that there are difficulties formulating the
corresponding mathematical theory. But let me quote from page 2 of
Wald's book on QFT in Curved Spacetime (1984):

The difficulties involved in explaining the formulation of quantum
field theory in curved spacetime to a reader familiar with standard
treatments of quantum field theory in flat spacetime is somewhat
analogous to the difficulties involved in explaning general
relativity to a reader familiar with special relativity in the manner
in which it normally is formulated---where primary emphasis is placed
upon the existence of global families of inertial observers and the
relationships between these families described by Poincare
transformations. Neither the notion of global inertial observers nor
Poincare transformations generalize in a meaningful way to curved
spacetime. However, when one recognizes that the structure of
spacetime in special relativity is most naturally and simply
described by a flat spacetime metric---and that the existence of
global families of inertial observers may be viewed as a secondary
consequence of the presence of this flat metric---the transition fo
the framework of general relativity is straightforward: One simply
allows the spacetime metric to be curved.

In a similar manner, in quantum field theory in a flat spacetime, the
Poincare group plays a key role in picking out a preferred vacuum
state and defining the notion of a "particle". In the past, much
attention has been devoted to the issue of how to generalize the
notion of "particles" to curved spacetime. One of the key points
which will be emphasized by our presentation here is that this issue
is irrelvant to the formulation of quantum field theory in curved
spacetime---in much the same manner as the issue of how to generalize
the definition of global inertial coordinates to curved spacetime is
irrelevant to the formulation of general relativity. Quantum field
theory is a quantum theory of /fields/, not particles. Although, in
appropriate circumstances a particle interpretation of the theory may
be available, the notion of "particles" plays no fundamental role
either in the formulation or interpretation of the theory.

As you can see, Wald has much less kind words about the formulation of
quantum particles on curved spacetime.

> >As soon as you have a Fock space, you have a field theory. This is a
> >mathematical fact, no matter how you constructed the Fock space.
> >Therefore, any consequence of Haag's theorem will apply to such a
> >theory equally well. And, unlike you claim, Haag's theorem is not a
> >major obstacle. The non-trivial representations of the operator
> >algebra do exist and they are implicitly (perturbatively)
> >constructed during any QFT calculation involving interactions.
>
> If you have a Fock space you have a field theory of "non-interacting
> fields". Haag's theorem is an obstacle to having a theory of
> "interacting fields". I inclined to agree with Eugene. "Interacting
> fields" are unnecessary and a fiction. They arise by taking a wrong
> approach to the subject, via quantisation of classical fields.

A Fock space is merely a Hilbert space with some extra structure
(it's closed under tensor products of states). A Fock space can arise
in two ways. The Hilbert space on which a theory of quantum fields is
formulated can be given a Fock space structure, in which case matrix
elements of the field operators give wave functions for free.
A quantum theory of an indefinite number of identitcal particles can
also be constructed on a Hilbert space with a Fock structure, in which
case single particle wave functions yield quantum field operators,
again for free. The equivalence of these two constructions is the main
theorem of second quantization. In other words, quantum fields are
unavoidable, even if you forget about the quantization of classical
fields.

Now, Haag's theorem says nothing about Fock spaces. It only talks of
Hilbert spaces and quantum fields. It says nothing about the
impossibility of constructing interacting fields. It only says that if
a free theory and an interacting theory are constructed, they cannot be
related by a unitary transformation (where both theories are Poincare
invariant). Compare to ordinary quantum mechanics where knowing the
matrix elements of the x and p operators between the momentum states of
a free particle and between the bound states of the harmonic
oscillator, we can connect one set of matrix elements to the other
through a unitary transformation whose own matrix elements are given by
the Fourier transforms of weighted Hermite polynomials.

Note that the paragraph on second quantization specified nothing of
interaction, while the paragraph on Haag's theorem specified nothing of
Fock spaces. Which means that you can put them together and draw your
own conclusions about the fictitiousness of interacting quantum fields
and the applicability of Haag's theorem to a particle description.

This theorem is usually invoked as an obstruction in the construction
of ineracting fields via the "interaction representation" applied to
free fields. However, due to the renormalization procedure,
regularization breaks at least one of the hypotheses of Haag's theorem.
Renormalization allows us to construct multipoint correlation
functions which are finite in the limit where the regulator is removed.
These correlation functions are in turn sufficient to reconstruct the
quantum fields (see Streater & Wightman), which will necessarily be
inequivalent to those obtained by quantizing a free theory. Of course,
all of this is done order by order in perturbation theory.

Igor

[Igor Khavkine]

Charles Francis wrote:
> In message <1127528209.775506.155930@o13g2000cwo.googlegroups. com>,
> Igor Khavkine <igor.kh@gmail.com> writes
> >Eugene Stefanovich wrote:
> >
> >Regarding your original post. Your argument falls through when you
> >declare that particles are "fundamental" in the way you define that
> >term. One can take fields as a starting point and never mention
> >particles at all, they will fall out automatically. Just as fields
> >will fall out automatically when you start with particles. Both
> >formulations are equally "fundamental" or equally not so, however
> >you want to consider them.
>
> Mathematically this is, of course, true, but for me it doesn't hold
> water when one tries to put a physical interpretation on the
> fundamental entities. A particle is a simple entity without extent
> and with minimal properties. A field takes a different value at each
> point in spacetime - in complexity it may be likened to a machine
> with an infinite number of moving parts. Of course these days the
> fashion is to say "oh we mustn't think about physical
> interpretation". So I guess I'm just pig headed, because I think it
> is the main thing we should think about if we are going to advance
> our understanding of nature.

I completely agree that physical interpretation must be considered.
However, since we are dealing with a scientific theory, the
interpretation must be held to as high a standard as any other part of
the theory. Namely, the interpretation must consist of a dictionary to
translate the properties of objects of a theory into measurable and
verifiable quantities, and vice versa. For a successful theory, the
dictionary is required to be as complete as possible going from
experiment to theory, but there is no such requirement going the
opposite way. Hence an incompleteness in this second half of the
dictionary does not have a lot of weight in the discussion. I think the
point that you bring up, a mechanical interpretation of a particle as
opposed to a field, belongs to this second half of the dictionary.

> >> I am trying to avoid discussion of gravity and curved spacetimes.
> >> There are enough troubles in understanding simple electro-magnetic
> >> interactions. Can we stick to "flat spacetimes" as you call them.
> >
> >First, I don't see much trouble understanding simple
> >electro-magnetic interactions. Moreover, if you wish to talk of
> >fundamentals, you must take the most general situation possible,
> >which includes curved backgrounds. If you stick to flat space-time,
> >you are stuck in a stale-mate since the particle and field
> >formulations are equaivalent. If you refuse to consider curved
> >backgrounds, you are simply refusing to acknoledge a failing of the
> >particle approach, since it's been known for decades that the field
> >approach comes out a clear winner there.
>
> Not as far as I know. Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both
> point to serious difficulties even in defining fields.

There is no question that there are difficulties formulating the
corresponding mathematical theory. But let me quote from page 2 of
Wald's book on QFT in Curved Spacetime (1984):

The difficulties involved in explaining the formulation of quantum
field theory in curved spacetime to a reader familiar with standard
treatments of quantum field theory in flat spacetime is somewhat
analogous to the difficulties involved in explaning general
relativity to a reader familiar with special relativity in the manner
in which it normally is formulated---where primary emphasis is placed
upon the existence of global families of inertial observers and the
relationships between these families described by Poincare
transformations. Neither the notion of global inertial observers nor
Poincare transformations generalize in a meaningful way to curved
spacetime. However, when one recognizes that the structure of
spacetime in special relativity is most naturally and simply
described by a flat spacetime metric---and that the existence of
global families of inertial observers may be viewed as a secondary
consequence of the presence of this flat metric---the transition fo
the framework of general relativity is straightforward: One simply
allows the spacetime metric to be curved.

In a similar manner, in quantum field theory in a flat spacetime, the
Poincare group plays a key role in picking out a preferred vacuum
state and defining the notion of a "particle". In the past, much
attention has been devoted to the issue of how to generalize the
notion of "particles" to curved spacetime. One of the key points
which will be emphasized by our presentation here is that this issue
is irrelvant to the formulation of quantum field theory in curved
spacetime---in much the same manner as the issue of how to generalize
the definition of global inertial coordinates to curved spacetime is
irrelevant to the formulation of general relativity. Quantum field
theory is a quantum theory of /fields/, not particles. Although, in
appropriate circumstances a particle interpretation of the theory may
be available, the notion of "particles" plays no fundamental role
either in the formulation or interpretation of the theory.

As you can see, Wald has much less kind words about the formulation of
quantum particles on curved spacetime.

> >As soon as you have a Fock space, you have a field theory. This is a
> >mathematical fact, no matter how you constructed the Fock space.
> >Therefore, any consequence of Haag's theorem will apply to such a
> >theory equally well. And, unlike you claim, Haag's theorem is not a
> >major obstacle. The non-trivial representations of the operator
> >algebra do exist and they are implicitly (perturbatively)
> >constructed during any QFT calculation involving interactions.
>
> If you have a Fock space you have a field theory of "non-interacting
> fields". Haag's theorem is an obstacle to having a theory of
> "interacting fields". I inclined to agree with Eugene. "Interacting
> fields" are unnecessary and a fiction. They arise by taking a wrong
> approach to the subject, via quantisation of classical fields.

A Fock space is merely a Hilbert space with some extra structure
(it's closed under tensor products of states). A Fock space can arise
in two ways. The Hilbert space on which a theory of quantum fields is
formulated can be given a Fock space structure, in which case matrix
elements of the field operators give wave functions for free.
A quantum theory of an indefinite number of identitcal particles can
also be constructed on a Hilbert space with a Fock structure, in which
case single particle wave functions yield quantum field operators,
again for free. The equivalence of these two constructions is the main
theorem of second quantization. In other words, quantum fields are
unavoidable, even if you forget about the quantization of classical
fields.

Now, Haag's theorem says nothing about Fock spaces. It only talks of
Hilbert spaces and quantum fields. It says nothing about the
impossibility of constructing interacting fields. It only says that if
a free theory and an interacting theory are constructed, they cannot be
related by a unitary transformation (where both theories are Poincare
invariant). Compare to ordinary quantum mechanics where knowing the
matrix elements of the x and p operators between the momentum states of
a free particle and between the bound states of the harmonic
oscillator, we can connect one set of matrix elements to the other
through a unitary transformation whose own matrix elements are given by
the Fourier transforms of weighted Hermite polynomials.

Note that the paragraph on second quantization specified nothing of
interaction, while the paragraph on Haag's theorem specified nothing of
Fock spaces. Which means that you can put them together and draw your
own conclusions about the fictitiousness of interacting quantum fields
and the applicability of Haag's theorem to a particle description.

This theorem is usually invoked as an obstruction in the construction
of ineracting fields via the "interaction representation" applied to
free fields. However, due to the renormalization procedure,
regularization breaks at least one of the hypotheses of Haag's theorem.
Renormalization allows us to construct multipoint correlation
functions which are finite in the limit where the regulator is removed.
These correlation functions are in turn sufficient to reconstruct the
quantum fields (see Streater & Wightman), which will necessarily be
inequivalent to those obtained by quantizing a free theory. Of course,
all of this is done order by order in perturbation theory.

Igor

[Igor Khavkine]

Charles Francis wrote:
> In message <1127528209.775506.155930@o13g2000cwo.googlegroups. com>,
> Igor Khavkine <igor.kh@gmail.com> writes
> >Eugene Stefanovich wrote:
> >
> >Regarding your original post. Your argument falls through when you
> >declare that particles are "fundamental" in the way you define that
> >term. One can take fields as a starting point and never mention
> >particles at all, they will fall out automatically. Just as fields
> >will fall out automatically when you start with particles. Both
> >formulations are equally "fundamental" or equally not so, however
> >you want to consider them.
>
> Mathematically this is, of course, true, but for me it doesn't hold
> water when one tries to put a physical interpretation on the
> fundamental entities. A particle is a simple entity without extent
> and with minimal properties. A field takes a different value at each
> point in spacetime - in complexity it may be likened to a machine
> with an infinite number of moving parts. Of course these days the
> fashion is to say "oh we mustn't think about physical
> interpretation". So I guess I'm just pig headed, because I think it
> is the main thing we should think about if we are going to advance
> our understanding of nature.

I completely agree that physical interpretation must be considered.
However, since we are dealing with a scientific theory, the
interpretation must be held to as high a standard as any other part of
the theory. Namely, the interpretation must consist of a dictionary to
translate the properties of objects of a theory into measurable and
verifiable quantities, and vice versa. For a successful theory, the
dictionary is required to be as complete as possible going from
experiment to theory, but there is no such requirement going the
opposite way. Hence an incompleteness in this second half of the
dictionary does not have a lot of weight in the discussion. I think the
point that you bring up, a mechanical interpretation of a particle as
opposed to a field, belongs to this second half of the dictionary.

> >> I am trying to avoid discussion of gravity and curved spacetimes.
> >> There are enough troubles in understanding simple electro-magnetic
> >> interactions. Can we stick to "flat spacetimes" as you call them.
> >
> >First, I don't see much trouble understanding simple
> >electro-magnetic interactions. Moreover, if you wish to talk of
> >fundamentals, you must take the most general situation possible,
> >which includes curved backgrounds. If you stick to flat space-time,
> >you are stuck in a stale-mate since the particle and field
> >formulations are equaivalent. If you refuse to consider curved
> >backgrounds, you are simply refusing to acknoledge a failing of the
> >particle approach, since it's been known for decades that the field
> >approach comes out a clear winner there.
>
> Not as far as I know. Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both
> point to serious difficulties even in defining fields.

There is no question that there are difficulties formulating the
corresponding mathematical theory. But let me quote from page 2 of
Wald's book on QFT in Curved Spacetime (1984):

The difficulties involved in explaining the formulation of quantum
field theory in curved spacetime to a reader familiar with standard
treatments of quantum field theory in flat spacetime is somewhat
analogous to the difficulties involved in explaning general
relativity to a reader familiar with special relativity in the manner
in which it normally is formulated---where primary emphasis is placed
upon the existence of global families of inertial observers and the
relationships between these families described by Poincare
transformations. Neither the notion of global inertial observers nor
Poincare transformations generalize in a meaningful way to curved
spacetime. However, when one recognizes that the structure of
spacetime in special relativity is most naturally and simply
described by a flat spacetime metric---and that the existence of
global families of inertial observers may be viewed as a secondary
consequence of the presence of this flat metric---the transition fo
the framework of general relativity is straightforward: One simply
allows the spacetime metric to be curved.

In a similar manner, in quantum field theory in a flat spacetime, the
Poincare group plays a key role in picking out a preferred vacuum
state and defining the notion of a "particle". In the past, much
attention has been devoted to the issue of how to generalize the
notion of "particles" to curved spacetime. One of the key points
which will be emphasized by our presentation here is that this issue
is irrelvant to the formulation of quantum field theory in curved
spacetime---in much the same manner as the issue of how to generalize
the definition of global inertial coordinates to curved spacetime is
irrelevant to the formulation of general relativity. Quantum field
theory is a quantum theory of /fields/, not particles. Although, in
appropriate circumstances a particle interpretation of the theory may
be available, the notion of "particles" plays no fundamental role
either in the formulation or interpretation of the theory.

As you can see, Wald has much less kind words about the formulation of
quantum particles on curved spacetime.

> >As soon as you have a Fock space, you have a field theory. This is a
> >mathematical fact, no matter how you constructed the Fock space.
> >Therefore, any consequence of Haag's theorem will apply to such a
> >theory equally well. And, unlike you claim, Haag's theorem is not a
> >major obstacle. The non-trivial representations of the operator
> >algebra do exist and they are implicitly (perturbatively)
> >constructed during any QFT calculation involving interactions.
>
> If you have a Fock space you have a field theory of "non-interacting
> fields". Haag's theorem is an obstacle to having a theory of
> "interacting fields". I inclined to agree with Eugene. "Interacting
> fields" are unnecessary and a fiction. They arise by taking a wrong
> approach to the subject, via quantisation of classical fields.

A Fock space is merely a Hilbert space with some extra structure
(it's closed under tensor products of states). A Fock space can arise
in two ways. The Hilbert space on which a theory of quantum fields is
formulated can be given a Fock space structure, in which case matrix
elements of the field operators give wave functions for free.
A quantum theory of an indefinite number of identitcal particles can
also be constructed on a Hilbert space with a Fock structure, in which
case single particle wave functions yield quantum field operators,
again for free. The equivalence of these two constructions is the main
theorem of second quantization. In other words, quantum fields are
unavoidable, even if you forget about the quantization of classical
fields.

Now, Haag's theorem says nothing about Fock spaces. It only talks of
Hilbert spaces and quantum fields. It says nothing about the
impossibility of constructing interacting fields. It only says that if
a free theory and an interacting theory are constructed, they cannot be
related by a unitary transformation (where both theories are Poincare
invariant). Compare to ordinary quantum mechanics where knowing the
matrix elements of the x and p operators between the momentum states of
a free particle and between the bound states of the harmonic
oscillator, we can connect one set of matrix elements to the other
through a unitary transformation whose own matrix elements are given by
the Fourier transforms of weighted Hermite polynomials.

Note that the paragraph on second quantization specified nothing of
interaction, while the paragraph on Haag's theorem specified nothing of
Fock spaces. Which means that you can put them together and draw your
own conclusions about the fictitiousness of interacting quantum fields
and the applicability of Haag's theorem to a particle description.

This theorem is usually invoked as an obstruction in the construction
of ineracting fields via the "interaction representation" applied to
free fields. However, due to the renormalization procedure,
regularization breaks at least one of the hypotheses of Haag's theorem.
Renormalization allows us to construct multipoint correlation
functions which are finite in the limit where the regulator is removed.
These correlation functions are in turn sufficient to reconstruct the
quantum fields (see Streater & Wightman), which will necessarily be
inequivalent to those obtained by quantizing a free theory. Of course,
all of this is done order by order in perturbation theory.

Igor

[Igor Khavkine]

Charles Francis wrote:
> In message <1127528209.775506.155930@o13g2000cwo.googlegroups. com>,
> Igor Khavkine <igor.kh@gmail.com> writes
> >Eugene Stefanovich wrote:
> >
> >Regarding your original post. Your argument falls through when you
> >declare that particles are "fundamental" in the way you define that
> >term. One can take fields as a starting point and never mention
> >particles at all, they will fall out automatically. Just as fields
> >will fall out automatically when you start with particles. Both
> >formulations are equally "fundamental" or equally not so, however
> >you want to consider them.
>
> Mathematically this is, of course, true, but for me it doesn't hold
> water when one tries to put a physical interpretation on the
> fundamental entities. A particle is a simple entity without extent
> and with minimal properties. A field takes a different value at each
> point in spacetime - in complexity it may be likened to a machine
> with an infinite number of moving parts. Of course these days the
> fashion is to say "oh we mustn't think about physical
> interpretation". So I guess I'm just pig headed, because I think it
> is the main thing we should think about if we are going to advance
> our understanding of nature.

I completely agree that physical interpretation must be considered.
However, since we are dealing with a scientific theory, the
interpretation must be held to as high a standard as any other part of
the theory. Namely, the interpretation must consist of a dictionary to
translate the properties of objects of a theory into measurable and
verifiable quantities, and vice versa. For a successful theory, the
dictionary is required to be as complete as possible going from
experiment to theory, but there is no such requirement going the
opposite way. Hence an incompleteness in this second half of the
dictionary does not have a lot of weight in the discussion. I think the
point that you bring up, a mechanical interpretation of a particle as
opposed to a field, belongs to this second half of the dictionary.

> >> I am trying to avoid discussion of gravity and curved spacetimes.
> >> There are enough troubles in understanding simple electro-magnetic
> >> interactions. Can we stick to "flat spacetimes" as you call them.
> >
> >First, I don't see much trouble understanding simple
> >electro-magnetic interactions. Moreover, if you wish to talk of
> >fundamentals, you must take the most general situation possible,
> >which includes curved backgrounds. If you stick to flat space-time,
> >you are stuck in a stale-mate since the particle and field
> >formulations are equaivalent. If you refuse to consider curved
> >backgrounds, you are simply refusing to acknoledge a failing of the
> >particle approach, since it's been known for decades that the field
> >approach comes out a clear winner there.
>
> Not as far as I know. Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both
> point to serious difficulties even in defining fields.

There is no question that there are difficulties formulating the
corresponding mathematical theory. But let me quote from page 2 of
Wald's book on QFT in Curved Spacetime (1984):

The difficulties involved in explaining the formulation of quantum
field theory in curved spacetime to a reader familiar with standard
treatments of quantum field theory in flat spacetime is somewhat
analogous to the difficulties involved in explaning general
relativity to a reader familiar with special relativity in the manner
in which it normally is formulated---where primary emphasis is placed
upon the existence of global families of inertial observers and the
relationships between these families described by Poincare
transformations. Neither the notion of global inertial observers nor
Poincare transformations generalize in a meaningful way to curved
spacetime. However, when one recognizes that the structure of
spacetime in special relativity is most naturally and simply
described by a flat spacetime metric---and that the existence of
global families of inertial observers may be viewed as a secondary
consequence of the presence of this flat metric---the transition fo
the framework of general relativity is straightforward: One simply
allows the spacetime metric to be curved.

In a similar manner, in quantum field theory in a flat spacetime, the
Poincare group plays a key role in picking out a preferred vacuum
state and defining the notion of a "particle". In the past, much
attention has been devoted to the issue of how to generalize the
notion of "particles" to curved spacetime. One of the key points
which will be emphasized by our presentation here is that this issue
is irrelvant to the formulation of quantum field theory in curved
spacetime---in much the same manner as the issue of how to generalize
the definition of global inertial coordinates to curved spacetime is
irrelevant to the formulation of general relativity. Quantum field
theory is a quantum theory of /fields/, not particles. Although, in
appropriate circumstances a particle interpretation of the theory may
be available, the notion of "particles" plays no fundamental role
either in the formulation or interpretation of the theory.

As you can see, Wald has much less kind words about the formulation of
quantum particles on curved spacetime.

> >As soon as you have a Fock space, you have a field theory. This is a
> >mathematical fact, no matter how you constructed the Fock space.
> >Therefore, any consequence of Haag's theorem will apply to such a
> >theory equally well. And, unlike you claim, Haag's theorem is not a
> >major obstacle. The non-trivial representations of the operator
> >algebra do exist and they are implicitly (perturbatively)
> >constructed during any QFT calculation involving interactions.
>
> If you have a Fock space you have a field theory of "non-interacting
> fields". Haag's theorem is an obstacle to having a theory of
> "interacting fields". I inclined to agree with Eugene. "Interacting
> fields" are unnecessary and a fiction. They arise by taking a wrong
> approach to the subject, via quantisation of classical fields.

A Fock space is merely a Hilbert space with some extra structure
(it's closed under tensor products of states). A Fock space can arise
in two ways. The Hilbert space on which a theory of quantum fields is
formulated can be given a Fock space structure, in which case matrix
elements of the field operators give wave functions for free.
A quantum theory of an indefinite number of identitcal particles can
also be constructed on a Hilbert space with a Fock structure, in which
case single particle wave functions yield quantum field operators,
again for free. The equivalence of these two constructions is the main
theorem of second quantization. In other words, quantum fields are
unavoidable, even if you forget about the quantization of classical
fields.

Now, Haag's theorem says nothing about Fock spaces. It only talks of
Hilbert spaces and quantum fields. It says nothing about the
impossibility of constructing interacting fields. It only says that if
a free theory and an interacting theory are constructed, they cannot be
related by a unitary transformation (where both theories are Poincare
invariant). Compare to ordinary quantum mechanics where knowing the
matrix elements of the x and p operators between the momentum states of
a free particle and between the bound states of the harmonic
oscillator, we can connect one set of matrix elements to the other
through a unitary transformation whose own matrix elements are given by
the Fourier transforms of weighted Hermite polynomials.

Note that the paragraph on second quantization specified nothing of
interaction, while the paragraph on Haag's theorem specified nothing of
Fock spaces. Which means that you can put them together and draw your
own conclusions about the fictitiousness of interacting quantum fields
and the applicability of Haag's theorem to a particle description.

This theorem is usually invoked as an obstruction in the construction
of ineracting fields via the "interaction representation" applied to
free fields. However, due to the renormalization procedure,
regularization breaks at least one of the hypotheses of Haag's theorem.
Renormalization allows us to construct multipoint correlation
functions which are finite in the limit where the regulator is removed.
These correlation functions are in turn sufficient to reconstruct the
quantum fields (see Streater & Wightman), which will necessarily be
inequivalent to those obtained by quantizing a free theory. Of course,
all of this is done order by order in perturbation theory.

Igor

[Igor Khavkine]

Charles Francis wrote:
> In message <1127528209.775506.155930@o13g2000cwo.googlegroups. com>,
> Igor Khavkine <igor.kh@gmail.com> writes
> >Eugene Stefanovich wrote:
> >
> >Regarding your original post. Your argument falls through when you
> >declare that particles are "fundamental" in the way you define that
> >term. One can take fields as a starting point and never mention
> >particles at all, they will fall out automatically. Just as fields
> >will fall out automatically when you start with particles. Both
> >formulations are equally "fundamental" or equally not so, however
> >you want to consider them.
>
> Mathematically this is, of course, true, but for me it doesn't hold
> water when one tries to put a physical interpretation on the
> fundamental entities. A particle is a simple entity without extent
> and with minimal properties. A field takes a different value at each
> point in spacetime - in complexity it may be likened to a machine
> with an infinite number of moving parts. Of course these days the
> fashion is to say "oh we mustn't think about physical
> interpretation". So I guess I'm just pig headed, because I think it
> is the main thing we should think about if we are going to advance
> our understanding of nature.

I completely agree that physical interpretation must be considered.
However, since we are dealing with a scientific theory, the
interpretation must be held to as high a standard as any other part of
the theory. Namely, the interpretation must consist of a dictionary to
translate the properties of objects of a theory into measurable and
verifiable quantities, and vice versa. For a successful theory, the
dictionary is required to be as complete as possible going from
experiment to theory, but there is no such requirement going the
opposite way. Hence an incompleteness in this second half of the
dictionary does not have a lot of weight in the discussion. I think the
point that you bring up, a mechanical interpretation of a particle as
opposed to a field, belongs to this second half of the dictionary.

> >> I am trying to avoid discussion of gravity and curved spacetimes.
> >> There are enough troubles in understanding simple electro-magnetic
> >> interactions. Can we stick to "flat spacetimes" as you call them.
> >
> >First, I don't see much trouble understanding simple
> >electro-magnetic interactions. Moreover, if you wish to talk of
> >fundamentals, you must take the most general situation possible,
> >which includes curved backgrounds. If you stick to flat space-time,
> >you are stuck in a stale-mate since the particle and field
> >formulations are equaivalent. If you refuse to consider curved
> >backgrounds, you are simply refusing to acknoledge a failing of the
> >particle approach, since it's been known for decades that the field
> >approach comes out a clear winner there.
>
> Not as far as I know. Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both
> point to serious difficulties even in defining fields.

There is no question that there are difficulties formulating the
corresponding mathematical theory. But let me quote from page 2 of
Wald's book on QFT in Curved Spacetime (1984):

The difficulties involved in explaining the formulation of quantum
field theory in curved spacetime to a reader familiar with standard
treatments of quantum field theory in flat spacetime is somewhat
analogous to the difficulties involved in explaning general
relativity to a reader familiar with special relativity in the manner
in which it normally is formulated---where primary emphasis is placed
upon the existence of global families of inertial observers and the
relationships between these families described by Poincare
transformations. Neither the notion of global inertial observers nor
Poincare transformations generalize in a meaningful way to curved
spacetime. However, when one recognizes that the structure of
spacetime in special relativity is most naturally and simply
described by a flat spacetime metric---and that the existence of
global families of inertial observers may be viewed as a secondary
consequence of the presence of this flat metric---the transition fo
the framework of general relativity is straightforward: One simply
allows the spacetime metric to be curved.

In a similar manner, in quantum field theory in a flat spacetime, the
Poincare group plays a key role in picking out a preferred vacuum
state and defining the notion of a "particle". In the past, much
attention has been devoted to the issue of how to generalize the
notion of "particles" to curved spacetime. One of the key points
which will be emphasized by our presentation here is that this issue
is irrelvant to the formulation of quantum field theory in curved
spacetime---in much the same manner as the issue of how to generalize
the definition of global inertial coordinates to curved spacetime is
irrelevant to the formulation of general relativity. Quantum field
theory is a quantum theory of /fields/, not particles. Although, in
appropriate circumstances a particle interpretation of the theory may
be available, the notion of "particles" plays no fundamental role
either in the formulation or interpretation of the theory.

As you can see, Wald has much less kind words about the formulation of
quantum particles on curved spacetime.

> >As soon as you have a Fock space, you have a field theory. This is a
> >mathematical fact, no matter how you constructed the Fock space.
> >Therefore, any consequence of Haag's theorem will apply to such a
> >theory equally well. And, unlike you claim, Haag's theorem is not a
> >major obstacle. The non-trivial representations of the operator
> >algebra do exist and they are implicitly (perturbatively)
> >constructed during any QFT calculation involving interactions.
>
> If you have a Fock space you have a field theory of "non-interacting
> fields". Haag's theorem is an obstacle to having a theory of
> "interacting fields". I inclined to agree with Eugene. "Interacting
> fields" are unnecessary and a fiction. They arise by taking a wrong
> approach to the subject, via quantisation of classical fields.

A Fock space is merely a Hilbert space with some extra structure
(it's closed under tensor products of states). A Fock space can arise
in two ways. The Hilbert space on which a theory of quantum fields is
formulated can be given a Fock space structure, in which case matrix
elements of the field operators give wave functions for free.
A quantum theory of an indefinite number of identitcal particles can
also be constructed on a Hilbert space with a Fock structure, in which
case single particle wave functions yield quantum field operators,
again for free. The equivalence of these two constructions is the main
theorem of second quantization. In other words, quantum fields are
unavoidable, even if you forget about the quantization of classical
fields.

Now, Haag's theorem says nothing about Fock spaces. It only talks of
Hilbert spaces and quantum fields. It says nothing about the
impossibility of constructing interacting fields. It only says that if
a free theory and an interacting theory are constructed, they cannot be
related by a unitary transformation (where both theories are Poincare
invariant). Compare to ordinary quantum mechanics where knowing the
matrix elements of the x and p operators between the momentum states of
a free particle and between the bound states of the harmonic
oscillator, we can connect one set of matrix elements to the other
through a unitary transformation whose own matrix elements are given by
the Fourier transforms of weighted Hermite polynomials.

Note that the paragraph on second quantization specified nothing of
interaction, while the paragraph on Haag's theorem specified nothing of
Fock spaces. Which means that you can put them together and draw your
own conclusions about the fictitiousness of interacting quantum fields
and the applicability of Haag's theorem to a particle description.

This theorem is usually invoked as an obstruction in the construction
of ineracting fields via the "interaction representation" applied to
free fields. However, due to the renormalization procedure,
regularization breaks at least one of the hypotheses of Haag's theorem.
Renormalization allows us to construct multipoint correlation
functions which are finite in the limit where the regulator is removed.
These correlation functions are in turn sufficient to reconstruct the
quantum fields (see Streater & Wightman), which will necessarily be
inequivalent to those obtained by quantizing a free theory. Of course,
all of this is done order by order in perturbation theory.

Igor

[Igor Khavkine]

Juan R. wrote:
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when you
> > declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields
> > will fall out automatically when you start with particles. Both
> > formulations are equally "fundamental" or equally not so, however
> > you want to consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments
> are particles, newer fields. As clearly stated by Weinberg in his
> volume 1, we know more about particles that about fields.

I don't know where you've seen this definition, neither do I know where
you've seen a definition that says that particles are observable. Oh,
and I can also say that one never measures particles, only field
correlation functions. And could you be more specific about what
Weinberg says on this topic?

> > > I am trying to avoid discussion of gravity and curved spacetimes.
> > > There are enough troubles in understanding simple
> > > electro-magnetic interactions. Can we stick to "flat spacetimes"
> > > as you call them.
> >
> > First, I don't see much trouble understanding simple
> > electro-magnetic interactions. Moreover, if you wish to talk of
> > fundamentals, you must take the most general situation possible,
> > which includes curved backgrounds. If you stick to flat space-time,
> > you are stuck in a stale-mate since the particle and field
> > formulations are equaivalent. If you refuse to consider curved
> > backgrounds, you are simply refusing to acknoledge a failing of the
> > particle approach, since it's been known for decades that the field
> > approach comes out a clear winner there.
>
> 1) curved spacetime is just a view, only that. As was claimed by
> Feynmann (in his lectures on gravitation) the curved spacetime
> interpretation of GR is unnecesary for physics. In fact, in the
> torsion re-geometrization of gravity, spacetime curvature is exactly
> zero. Therefore, i think that Stefanovich could generalize his work
> to torsion gravity using the flat spacetime he uses for QED.
>
> 2) That in curved spacetime there is no posibility for description of
> particles is not very convincing just as shows the curved spacetime
> generalization of the FFW action (See Hoyle and Narkilar
> gravitational theory).

I am not familiar with either "torsion re-geometrization" nor with the
work of Hoyle and Narkilar. But I do know that every successful
physical theory has been generalized to the case of a curved classical
background. If Eugene's theory is to be successful, why should it be
any different? There is always the option of picking an alternative
description of gravity. But then you have to justify the choice and
explain why that generalization was possible, yet the generalization to
GR (even in the limit in which it is known to apply) was not.

> I am not sure i say next, but in string heory one uses the Fock space
> for the representation of states of vibrating string and one has not
> fields.

One does have fields. They are the embedding space-time coordinates X^u
which are functions of the internal (world-sheet) coordinates (s,t) on
the string. In fact, if we take X^u to be the embedding spac-time
coordinates of a particle as a function of some parameter t that
parametrizes its world-line. These coordinates can also be considered
as fields defined on a 0+1 dimensional manifold.

> > > You are saying that all issues "have been studied and delineated
> > > in exhaustive depth" What about the time evolution of
> > > interacting systems? Take the simplest interacting system - two
> > > electrons. Suppose that I gave you a full description (wave
> > > function) of this system at time t=0. How would you find the
> > > state of the system at a later time t? The amazing thing (to me)
> > > is that field-based quantum electrodynamics doesn't have a clue
> > > how to do that. I mean a rigorous approach. I don't ask you to
> > > solve the equations of motion. I even don't ask you to write the
> > > full set of equations to be solved. I am just asking about a
> > > general algorithm how to do that. Which steps in what order
> > > should be taken? So that, if we had a supercomputer we could give
> > > it all necessary instructions.
> >
> > This algorithm is called the closed time path formalism. It exist,
> > it works, and it has been extensively discussed here in the past.
>
> Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
> the FULL two body problem in QFT still.

Hmm, on page 569 of his vol. I on QFT, Weinberg talks about the
spherical harmonic decomposition of solutions to the Dirac equation.
I fail to see the relevance. But, in a similar vein, no-body has solved
the FULL Newtonian 4-body problem (there is actually an infinite series
solution to the 3-body problem). The latter, however, is not a
testament to the incompleteness of Newtonian mechanics but the
technical difficulties associated with the given problem.

> Moreover it is well-known
> that path formalism has several flaws. For example, the standard
> version of the PF offers incorrect S matrices in 'complex' problems:
> a typical example is the nonlinear sigma model.

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when you
> > declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields
> > will fall out automatically when you start with particles. Both
> > formulations are equally "fundamental" or equally not so, however
> > you want to consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments
> are particles, newer fields. As clearly stated by Weinberg in his
> volume 1, we know more about particles that about fields.

I don't know where you've seen this definition, neither do I know where
you've seen a definition that says that particles are observable. Oh,
and I can also say that one never measures particles, only field
correlation functions. And could you be more specific about what
Weinberg says on this topic?

> > > I am trying to avoid discussion of gravity and curved spacetimes.
> > > There are enough troubles in understanding simple
> > > electro-magnetic interactions. Can we stick to "flat spacetimes"
> > > as you call them.
> >
> > First, I don't see much trouble understanding simple
> > electro-magnetic interactions. Moreover, if you wish to talk of
> > fundamentals, you must take the most general situation possible,
> > which includes curved backgrounds. If you stick to flat space-time,
> > you are stuck in a stale-mate since the particle and field
> > formulations are equaivalent. If you refuse to consider curved
> > backgrounds, you are simply refusing to acknoledge a failing of the
> > particle approach, since it's been known for decades that the field
> > approach comes out a clear winner there.
>
> 1) curved spacetime is just a view, only that. As was claimed by
> Feynmann (in his lectures on gravitation) the curved spacetime
> interpretation of GR is unnecesary for physics. In fact, in the
> torsion re-geometrization of gravity, spacetime curvature is exactly
> zero. Therefore, i think that Stefanovich could generalize his work
> to torsion gravity using the flat spacetime he uses for QED.
>
> 2) That in curved spacetime there is no posibility for description of
> particles is not very convincing just as shows the curved spacetime
> generalization of the FFW action (See Hoyle and Narkilar
> gravitational theory).

I am not familiar with either "torsion re-geometrization" nor with the
work of Hoyle and Narkilar. But I do know that every successful
physical theory has been generalized to the case of a curved classical
background. If Eugene's theory is to be successful, why should it be
any different? There is always the option of picking an alternative
description of gravity. But then you have to justify the choice and
explain why that generalization was possible, yet the generalization to
GR (even in the limit in which it is known to apply) was not.

> I am not sure i say next, but in string heory one uses the Fock space
> for the representation of states of vibrating string and one has not
> fields.

One does have fields. They are the embedding space-time coordinates X^u
which are functions of the internal (world-sheet) coordinates (s,t) on
the string. In fact, if we take X^u to be the embedding spac-time
coordinates of a particle as a function of some parameter t that
parametrizes its world-line. These coordinates can also be considered
as fields defined on a 0+1 dimensional manifold.

> > > You are saying that all issues "have been studied and delineated
> > > in exhaustive depth" What about the time evolution of
> > > interacting systems? Take the simplest interacting system - two
> > > electrons. Suppose that I gave you a full description (wave
> > > function) of this system at time t=0. How would you find the
> > > state of the system at a later time t? The amazing thing (to me)
> > > is that field-based quantum electrodynamics doesn't have a clue
> > > how to do that. I mean a rigorous approach. I don't ask you to
> > > solve the equations of motion. I even don't ask you to write the
> > > full set of equations to be solved. I am just asking about a
> > > general algorithm how to do that. Which steps in what order
> > > should be taken? So that, if we had a supercomputer we could give
> > > it all necessary instructions.
> >
> > This algorithm is called the closed time path formalism. It exist,
> > it works, and it has been extensively discussed here in the past.
>
> Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
> the FULL two body problem in QFT still.

Hmm, on page 569 of his vol. I on QFT, Weinberg talks about the
spherical harmonic decomposition of solutions to the Dirac equation.
I fail to see the relevance. But, in a similar vein, no-body has solved
the FULL Newtonian 4-body problem (there is actually an infinite series
solution to the 3-body problem). The latter, however, is not a
testament to the incompleteness of Newtonian mechanics but the
technical difficulties associated with the given problem.

> Moreover it is well-known
> that path formalism has several flaws. For example, the standard
> version of the PF offers incorrect S matrices in 'complex' problems:
> a typical example is the nonlinear sigma model.

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when you
> > declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields
> > will fall out automatically when you start with particles. Both
> > formulations are equally "fundamental" or equally not so, however
> > you want to consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments
> are particles, newer fields. As clearly stated by Weinberg in his
> volume 1, we know more about particles that about fields.

I don't know where you've seen this definition, neither do I know where
you've seen a definition that says that particles are observable. Oh,
and I can also say that one never measures particles, only field
correlation functions. And could you be more specific about what
Weinberg says on this topic?

> > > I am trying to avoid discussion of gravity and curved spacetimes.
> > > There are enough troubles in understanding simple
> > > electro-magnetic interactions. Can we stick to "flat spacetimes"
> > > as you call them.
> >
> > First, I don't see much trouble understanding simple
> > electro-magnetic interactions. Moreover, if you wish to talk of
> > fundamentals, you must take the most general situation possible,
> > which includes curved backgrounds. If you stick to flat space-time,
> > you are stuck in a stale-mate since the particle and field
> > formulations are equaivalent. If you refuse to consider curved
> > backgrounds, you are simply refusing to acknoledge a failing of the
> > particle approach, since it's been known for decades that the field
> > approach comes out a clear winner there.
>
> 1) curved spacetime is just a view, only that. As was claimed by
> Feynmann (in his lectures on gravitation) the curved spacetime
> interpretation of GR is unnecesary for physics. In fact, in the
> torsion re-geometrization of gravity, spacetime curvature is exactly
> zero. Therefore, i think that Stefanovich could generalize his work
> to torsion gravity using the flat spacetime he uses for QED.
>
> 2) That in curved spacetime there is no posibility for description of
> particles is not very convincing just as shows the curved spacetime
> generalization of the FFW action (See Hoyle and Narkilar
> gravitational theory).

I am not familiar with either "torsion re-geometrization" nor with the
work of Hoyle and Narkilar. But I do know that every successful
physical theory has been generalized to the case of a curved classical
background. If Eugene's theory is to be successful, why should it be
any different? There is always the option of picking an alternative
description of gravity. But then you have to justify the choice and
explain why that generalization was possible, yet the generalization to
GR (even in the limit in which it is known to apply) was not.

> I am not sure i say next, but in string heory one uses the Fock space
> for the representation of states of vibrating string and one has not
> fields.

One does have fields. They are the embedding space-time coordinates X^u
which are functions of the internal (world-sheet) coordinates (s,t) on
the string. In fact, if we take X^u to be the embedding spac-time
coordinates of a particle as a function of some parameter t that
parametrizes its world-line. These coordinates can also be considered
as fields defined on a 0+1 dimensional manifold.

> > > You are saying that all issues "have been studied and delineated
> > > in exhaustive depth" What about the time evolution of
> > > interacting systems? Take the simplest interacting system - two
> > > electrons. Suppose that I gave you a full description (wave
> > > function) of this system at time t=0. How would you find the
> > > state of the system at a later time t? The amazing thing (to me)
> > > is that field-based quantum electrodynamics doesn't have a clue
> > > how to do that. I mean a rigorous approach. I don't ask you to
> > > solve the equations of motion. I even don't ask you to write the
> > > full set of equations to be solved. I am just asking about a
> > > general algorithm how to do that. Which steps in what order
> > > should be taken? So that, if we had a supercomputer we could give
> > > it all necessary instructions.
> >
> > This algorithm is called the closed time path formalism. It exist,
> > it works, and it has been extensively discussed here in the past.
>
> Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
> the FULL two body problem in QFT still.

Hmm, on page 569 of his vol. I on QFT, Weinberg talks about the
spherical harmonic decomposition of solutions to the Dirac equation.
I fail to see the relevance. But, in a similar vein, no-body has solved
the FULL Newtonian 4-body problem (there is actually an infinite series
solution to the 3-body problem). The latter, however, is not a
testament to the incompleteness of Newtonian mechanics but the
technical difficulties associated with the given problem.

> Moreover it is well-known
> that path formalism has several flaws. For example, the standard
> version of the PF offers incorrect S matrices in 'complex' problems:
> a typical example is the nonlinear sigma model.

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when you
> > declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields
> > will fall out automatically when you start with particles. Both
> > formulations are equally "fundamental" or equally not so, however
> > you want to consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments
> are particles, newer fields. As clearly stated by Weinberg in his
> volume 1, we know more about particles that about fields.

I don't know where you've seen this definition, neither do I know where
you've seen a definition that says that particles are observable. Oh,
and I can also say that one never measures particles, only field
correlation functions. And could you be more specific about what
Weinberg says on this topic?

> > > I am trying to avoid discussion of gravity and curved spacetimes.
> > > There are enough troubles in understanding simple
> > > electro-magnetic interactions. Can we stick to "flat spacetimes"
> > > as you call them.
> >
> > First, I don't see much trouble understanding simple
> > electro-magnetic interactions. Moreover, if you wish to talk of
> > fundamentals, you must take the most general situation possible,
> > which includes curved backgrounds. If you stick to flat space-time,
> > you are stuck in a stale-mate since the particle and field
> > formulations are equaivalent. If you refuse to consider curved
> > backgrounds, you are simply refusing to acknoledge a failing of the
> > particle approach, since it's been known for decades that the field
> > approach comes out a clear winner there.
>
> 1) curved spacetime is just a view, only that. As was claimed by
> Feynmann (in his lectures on gravitation) the curved spacetime
> interpretation of GR is unnecesary for physics. In fact, in the
> torsion re-geometrization of gravity, spacetime curvature is exactly
> zero. Therefore, i think that Stefanovich could generalize his work
> to torsion gravity using the flat spacetime he uses for QED.
>
> 2) That in curved spacetime there is no posibility for description of
> particles is not very convincing just as shows the curved spacetime
> generalization of the FFW action (See Hoyle and Narkilar
> gravitational theory).

I am not familiar with either "torsion re-geometrization" nor with the
work of Hoyle and Narkilar. But I do know that every successful
physical theory has been generalized to the case of a curved classical
background. If Eugene's theory is to be successful, why should it be
any different? There is always the option of picking an alternative
description of gravity. But then you have to justify the choice and
explain why that generalization was possible, yet the generalization to
GR (even in the limit in which it is known to apply) was not.

> I am not sure i say next, but in string heory one uses the Fock space
> for the representation of states of vibrating string and one has not
> fields.

One does have fields. They are the embedding space-time coordinates X^u
which are functions of the internal (world-sheet) coordinates (s,t) on
the string. In fact, if we take X^u to be the embedding spac-time
coordinates of a particle as a function of some parameter t that
parametrizes its world-line. These coordinates can also be considered
as fields defined on a 0+1 dimensional manifold.

> > > You are saying that all issues "have been studied and delineated
> > > in exhaustive depth" What about the time evolution of
> > > interacting systems? Take the simplest interacting system - two
> > > electrons. Suppose that I gave you a full description (wave
> > > function) of this system at time t=0. How would you find the
> > > state of the system at a later time t? The amazing thing (to me)
> > > is that field-based quantum electrodynamics doesn't have a clue
> > > how to do that. I mean a rigorous approach. I don't ask you to
> > > solve the equations of motion. I even don't ask you to write the
> > > full set of equations to be solved. I am just asking about a
> > > general algorithm how to do that. Which steps in what order
> > > should be taken? So that, if we had a supercomputer we could give
> > > it all necessary instructions.
> >
> > This algorithm is called the closed time path formalism. It exist,
> > it works, and it has been extensively discussed here in the past.
>
> Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
> the FULL two body problem in QFT still.

Hmm, on page 569 of his vol. I on QFT, Weinberg talks about the
spherical harmonic decomposition of solutions to the Dirac equation.
I fail to see the relevance. But, in a similar vein, no-body has solved
the FULL Newtonian 4-body problem (there is actually an infinite series
solution to the 3-body problem). The latter, however, is not a
testament to the incompleteness of Newtonian mechanics but the
technical difficulties associated with the given problem.

> Moreover it is well-known
> that path formalism has several flaws. For example, the standard
> version of the PF offers incorrect S matrices in 'complex' problems:
> a typical example is the nonlinear sigma model.

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when you
> > declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields
> > will fall out automatically when you start with particles. Both
> > formulations are equally "fundamental" or equally not so, however
> > you want to consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments
> are particles, newer fields. As clearly stated by Weinberg in his
> volume 1, we know more about particles that about fields.

I don't know where you've seen this definition, neither do I know where
you've seen a definition that says that particles are observable. Oh,
and I can also say that one never measures particles, only field
correlation functions. And could you be more specific about what
Weinberg says on this topic?

> > > I am trying to avoid discussion of gravity and curved spacetimes.
> > > There are enough troubles in understanding simple
> > > electro-magnetic interactions. Can we stick to "flat spacetimes"
> > > as you call them.
> >
> > First, I don't see much trouble understanding simple
> > electro-magnetic interactions. Moreover, if you wish to talk of
> > fundamentals, you must take the most general situation possible,
> > which includes curved backgrounds. If you stick to flat space-time,
> > you are stuck in a stale-mate since the particle and field
> > formulations are equaivalent. If you refuse to consider curved
> > backgrounds, you are simply refusing to acknoledge a failing of the
> > particle approach, since it's been known for decades that the field
> > approach comes out a clear winner there.
>
> 1) curved spacetime is just a view, only that. As was claimed by
> Feynmann (in his lectures on gravitation) the curved spacetime
> interpretation of GR is unnecesary for physics. In fact, in the
> torsion re-geometrization of gravity, spacetime curvature is exactly
> zero. Therefore, i think that Stefanovich could generalize his work
> to torsion gravity using the flat spacetime he uses for QED.
>
> 2) That in curved spacetime there is no posibility for description of
> particles is not very convincing just as shows the curved spacetime
> generalization of the FFW action (See Hoyle and Narkilar
> gravitational theory).

I am not familiar with either "torsion re-geometrization" nor with the
work of Hoyle and Narkilar. But I do know that every successful
physical theory has been generalized to the case of a curved classical
background. If Eugene's theory is to be successful, why should it be
any different? There is always the option of picking an alternative
description of gravity. But then you have to justify the choice and
explain why that generalization was possible, yet the generalization to
GR (even in the limit in which it is known to apply) was not.

> I am not sure i say next, but in string heory one uses the Fock space
> for the representation of states of vibrating string and one has not
> fields.

One does have fields. They are the embedding space-time coordinates X^u
which are functions of the internal (world-sheet) coordinates (s,t) on
the string. In fact, if we take X^u to be the embedding spac-time
coordinates of a particle as a function of some parameter t that
parametrizes its world-line. These coordinates can also be considered
as fields defined on a 0+1 dimensional manifold.

> > > You are saying that all issues "have been studied and delineated
> > > in exhaustive depth" What about the time evolution of
> > > interacting systems? Take the simplest interacting system - two
> > > electrons. Suppose that I gave you a full description (wave
> > > function) of this system at time t=0. How would you find the
> > > state of the system at a later time t? The amazing thing (to me)
> > > is that field-based quantum electrodynamics doesn't have a clue
> > > how to do that. I mean a rigorous approach. I don't ask you to
> > > solve the equations of motion. I even don't ask you to write the
> > > full set of equations to be solved. I am just asking about a
> > > general algorithm how to do that. Which steps in what order
> > > should be taken? So that, if we had a supercomputer we could give
> > > it all necessary instructions.
> >
> > This algorithm is called the closed time path formalism. It exist,
> > it works, and it has been extensively discussed here in the past.
>
> Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
> the FULL two body problem in QFT still.

Hmm, on page 569 of his vol. I on QFT, Weinberg talks about the
spherical harmonic decomposition of solutions to the Dirac equation.
I fail to see the relevance. But, in a similar vein, no-body has solved
the FULL Newtonian 4-body problem (there is actually an infinite series
solution to the 3-body problem). The latter, however, is not a
testament to the incompleteness of Newtonian mechanics but the
technical difficulties associated with the given problem.

> Moreover it is well-known
> that path formalism has several flaws. For example, the standard
> version of the PF offers incorrect S matrices in 'complex' problems:
> a typical example is the nonlinear sigma model.

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when you
> > declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields
> > will fall out automatically when you start with particles. Both
> > formulations are equally "fundamental" or equally not so, however
> > you want to consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments
> are particles, newer fields. As clearly stated by Weinberg in his
> volume 1, we know more about particles that about fields.

I don't know where you've seen this definition, neither do I know where
you've seen a definition that says that particles are observable. Oh,
and I can also say that one never measures particles, only field
correlation functions. And could you be more specific about what
Weinberg says on this topic?

> > > I am trying to avoid discussion of gravity and curved spacetimes.
> > > There are enough troubles in understanding simple
> > > electro-magnetic interactions. Can we stick to "flat spacetimes"
> > > as you call them.
> >
> > First, I don't see much trouble understanding simple
> > electro-magnetic interactions. Moreover, if you wish to talk of
> > fundamentals, you must take the most general situation possible,
> > which includes curved backgrounds. If you stick to flat space-time,
> > you are stuck in a stale-mate since the particle and field
> > formulations are equaivalent. If you refuse to consider curved
> > backgrounds, you are simply refusing to acknoledge a failing of the
> > particle approach, since it's been known for decades that the field
> > approach comes out a clear winner there.
>
> 1) curved spacetime is just a view, only that. As was claimed by
> Feynmann (in his lectures on gravitation) the curved spacetime
> interpretation of GR is unnecesary for physics. In fact, in the
> torsion re-geometrization of gravity, spacetime curvature is exactly
> zero. Therefore, i think that Stefanovich could generalize his work
> to torsion gravity using the flat spacetime he uses for QED.
>
> 2) That in curved spacetime there is no posibility for description of
> particles is not very convincing just as shows the curved spacetime
> generalization of the FFW action (See Hoyle and Narkilar
> gravitational theory).

I am not familiar with either "torsion re-geometrization" nor with the
work of Hoyle and Narkilar. But I do know that every successful
physical theory has been generalized to the case of a curved classical
background. If Eugene's theory is to be successful, why should it be
any different? There is always the option of picking an alternative
description of gravity. But then you have to justify the choice and
explain why that generalization was possible, yet the generalization to
GR (even in the limit in which it is known to apply) was not.

> I am not sure i say next, but in string heory one uses the Fock space
> for the representation of states of vibrating string and one has not
> fields.

One does have fields. They are the embedding space-time coordinates X^u
which are functions of the internal (world-sheet) coordinates (s,t) on
the string. In fact, if we take X^u to be the embedding spac-time
coordinates of a particle as a function of some parameter t that
parametrizes its world-line. These coordinates can also be considered
as fields defined on a 0+1 dimensional manifold.

> > > You are saying that all issues "have been studied and delineated
> > > in exhaustive depth" What about the time evolution of
> > > interacting systems? Take the simplest interacting system - two
> > > electrons. Suppose that I gave you a full description (wave
> > > function) of this system at time t=0. How would you find the
> > > state of the system at a later time t? The amazing thing (to me)
> > > is that field-based quantum electrodynamics doesn't have a clue
> > > how to do that. I mean a rigorous approach. I don't ask you to
> > > solve the equations of motion. I even don't ask you to write the
> > > full set of equations to be solved. I am just asking about a
> > > general algorithm how to do that. Which steps in what order
> > > should be taken? So that, if we had a supercomputer we could give
> > > it all necessary instructions.
> >
> > This algorithm is called the closed time path formalism. It exist,
> > it works, and it has been extensively discussed here in the past.
>
> Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
> the FULL two body problem in QFT still.

Hmm, on page 569 of his vol. I on QFT, Weinberg talks about the
spherical harmonic decomposition of solutions to the Dirac equation.
I fail to see the relevance. But, in a similar vein, no-body has solved
the FULL Newtonian 4-body problem (there is actually an infinite series
solution to the 3-body problem). The latter, however, is not a
testament to the incompleteness of Newtonian mechanics but the
technical difficulties associated with the given problem.

> Moreover it is well-known
> that path formalism has several flaws. For example, the standard
> version of the PF offers incorrect S matrices in 'complex' problems:
> a typical example is the nonlinear sigma model.

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when you
> > declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields
> > will fall out automatically when you start with particles. Both
> > formulations are equally "fundamental" or equally not so, however
> > you want to consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments
> are particles, newer fields. As clearly stated by Weinberg in his
> volume 1, we know more about particles that about fields.

I don't know where you've seen this definition, neither do I know where
you've seen a definition that says that particles are observable. Oh,
and I can also say that one never measures particles, only field
correlation functions. And could you be more specific about what
Weinberg says on this topic?

> > > I am trying to avoid discussion of gravity and curved spacetimes.
> > > There are enough troubles in understanding simple
> > > electro-magnetic interactions. Can we stick to "flat spacetimes"
> > > as you call them.
> >
> > First, I don't see much trouble understanding simple
> > electro-magnetic interactions. Moreover, if you wish to talk of
> > fundamentals, you must take the most general situation possible,
> > which includes curved backgrounds. If you stick to flat space-time,
> > you are stuck in a stale-mate since the particle and field
> > formulations are equaivalent. If you refuse to consider curved
> > backgrounds, you are simply refusing to acknoledge a failing of the
> > particle approach, since it's been known for decades that the field
> > approach comes out a clear winner there.
>
> 1) curved spacetime is just a view, only that. As was claimed by
> Feynmann (in his lectures on gravitation) the curved spacetime
> interpretation of GR is unnecesary for physics. In fact, in the
> torsion re-geometrization of gravity, spacetime curvature is exactly
> zero. Therefore, i think that Stefanovich could generalize his work
> to torsion gravity using the flat spacetime he uses for QED.
>
> 2) That in curved spacetime there is no posibility for description of
> particles is not very convincing just as shows the curved spacetime
> generalization of the FFW action (See Hoyle and Narkilar
> gravitational theory).

I am not familiar with either "torsion re-geometrization" nor with the
work of Hoyle and Narkilar. But I do know that every successful
physical theory has been generalized to the case of a curved classical
background. If Eugene's theory is to be successful, why should it be
any different? There is always the option of picking an alternative
description of gravity. But then you have to justify the choice and
explain why that generalization was possible, yet the generalization to
GR (even in the limit in which it is known to apply) was not.

> I am not sure i say next, but in string heory one uses the Fock space
> for the representation of states of vibrating string and one has not
> fields.

One does have fields. They are the embedding space-time coordinates X^u
which are functions of the internal (world-sheet) coordinates (s,t) on
the string. In fact, if we take X^u to be the embedding spac-time
coordinates of a particle as a function of some parameter t that
parametrizes its world-line. These coordinates can also be considered
as fields defined on a 0+1 dimensional manifold.

> > > You are saying that all issues "have been studied and delineated
> > > in exhaustive depth" What about the time evolution of
> > > interacting systems? Take the simplest interacting system - two
> > > electrons. Suppose that I gave you a full description (wave
> > > function) of this system at time t=0. How would you find the
> > > state of the system at a later time t? The amazing thing (to me)
> > > is that field-based quantum electrodynamics doesn't have a clue
> > > how to do that. I mean a rigorous approach. I don't ask you to
> > > solve the equations of motion. I even don't ask you to write the
> > > full set of equations to be solved. I am just asking about a
> > > general algorithm how to do that. Which steps in what order
> > > should be taken? So that, if we had a supercomputer we could give
> > > it all necessary instructions.
> >
> > This algorithm is called the closed time path formalism. It exist,
> > it works, and it has been extensively discussed here in the past.
>
> Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
> the FULL two body problem in QFT still.

Hmm, on page 569 of his vol. I on QFT, Weinberg talks about the
spherical harmonic decomposition of solutions to the Dirac equation.
I fail to see the relevance. But, in a similar vein, no-body has solved
the FULL Newtonian 4-body problem (there is actually an infinite series
solution to the 3-body problem). The latter, however, is not a
testament to the incompleteness of Newtonian mechanics but the
technical difficulties associated with the given problem.

> Moreover it is well-known
> that path formalism has several flaws. For example, the standard
> version of the PF offers incorrect S matrices in 'complex' problems:
> a typical example is the nonlinear sigma model.

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when you
> > declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields
> > will fall out automatically when you start with particles. Both
> > formulations are equally "fundamental" or equally not so, however
> > you want to consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments
> are particles, newer fields. As clearly stated by Weinberg in his
> volume 1, we know more about particles that about fields.

I don't know where you've seen this definition, neither do I know where
you've seen a definition that says that particles are observable. Oh,
and I can also say that one never measures particles, only field
correlation functions. And could you be more specific about what
Weinberg says on this topic?

> > > I am trying to avoid discussion of gravity and curved spacetimes.
> > > There are enough troubles in understanding simple
> > > electro-magnetic interactions. Can we stick to "flat spacetimes"
> > > as you call them.
> >
> > First, I don't see much trouble understanding simple
> > electro-magnetic interactions. Moreover, if you wish to talk of
> > fundamentals, you must take the most general situation possible,
> > which includes curved backgrounds. If you stick to flat space-time,
> > you are stuck in a stale-mate since the particle and field
> > formulations are equaivalent. If you refuse to consider curved
> > backgrounds, you are simply refusing to acknoledge a failing of the
> > particle approach, since it's been known for decades that the field
> > approach comes out a clear winner there.
>
> 1) curved spacetime is just a view, only that. As was claimed by
> Feynmann (in his lectures on gravitation) the curved spacetime
> interpretation of GR is unnecesary for physics. In fact, in the
> torsion re-geometrization of gravity, spacetime curvature is exactly
> zero. Therefore, i think that Stefanovich could generalize his work
> to torsion gravity using the flat spacetime he uses for QED.
>
> 2) That in curved spacetime there is no posibility for description of
> particles is not very convincing just as shows the curved spacetime
> generalization of the FFW action (See Hoyle and Narkilar
> gravitational theory).

I am not familiar with either "torsion re-geometrization" nor with the
work of Hoyle and Narkilar. But I do know that every successful
physical theory has been generalized to the case of a curved classical
background. If Eugene's theory is to be successful, why should it be
any different? There is always the option of picking an alternative
description of gravity. But then you have to justify the choice and
explain why that generalization was possible, yet the generalization to
GR (even in the limit in which it is known to apply) was not.

> I am not sure i say next, but in string heory one uses the Fock space
> for the representation of states of vibrating string and one has not
> fields.

One does have fields. They are the embedding space-time coordinates X^u
which are functions of the internal (world-sheet) coordinates (s,t) on
the string. In fact, if we take X^u to be the embedding spac-time
coordinates of a particle as a function of some parameter t that
parametrizes its world-line. These coordinates can also be considered
as fields defined on a 0+1 dimensional manifold.

> > > You are saying that all issues "have been studied and delineated
> > > in exhaustive depth" What about the time evolution of
> > > interacting systems? Take the simplest interacting system - two
> > > electrons. Suppose that I gave you a full description (wave
> > > function) of this system at time t=0. How would you find the
> > > state of the system at a later time t? The amazing thing (to me)
> > > is that field-based quantum electrodynamics doesn't have a clue
> > > how to do that. I mean a rigorous approach. I don't ask you to
> > > solve the equations of motion. I even don't ask you to write the
> > > full set of equations to be solved. I am just asking about a
> > > general algorithm how to do that. Which steps in what order
> > > should be taken? So that, if we had a supercomputer we could give
> > > it all necessary instructions.
> >
> > This algorithm is called the closed time path formalism. It exist,
> > it works, and it has been extensively discussed here in the past.
>
> Curiously Weinberg (p569) agrees with Stefanovich: nobody has solved
> the FULL two body problem in QFT still.

Hmm, on page 569 of his vol. I on QFT, Weinberg talks about the
spherical harmonic decomposition of solutions to the Dirac equation.
I fail to see the relevance. But, in a similar vein, no-body has solved
the FULL Newtonian 4-body problem (there is actually an infinite series
solution to the 3-body problem). The latter, however, is not a
testament to the incompleteness of Newtonian mechanics but the
technical difficulties associated with the given problem.

> Moreover it is well-known
> that path formalism has several flaws. For example, the standard
> version of the PF offers incorrect S matrices in 'complex' problems:
> a typical example is the nonlinear sigma model.

Reference please.

Igor

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1127902766.738417.90890@g49g2000cwa.googlegro ups.com...
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when
> > you declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields will
> > fall out automatically when you start with particles. Both formulations
> > are equally "fundamental" or equally not so, however you want to
> > consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.

You are right.
I found Weinberg's book much closer to my position than any other book.
Weinberg still speaks about the fundamental character of quantum fields, but
if
you read his book carefully, you'll notice that fields are introduced there
for one and only one reason: to facilitate the construction of the Lorentz
invariant S-matrix. If one can do that without invoking quantum fields
(I am pretty sure it is possible), then we can forget about fields
altogether.

It is also interesting to note Weinberg's attitude to local gauge
invariance.
(he briefly mentions about that in the book, there is more in his
1960's papers). He does not present the local gauge invarance
as a fundamental physical principle (as many other textbooks do).
Instead, he notices that the quantum field of photons does not transform
covariantly with respect to boosts. Therefore, in order to get the Lorentz
invariant S-matrix (which is Weinberg's primary concern) one must build
interaction in a specific way, i.e., to
couple the photon field with the "conserved current" built of charged
particle
fields. This is known as "minimal coupling".

In summary: the fundamental requirement is to build interacting generators
of the Poincare group in the Fock space and their commutators with particle
observables. Once we have that, we have all physics in our hands.
Unfortunately, the only known way to build the generators is to use
certain linear combinations of particle creation and annihilation operators,
known as fields. Luckily, this construction leads to perfect agreement with
experiment. However, there is no proof that this is the only way to build
the generators.
There is no indication that the fields have any fundamental role
(except as convenient prefabricated "building blocks" for interaction).

Again, I am not talking about curved spaces, solitons, broken symmetries.
I am talking only about QED, as the title of this thread shows.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1127902766.738417.90890@g49g2000cwa.googlegro ups.com...
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when
> > you declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields will
> > fall out automatically when you start with particles. Both formulations
> > are equally "fundamental" or equally not so, however you want to
> > consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.

You are right.
I found Weinberg's book much closer to my position than any other book.
Weinberg still speaks about the fundamental character of quantum fields, but
if
you read his book carefully, you'll notice that fields are introduced there
for one and only one reason: to facilitate the construction of the Lorentz
invariant S-matrix. If one can do that without invoking quantum fields
(I am pretty sure it is possible), then we can forget about fields
altogether.

It is also interesting to note Weinberg's attitude to local gauge
invariance.
(he briefly mentions about that in the book, there is more in his
1960's papers). He does not present the local gauge invarance
as a fundamental physical principle (as many other textbooks do).
Instead, he notices that the quantum field of photons does not transform
covariantly with respect to boosts. Therefore, in order to get the Lorentz
invariant S-matrix (which is Weinberg's primary concern) one must build
interaction in a specific way, i.e., to
couple the photon field with the "conserved current" built of charged
particle
fields. This is known as "minimal coupling".

In summary: the fundamental requirement is to build interacting generators
of the Poincare group in the Fock space and their commutators with particle
observables. Once we have that, we have all physics in our hands.
Unfortunately, the only known way to build the generators is to use
certain linear combinations of particle creation and annihilation operators,
known as fields. Luckily, this construction leads to perfect agreement with
experiment. However, there is no proof that this is the only way to build
the generators.
There is no indication that the fields have any fundamental role
(except as convenient prefabricated "building blocks" for interaction).

Again, I am not talking about curved spaces, solitons, broken symmetries.
I am talking only about QED, as the title of this thread shows.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1127902766.738417.90890@g49g2000cwa.googlegro ups.com...
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when
> > you declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields will
> > fall out automatically when you start with particles. Both formulations
> > are equally "fundamental" or equally not so, however you want to
> > consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.

You are right.
I found Weinberg's book much closer to my position than any other book.
Weinberg still speaks about the fundamental character of quantum fields, but
if
you read his book carefully, you'll notice that fields are introduced there
for one and only one reason: to facilitate the construction of the Lorentz
invariant S-matrix. If one can do that without invoking quantum fields
(I am pretty sure it is possible), then we can forget about fields
altogether.

It is also interesting to note Weinberg's attitude to local gauge
invariance.
(he briefly mentions about that in the book, there is more in his
1960's papers). He does not present the local gauge invarance
as a fundamental physical principle (as many other textbooks do).
Instead, he notices that the quantum field of photons does not transform
covariantly with respect to boosts. Therefore, in order to get the Lorentz
invariant S-matrix (which is Weinberg's primary concern) one must build
interaction in a specific way, i.e., to
couple the photon field with the "conserved current" built of charged
particle
fields. This is known as "minimal coupling".

In summary: the fundamental requirement is to build interacting generators
of the Poincare group in the Fock space and their commutators with particle
observables. Once we have that, we have all physics in our hands.
Unfortunately, the only known way to build the generators is to use
certain linear combinations of particle creation and annihilation operators,
known as fields. Luckily, this construction leads to perfect agreement with
experiment. However, there is no proof that this is the only way to build
the generators.
There is no indication that the fields have any fundamental role
(except as convenient prefabricated "building blocks" for interaction).

Again, I am not talking about curved spaces, solitons, broken symmetries.
I am talking only about QED, as the title of this thread shows.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1127902766.738417.90890@g49g2000cwa.googlegro ups.com...
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when
> > you declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields will
> > fall out automatically when you start with particles. Both formulations
> > are equally "fundamental" or equally not so, however you want to
> > consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.

You are right.
I found Weinberg's book much closer to my position than any other book.
Weinberg still speaks about the fundamental character of quantum fields, but
if
you read his book carefully, you'll notice that fields are introduced there
for one and only one reason: to facilitate the construction of the Lorentz
invariant S-matrix. If one can do that without invoking quantum fields
(I am pretty sure it is possible), then we can forget about fields
altogether.

It is also interesting to note Weinberg's attitude to local gauge
invariance.
(he briefly mentions about that in the book, there is more in his
1960's papers). He does not present the local gauge invarance
as a fundamental physical principle (as many other textbooks do).
Instead, he notices that the quantum field of photons does not transform
covariantly with respect to boosts. Therefore, in order to get the Lorentz
invariant S-matrix (which is Weinberg's primary concern) one must build
interaction in a specific way, i.e., to
couple the photon field with the "conserved current" built of charged
particle
fields. This is known as "minimal coupling".

In summary: the fundamental requirement is to build interacting generators
of the Poincare group in the Fock space and their commutators with particle
observables. Once we have that, we have all physics in our hands.
Unfortunately, the only known way to build the generators is to use
certain linear combinations of particle creation and annihilation operators,
known as fields. Luckily, this construction leads to perfect agreement with
experiment. However, there is no proof that this is the only way to build
the generators.
There is no indication that the fields have any fundamental role
(except as convenient prefabricated "building blocks" for interaction).

Again, I am not talking about curved spaces, solitons, broken symmetries.
I am talking only about QED, as the title of this thread shows.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1127902766.738417.90890@g49g2000cwa.googlegro ups.com...
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when
> > you declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields will
> > fall out automatically when you start with particles. Both formulations
> > are equally "fundamental" or equally not so, however you want to
> > consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.

You are right.
I found Weinberg's book much closer to my position than any other book.
Weinberg still speaks about the fundamental character of quantum fields, but
if
you read his book carefully, you'll notice that fields are introduced there
for one and only one reason: to facilitate the construction of the Lorentz
invariant S-matrix. If one can do that without invoking quantum fields
(I am pretty sure it is possible), then we can forget about fields
altogether.

It is also interesting to note Weinberg's attitude to local gauge
invariance.
(he briefly mentions about that in the book, there is more in his
1960's papers). He does not present the local gauge invarance
as a fundamental physical principle (as many other textbooks do).
Instead, he notices that the quantum field of photons does not transform
covariantly with respect to boosts. Therefore, in order to get the Lorentz
invariant S-matrix (which is Weinberg's primary concern) one must build
interaction in a specific way, i.e., to
couple the photon field with the "conserved current" built of charged
particle
fields. This is known as "minimal coupling".

In summary: the fundamental requirement is to build interacting generators
of the Poincare group in the Fock space and their commutators with particle
observables. Once we have that, we have all physics in our hands.
Unfortunately, the only known way to build the generators is to use
certain linear combinations of particle creation and annihilation operators,
known as fields. Luckily, this construction leads to perfect agreement with
experiment. However, there is no proof that this is the only way to build
the generators.
There is no indication that the fields have any fundamental role
(except as convenient prefabricated "building blocks" for interaction).

Again, I am not talking about curved spaces, solitons, broken symmetries.
I am talking only about QED, as the title of this thread shows.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1127902766.738417.90890@g49g2000cwa.googlegro ups.com...
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when
> > you declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields will
> > fall out automatically when you start with particles. Both formulations
> > are equally "fundamental" or equally not so, however you want to
> > consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.

You are right.
I found Weinberg's book much closer to my position than any other book.
Weinberg still speaks about the fundamental character of quantum fields, but
if
you read his book carefully, you'll notice that fields are introduced there
for one and only one reason: to facilitate the construction of the Lorentz
invariant S-matrix. If one can do that without invoking quantum fields
(I am pretty sure it is possible), then we can forget about fields
altogether.

It is also interesting to note Weinberg's attitude to local gauge
invariance.
(he briefly mentions about that in the book, there is more in his
1960's papers). He does not present the local gauge invarance
as a fundamental physical principle (as many other textbooks do).
Instead, he notices that the quantum field of photons does not transform
covariantly with respect to boosts. Therefore, in order to get the Lorentz
invariant S-matrix (which is Weinberg's primary concern) one must build
interaction in a specific way, i.e., to
couple the photon field with the "conserved current" built of charged
particle
fields. This is known as "minimal coupling".

In summary: the fundamental requirement is to build interacting generators
of the Poincare group in the Fock space and their commutators with particle
observables. Once we have that, we have all physics in our hands.
Unfortunately, the only known way to build the generators is to use
certain linear combinations of particle creation and annihilation operators,
known as fields. Luckily, this construction leads to perfect agreement with
experiment. However, there is no proof that this is the only way to build
the generators.
There is no indication that the fields have any fundamental role
(except as convenient prefabricated "building blocks" for interaction).

Again, I am not talking about curved spaces, solitons, broken symmetries.
I am talking only about QED, as the title of this thread shows.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1127902766.738417.90890@g49g2000cwa.googlegro ups.com...
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when
> > you declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields will
> > fall out automatically when you start with particles. Both formulations
> > are equally "fundamental" or equally not so, however you want to
> > consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.

You are right.
I found Weinberg's book much closer to my position than any other book.
Weinberg still speaks about the fundamental character of quantum fields, but
if
you read his book carefully, you'll notice that fields are introduced there
for one and only one reason: to facilitate the construction of the Lorentz
invariant S-matrix. If one can do that without invoking quantum fields
(I am pretty sure it is possible), then we can forget about fields
altogether.

It is also interesting to note Weinberg's attitude to local gauge
invariance.
(he briefly mentions about that in the book, there is more in his
1960's papers). He does not present the local gauge invarance
as a fundamental physical principle (as many other textbooks do).
Instead, he notices that the quantum field of photons does not transform
covariantly with respect to boosts. Therefore, in order to get the Lorentz
invariant S-matrix (which is Weinberg's primary concern) one must build
interaction in a specific way, i.e., to
couple the photon field with the "conserved current" built of charged
particle
fields. This is known as "minimal coupling".

In summary: the fundamental requirement is to build interacting generators
of the Poincare group in the Fock space and their commutators with particle
observables. Once we have that, we have all physics in our hands.
Unfortunately, the only known way to build the generators is to use
certain linear combinations of particle creation and annihilation operators,
known as fields. Luckily, this construction leads to perfect agreement with
experiment. However, there is no proof that this is the only way to build
the generators.
There is no indication that the fields have any fundamental role
(except as convenient prefabricated "building blocks" for interaction).

Again, I am not talking about curved spaces, solitons, broken symmetries.
I am talking only about QED, as the title of this thread shows.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1127902766.738417.90890@g49g2000cwa.googlegro ups.com...
> Igor Khavkine wrote:
> >
> > Regarding your original post. Your argument falls through when
> > you declare that particles are "fundamental" in the way you define that
> > term. One can take fields as a starting point and never mention
> > particles at all, they will fall out automatically. Just as fields will
> > fall out automatically when you start with particles. Both formulations
> > are equally "fundamental" or equally not so, however you want to
> > consider them.
>
> Particles are more fundamental that fields because fields are ALWAYS
> -by definition- unobserved, one measures in scattering experiments are
> particles, newer fields. As clearly stated by Weinberg in his volume 1,
> we know more about particles that about fields.

You are right.
I found Weinberg's book much closer to my position than any other book.
Weinberg still speaks about the fundamental character of quantum fields, but
if
you read his book carefully, you'll notice that fields are introduced there
for one and only one reason: to facilitate the construction of the Lorentz
invariant S-matrix. If one can do that without invoking quantum fields
(I am pretty sure it is possible), then we can forget about fields
altogether.

It is also interesting to note Weinberg's attitude to local gauge
invariance.
(he briefly mentions about that in the book, there is more in his
1960's papers). He does not present the local gauge invarance
as a fundamental physical principle (as many other textbooks do).
Instead, he notices that the quantum field of photons does not transform
covariantly with respect to boosts. Therefore, in order to get the Lorentz
invariant S-matrix (which is Weinberg's primary concern) one must build
interaction in a specific way, i.e., to
couple the photon field with the "conserved current" built of charged
particle
fields. This is known as "minimal coupling".

In summary: the fundamental requirement is to build interacting generators
of the Poincare group in the Fock space and their commutators with particle
observables. Once we have that, we have all physics in our hands.
Unfortunately, the only known way to build the generators is to use
certain linear combinations of particle creation and annihilation operators,
known as fields. Luckily, this construction leads to perfect agreement with
experiment. However, there is no proof that this is the only way to build
the generators.
There is no indication that the fields have any fundamental role
(except as convenient prefabricated "building blocks" for interaction).

Again, I am not talking about curved spaces, solitons, broken symmetries.
I am talking only about QED, as the title of this thread shows.

Eugene.

[carlip-nospam@physics.ucdavis.edu]

Charles Francis <charles@clef.demon.co.uk> wrote:

[...]
> Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both point
> to serious difficulties even in defining fields.

I have Wald's book (_Quantum Field Theory in Curved Spacetime and
Black Hole Thermodynamics_) in front of me, and I think you have
this backwards. In the introduction, he writes:

In these notes, we shall present quantum field theory
directly as a theory resulting from applying the principles
of quantum theory to a classical field system.... In this
manner, it will be clear from the outset that the fundamental
observables of the theory are field amplitudes and momenta.
As we shall see, in flat spacetime -- and, more generally,
in a curved stationary spacetime -- a natural particle
interpretation will emerge when we couple the field to a
simple model quantum mechanical system (i.e., a "particle
detector") and investigate the effects of the interaction.
In curved spacetimes which are asymptotically stationary
in the past and/or future, natural particle interpretations
are also available in these asymptotic regions. However,
in other circumstances, the notion of "particle" is, at best,
of very limited utility.

A running theme of Wald's book is that it is *not* hard to define
quantum fields in a curved spacetime, at least as long as the
spacetime is globally hyperbolic. The problems are instead in
the particle formulation.

Steve Carlip

[carlip-nospam@physics.ucdavis.edu]

Charles Francis <charles@clef.demon.co.uk> wrote:

[...]
> Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both point
> to serious difficulties even in defining fields.

I have Wald's book (_Quantum Field Theory in Curved Spacetime and
Black Hole Thermodynamics_) in front of me, and I think you have
this backwards. In the introduction, he writes:

In these notes, we shall present quantum field theory
directly as a theory resulting from applying the principles
of quantum theory to a classical field system.... In this
manner, it will be clear from the outset that the fundamental
observables of the theory are field amplitudes and momenta.
As we shall see, in flat spacetime -- and, more generally,
in a curved stationary spacetime -- a natural particle
interpretation will emerge when we couple the field to a
simple model quantum mechanical system (i.e., a "particle
detector") and investigate the effects of the interaction.
In curved spacetimes which are asymptotically stationary
in the past and/or future, natural particle interpretations
are also available in these asymptotic regions. However,
in other circumstances, the notion of "particle" is, at best,
of very limited utility.

A running theme of Wald's book is that it is *not* hard to define
quantum fields in a curved spacetime, at least as long as the
spacetime is globally hyperbolic. The problems are instead in
the particle formulation.

Steve Carlip

[carlip-nospam@physics.ucdavis.edu]

Charles Francis <charles@clef.demon.co.uk> wrote:

[...]
> Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both point
> to serious difficulties even in defining fields.

I have Wald's book (_Quantum Field Theory in Curved Spacetime and
Black Hole Thermodynamics_) in front of me, and I think you have
this backwards. In the introduction, he writes:

In these notes, we shall present quantum field theory
directly as a theory resulting from applying the principles
of quantum theory to a classical field system.... In this
manner, it will be clear from the outset that the fundamental
observables of the theory are field amplitudes and momenta.
As we shall see, in flat spacetime -- and, more generally,
in a curved stationary spacetime -- a natural particle
interpretation will emerge when we couple the field to a
simple model quantum mechanical system (i.e., a "particle
detector") and investigate the effects of the interaction.
In curved spacetimes which are asymptotically stationary
in the past and/or future, natural particle interpretations
are also available in these asymptotic regions. However,
in other circumstances, the notion of "particle" is, at best,
of very limited utility.

A running theme of Wald's book is that it is *not* hard to define
quantum fields in a curved spacetime, at least as long as the
spacetime is globally hyperbolic. The problems are instead in
the particle formulation.

Steve Carlip

[carlip-nospam@physics.ucdavis.edu]

Charles Francis <charles@clef.demon.co.uk> wrote:

[...]
> Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both point
> to serious difficulties even in defining fields.

I have Wald's book (_Quantum Field Theory in Curved Spacetime and
Black Hole Thermodynamics_) in front of me, and I think you have
this backwards. In the introduction, he writes:

In these notes, we shall present quantum field theory
directly as a theory resulting from applying the principles
of quantum theory to a classical field system.... In this
manner, it will be clear from the outset that the fundamental
observables of the theory are field amplitudes and momenta.
As we shall see, in flat spacetime -- and, more generally,
in a curved stationary spacetime -- a natural particle
interpretation will emerge when we couple the field to a
simple model quantum mechanical system (i.e., a "particle
detector") and investigate the effects of the interaction.
In curved spacetimes which are asymptotically stationary
in the past and/or future, natural particle interpretations
are also available in these asymptotic regions. However,
in other circumstances, the notion of "particle" is, at best,
of very limited utility.

A running theme of Wald's book is that it is *not* hard to define
quantum fields in a curved spacetime, at least as long as the
spacetime is globally hyperbolic. The problems are instead in
the particle formulation.

Steve Carlip

[carlip-nospam@physics.ucdavis.edu]

Charles Francis <charles@clef.demon.co.uk> wrote:

[...]
> Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both point
> to serious difficulties even in defining fields.

I have Wald's book (_Quantum Field Theory in Curved Spacetime and
Black Hole Thermodynamics_) in front of me, and I think you have
this backwards. In the introduction, he writes:

In these notes, we shall present quantum field theory
directly as a theory resulting from applying the principles
of quantum theory to a classical field system.... In this
manner, it will be clear from the outset that the fundamental
observables of the theory are field amplitudes and momenta.
As we shall see, in flat spacetime -- and, more generally,
in a curved stationary spacetime -- a natural particle
interpretation will emerge when we couple the field to a
simple model quantum mechanical system (i.e., a "particle
detector") and investigate the effects of the interaction.
In curved spacetimes which are asymptotically stationary
in the past and/or future, natural particle interpretations
are also available in these asymptotic regions. However,
in other circumstances, the notion of "particle" is, at best,
of very limited utility.

A running theme of Wald's book is that it is *not* hard to define
quantum fields in a curved spacetime, at least as long as the
spacetime is globally hyperbolic. The problems are instead in
the particle formulation.

Steve Carlip

[carlip-nospam@physics.ucdavis.edu]

Charles Francis <charles@clef.demon.co.uk> wrote:

[...]
> Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both point
> to serious difficulties even in defining fields.

I have Wald's book (_Quantum Field Theory in Curved Spacetime and
Black Hole Thermodynamics_) in front of me, and I think you have
this backwards. In the introduction, he writes:

In these notes, we shall present quantum field theory
directly as a theory resulting from applying the principles
of quantum theory to a classical field system.... In this
manner, it will be clear from the outset that the fundamental
observables of the theory are field amplitudes and momenta.
As we shall see, in flat spacetime -- and, more generally,
in a curved stationary spacetime -- a natural particle
interpretation will emerge when we couple the field to a
simple model quantum mechanical system (i.e., a "particle
detector") and investigate the effects of the interaction.
In curved spacetimes which are asymptotically stationary
in the past and/or future, natural particle interpretations
are also available in these asymptotic regions. However,
in other circumstances, the notion of "particle" is, at best,
of very limited utility.

A running theme of Wald's book is that it is *not* hard to define
quantum fields in a curved spacetime, at least as long as the
spacetime is globally hyperbolic. The problems are instead in
the particle formulation.

Steve Carlip

[carlip-nospam@physics.ucdavis.edu]

Charles Francis <charles@clef.demon.co.uk> wrote:

[...]
> Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both point
> to serious difficulties even in defining fields.

I have Wald's book (_Quantum Field Theory in Curved Spacetime and
Black Hole Thermodynamics_) in front of me, and I think you have
this backwards. In the introduction, he writes:

In these notes, we shall present quantum field theory
directly as a theory resulting from applying the principles
of quantum theory to a classical field system.... In this
manner, it will be clear from the outset that the fundamental
observables of the theory are field amplitudes and momenta.
As we shall see, in flat spacetime -- and, more generally,
in a curved stationary spacetime -- a natural particle
interpretation will emerge when we couple the field to a
simple model quantum mechanical system (i.e., a "particle
detector") and investigate the effects of the interaction.
In curved spacetimes which are asymptotically stationary
in the past and/or future, natural particle interpretations
are also available in these asymptotic regions. However,
in other circumstances, the notion of "particle" is, at best,
of very limited utility.

A running theme of Wald's book is that it is *not* hard to define
quantum fields in a curved spacetime, at least as long as the
spacetime is globally hyperbolic. The problems are instead in
the particle formulation.

Steve Carlip

[carlip-nospam@physics.ucdavis.edu]

Charles Francis <charles@clef.demon.co.uk> wrote:

[...]
> Wald and Fulling have both written books on
> quantum field theory in curved space-time, as I recall they both point
> to serious difficulties even in defining fields.

I have Wald's book (_Quantum Field Theory in Curved Spacetime and
Black Hole Thermodynamics_) in front of me, and I think you have
this backwards. In the introduction, he writes:

In these notes, we shall present quantum field theory
directly as a theory resulting from applying the principles
of quantum theory to a classical field system.... In this
manner, it will be clear from the outset that the fundamental
observables of the theory are field amplitudes and momenta.
As we shall see, in flat spacetime -- and, more generally,
in a curved stationary spacetime -- a natural particle
interpretation will emerge when we couple the field to a
simple model quantum mechanical system (i.e., a "particle
detector") and investigate the effects of the interaction.
In curved spacetimes which are asymptotically stationary
in the past and/or future, natural particle interpretations
are also available in these asymptotic regions. However,
in other circumstances, the notion of "particle" is, at best,
of very limited utility.

A running theme of Wald's book is that it is *not* hard to define
quantum fields in a curved spacetime, at least as long as the
spacetime is globally hyperbolic. The problems are instead in
the particle formulation.

Steve Carlip

[Eugene Stefanovich]

J. Horta wrote:

> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.
> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.


The particle approach easily describes the hydrogen atom: this is a
two-particle state with certain wave-function that determines the
probabilities of measurements of various observables. I don't need
to mention "quantum fields" in this description. How would you
describe the hydrogen atom using quantum fields and not mentioning
the word "particle"?

Eugene.

[Eugene Stefanovich]

J. Horta wrote:

> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.
> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.


The particle approach easily describes the hydrogen atom: this is a
two-particle state with certain wave-function that determines the
probabilities of measurements of various observables. I don't need
to mention "quantum fields" in this description. How would you
describe the hydrogen atom using quantum fields and not mentioning
the word "particle"?

Eugene.

[Eugene Stefanovich]

J. Horta wrote:

> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.
> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.


The particle approach easily describes the hydrogen atom: this is a
two-particle state with certain wave-function that determines the
probabilities of measurements of various observables. I don't need
to mention "quantum fields" in this description. How would you
describe the hydrogen atom using quantum fields and not mentioning
the word "particle"?

Eugene.

[Eugene Stefanovich]

J. Horta wrote:

> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.
> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.


The particle approach easily describes the hydrogen atom: this is a
two-particle state with certain wave-function that determines the
probabilities of measurements of various observables. I don't need
to mention "quantum fields" in this description. How would you
describe the hydrogen atom using quantum fields and not mentioning
the word "particle"?

Eugene.

[Eugene Stefanovich]

J. Horta wrote:

> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.
> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.


The particle approach easily describes the hydrogen atom: this is a
two-particle state with certain wave-function that determines the
probabilities of measurements of various observables. I don't need
to mention "quantum fields" in this description. How would you
describe the hydrogen atom using quantum fields and not mentioning
the word "particle"?

Eugene.

[Eugene Stefanovich]

J. Horta wrote:

> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.
> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.


The particle approach easily describes the hydrogen atom: this is a
two-particle state with certain wave-function that determines the
probabilities of measurements of various observables. I don't need
to mention "quantum fields" in this description. How would you
describe the hydrogen atom using quantum fields and not mentioning
the word "particle"?

Eugene.

[Eugene Stefanovich]

J. Horta wrote:

> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.
> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.


The particle approach easily describes the hydrogen atom: this is a
two-particle state with certain wave-function that determines the
probabilities of measurements of various observables. I don't need
to mention "quantum fields" in this description. How would you
describe the hydrogen atom using quantum fields and not mentioning
the word "particle"?

Eugene.

[Eugene Stefanovich]

J. Horta wrote:

> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.
> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.


The particle approach easily describes the hydrogen atom: this is a
two-particle state with certain wave-function that determines the
probabilities of measurements of various observables. I don't need
to mention "quantum fields" in this description. How would you
describe the hydrogen atom using quantum fields and not mentioning
the word "particle"?

Eugene.

[J. Horta]

On Sun, 02 Oct 2005 19:55:32 +0000, Eugene Stefanovich wrote:

>
>
> J. Horta wrote:
>
>> I very much agree with Igor. What is or is not observable depends on
>> the situation. At very low energies electric fields may be measured
>> with a volt meter and magnetic fields with a magnetometer. A radio
>> picks up em waves. Sure, each one of these may be formulated as
>> interaction with quantum fields or, as follows from the formalism,
>> absorption and emission of particles. The physics should be the same
>> independent of how it is formulated.
>> If you should find, however,
>> a formulation of the physics using a particle approach which includes
>> phenomena which a quantum field approach can't then you could well
>> claim a more fundamental theory.
>
>
> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?
>
> Eugene.

To answer your question QF theory wouldn't be accepted
if the hydrogen atom were beyond its grasp. In fact the
Lamb shift prediction as I recall is touted as one of
the major achievements of QFT.

BTW, I could equally well ask you to describe the reception of
a 400 kHz wave with a radio receiver using just a particle
picture. Sure, it can be done but why bother. You assert
the particle aspect of field theory is more "fundamental"
than the field aspect. I don't see this. At 400 kHz photons
are not detectable as particles by any technology I am aware
of. At higher energies when all is said and done the detector
goes click and the computer registers another count. You
prefer to view this as particle interactions. I prefer to
think of it as the exchange of field quanta. In the end all
that is there is the math.

Paul C.

[J. Horta]

On Sun, 02 Oct 2005 19:55:32 +0000, Eugene Stefanovich wrote:

>
>
> J. Horta wrote:
>
>> I very much agree with Igor. What is or is not observable depends on
>> the situation. At very low energies electric fields may be measured
>> with a volt meter and magnetic fields with a magnetometer. A radio
>> picks up em waves. Sure, each one of these may be formulated as
>> interaction with quantum fields or, as follows from the formalism,
>> absorption and emission of particles. The physics should be the same
>> independent of how it is formulated.
>> If you should find, however,
>> a formulation of the physics using a particle approach which includes
>> phenomena which a quantum field approach can't then you could well
>> claim a more fundamental theory.
>
>
> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?
>
> Eugene.

To answer your question QF theory wouldn't be accepted
if the hydrogen atom were beyond its grasp. In fact the
Lamb shift prediction as I recall is touted as one of
the major achievements of QFT.

BTW, I could equally well ask you to describe the reception of
a 400 kHz wave with a radio receiver using just a particle
picture. Sure, it can be done but why bother. You assert
the particle aspect of field theory is more "fundamental"
than the field aspect. I don't see this. At 400 kHz photons
are not detectable as particles by any technology I am aware
of. At higher energies when all is said and done the detector
goes click and the computer registers another count. You
prefer to view this as particle interactions. I prefer to
think of it as the exchange of field quanta. In the end all
that is there is the math.

Paul C.

[J. Horta]

On Sun, 02 Oct 2005 19:55:32 +0000, Eugene Stefanovich wrote:

>
>
> J. Horta wrote:
>
>> I very much agree with Igor. What is or is not observable depends on
>> the situation. At very low energies electric fields may be measured
>> with a volt meter and magnetic fields with a magnetometer. A radio
>> picks up em waves. Sure, each one of these may be formulated as
>> interaction with quantum fields or, as follows from the formalism,
>> absorption and emission of particles. The physics should be the same
>> independent of how it is formulated.
>> If you should find, however,
>> a formulation of the physics using a particle approach which includes
>> phenomena which a quantum field approach can't then you could well
>> claim a more fundamental theory.
>
>
> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?
>
> Eugene.

To answer your question QF theory wouldn't be accepted
if the hydrogen atom were beyond its grasp. In fact the
Lamb shift prediction as I recall is touted as one of
the major achievements of QFT.

BTW, I could equally well ask you to describe the reception of
a 400 kHz wave with a radio receiver using just a particle
picture. Sure, it can be done but why bother. You assert
the particle aspect of field theory is more "fundamental"
than the field aspect. I don't see this. At 400 kHz photons
are not detectable as particles by any technology I am aware
of. At higher energies when all is said and done the detector
goes click and the computer registers another count. You
prefer to view this as particle interactions. I prefer to
think of it as the exchange of field quanta. In the end all
that is there is the math.

Paul C.

[J. Horta]

On Sun, 02 Oct 2005 19:55:32 +0000, Eugene Stefanovich wrote:

>
>
> J. Horta wrote:
>
>> I very much agree with Igor. What is or is not observable depends on
>> the situation. At very low energies electric fields may be measured
>> with a volt meter and magnetic fields with a magnetometer. A radio
>> picks up em waves. Sure, each one of these may be formulated as
>> interaction with quantum fields or, as follows from the formalism,
>> absorption and emission of particles. The physics should be the same
>> independent of how it is formulated.
>> If you should find, however,
>> a formulation of the physics using a particle approach which includes
>> phenomena which a quantum field approach can't then you could well
>> claim a more fundamental theory.
>
>
> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?
>
> Eugene.

To answer your question QF theory wouldn't be accepted
if the hydrogen atom were beyond its grasp. In fact the
Lamb shift prediction as I recall is touted as one of
the major achievements of QFT.

BTW, I could equally well ask you to describe the reception of
a 400 kHz wave with a radio receiver using just a particle
picture. Sure, it can be done but why bother. You assert
the particle aspect of field theory is more "fundamental"
than the field aspect. I don't see this. At 400 kHz photons
are not detectable as particles by any technology I am aware
of. At higher energies when all is said and done the detector
goes click and the computer registers another count. You
prefer to view this as particle interactions. I prefer to
think of it as the exchange of field quanta. In the end all
that is there is the math.

Paul C.

[J. Horta]

On Sun, 02 Oct 2005 19:55:32 +0000, Eugene Stefanovich wrote:

>
>
> J. Horta wrote:
>
>> I very much agree with Igor. What is or is not observable depends on
>> the situation. At very low energies electric fields may be measured
>> with a volt meter and magnetic fields with a magnetometer. A radio
>> picks up em waves. Sure, each one of these may be formulated as
>> interaction with quantum fields or, as follows from the formalism,
>> absorption and emission of particles. The physics should be the same
>> independent of how it is formulated.
>> If you should find, however,
>> a formulation of the physics using a particle approach which includes
>> phenomena which a quantum field approach can't then you could well
>> claim a more fundamental theory.
>
>
> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?
>
> Eugene.

To answer your question QF theory wouldn't be accepted
if the hydrogen atom were beyond its grasp. In fact the
Lamb shift prediction as I recall is touted as one of
the major achievements of QFT.

BTW, I could equally well ask you to describe the reception of
a 400 kHz wave with a radio receiver using just a particle
picture. Sure, it can be done but why bother. You assert
the particle aspect of field theory is more "fundamental"
than the field aspect. I don't see this. At 400 kHz photons
are not detectable as particles by any technology I am aware
of. At higher energies when all is said and done the detector
goes click and the computer registers another count. You
prefer to view this as particle interactions. I prefer to
think of it as the exchange of field quanta. In the end all
that is there is the math.

Paul C.

[J. Horta]

On Sun, 02 Oct 2005 19:55:32 +0000, Eugene Stefanovich wrote:

>
>
> J. Horta wrote:
>
>> I very much agree with Igor. What is or is not observable depends on
>> the situation. At very low energies electric fields may be measured
>> with a volt meter and magnetic fields with a magnetometer. A radio
>> picks up em waves. Sure, each one of these may be formulated as
>> interaction with quantum fields or, as follows from the formalism,
>> absorption and emission of particles. The physics should be the same
>> independent of how it is formulated.
>> If you should find, however,
>> a formulation of the physics using a particle approach which includes
>> phenomena which a quantum field approach can't then you could well
>> claim a more fundamental theory.
>
>
> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?
>
> Eugene.

To answer your question QF theory wouldn't be accepted
if the hydrogen atom were beyond its grasp. In fact the
Lamb shift prediction as I recall is touted as one of
the major achievements of QFT.

BTW, I could equally well ask you to describe the reception of
a 400 kHz wave with a radio receiver using just a particle
picture. Sure, it can be done but why bother. You assert
the particle aspect of field theory is more "fundamental"
than the field aspect. I don't see this. At 400 kHz photons
are not detectable as particles by any technology I am aware
of. At higher energies when all is said and done the detector
goes click and the computer registers another count. You
prefer to view this as particle interactions. I prefer to
think of it as the exchange of field quanta. In the end all
that is there is the math.

Paul C.

[J. Horta]

On Sun, 02 Oct 2005 19:55:32 +0000, Eugene Stefanovich wrote:

>
>
> J. Horta wrote:
>
>> I very much agree with Igor. What is or is not observable depends on
>> the situation. At very low energies electric fields may be measured
>> with a volt meter and magnetic fields with a magnetometer. A radio
>> picks up em waves. Sure, each one of these may be formulated as
>> interaction with quantum fields or, as follows from the formalism,
>> absorption and emission of particles. The physics should be the same
>> independent of how it is formulated.
>> If you should find, however,
>> a formulation of the physics using a particle approach which includes
>> phenomena which a quantum field approach can't then you could well
>> claim a more fundamental theory.
>
>
> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?
>
> Eugene.

To answer your question QF theory wouldn't be accepted
if the hydrogen atom were beyond its grasp. In fact the
Lamb shift prediction as I recall is touted as one of
the major achievements of QFT.

BTW, I could equally well ask you to describe the reception of
a 400 kHz wave with a radio receiver using just a particle
picture. Sure, it can be done but why bother. You assert
the particle aspect of field theory is more "fundamental"
than the field aspect. I don't see this. At 400 kHz photons
are not detectable as particles by any technology I am aware
of. At higher energies when all is said and done the detector
goes click and the computer registers another count. You
prefer to view this as particle interactions. I prefer to
think of it as the exchange of field quanta. In the end all
that is there is the math.

Paul C.

[J. Horta]

On Sun, 02 Oct 2005 19:55:32 +0000, Eugene Stefanovich wrote:

>
>
> J. Horta wrote:
>
>> I very much agree with Igor. What is or is not observable depends on
>> the situation. At very low energies electric fields may be measured
>> with a volt meter and magnetic fields with a magnetometer. A radio
>> picks up em waves. Sure, each one of these may be formulated as
>> interaction with quantum fields or, as follows from the formalism,
>> absorption and emission of particles. The physics should be the same
>> independent of how it is formulated.
>> If you should find, however,
>> a formulation of the physics using a particle approach which includes
>> phenomena which a quantum field approach can't then you could well
>> claim a more fundamental theory.
>
>
> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?
>
> Eugene.

To answer your question QF theory wouldn't be accepted
if the hydrogen atom were beyond its grasp. In fact the
Lamb shift prediction as I recall is touted as one of
the major achievements of QFT.

BTW, I could equally well ask you to describe the reception of
a 400 kHz wave with a radio receiver using just a particle
picture. Sure, it can be done but why bother. You assert
the particle aspect of field theory is more "fundamental"
than the field aspect. I don't see this. At 400 kHz photons
are not detectable as particles by any technology I am aware
of. At higher energies when all is said and done the detector
goes click and the computer registers another count. You
prefer to view this as particle interactions. I prefer to
think of it as the exchange of field quanta. In the end all
that is there is the math.

Paul C.

[J. Horta]

On Sun, 02 Oct 2005 19:55:32 +0000, Eugene Stefanovich wrote:

>
>
> J. Horta wrote:
>
>> I very much agree with Igor. What is or is not observable depends on
>> the situation. At very low energies electric fields may be measured
>> with a volt meter and magnetic fields with a magnetometer. A radio
>> picks up em waves. Sure, each one of these may be formulated as
>> interaction with quantum fields or, as follows from the formalism,
>> absorption and emission of particles. The physics should be the same
>> independent of how it is formulated.
>> If you should find, however,
>> a formulation of the physics using a particle approach which includes
>> phenomena which a quantum field approach can't then you could well
>> claim a more fundamental theory.
>
>
> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?
>
> Eugene.

To answer your question QF theory wouldn't be accepted
if the hydrogen atom were beyond its grasp. In fact the
Lamb shift prediction as I recall is touted as one of
the major achievements of QFT.

BTW, I could equally well ask you to describe the reception of
a 400 kHz wave with a radio receiver using just a particle
picture. Sure, it can be done but why bother. You assert
the particle aspect of field theory is more "fundamental"
than the field aspect. I don't see this. At 400 kHz photons
are not detectable as particles by any technology I am aware
of. At higher energies when all is said and done the detector
goes click and the computer registers another count. You
prefer to view this as particle interactions. I prefer to
think of it as the exchange of field quanta. In the end all
that is there is the math.

Paul C.

[Juan R.]

Eugene Stefanovich wrote:
> You are right.
> I found Weinberg's book much closer to my position than any other book.
> Weinberg still speaks about the fundamental character of quantum fields, but
> if
> you read his book carefully, you'll notice that fields are introduced there
> for one and only one reason: to facilitate the construction of the Lorentz
> invariant S-matrix. If one can do that without invoking quantum fields
> (I am pretty sure it is possible), then we can forget about fields
> altogether.

Already Feynman presented his spacetime view of QED based in a
no-fields picture. Hoyle and Narkilar, for instance, have improved that
view. The concept of fields like fundamental elements is unnecesary in
their approach (fields enter like a mathematical aid).

> It is also interesting to note Weinberg's attitude to local gauge
> invariance.
> (he briefly mentions about that in the book, there is more in his
> 1960's papers). He does not present the local gauge invarance
> as a fundamental physical principle (as many other textbooks do).
> Instead, he notices that the quantum field of photons does not transform
> covariantly with respect to boosts. Therefore, in order to get the Lorentz
> invariant S-matrix (which is Weinberg's primary concern) one must build
> interaction in a specific way, i.e., to
> couple the photon field with the "conserved current" built of charged
> particle
> fields. This is known as "minimal coupling".

Yes, but does that 'minimal coupling' rule work for two-particle
states? Note that 'multiparticle' states that Weinberg works are really
ONE-particle states via aplication of the cluster decomposition
principle.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> You are right.
> I found Weinberg's book much closer to my position than any other book.
> Weinberg still speaks about the fundamental character of quantum fields, but
> if
> you read his book carefully, you'll notice that fields are introduced there
> for one and only one reason: to facilitate the construction of the Lorentz
> invariant S-matrix. If one can do that without invoking quantum fields
> (I am pretty sure it is possible), then we can forget about fields
> altogether.

Already Feynman presented his spacetime view of QED based in a
no-fields picture. Hoyle and Narkilar, for instance, have improved that
view. The concept of fields like fundamental elements is unnecesary in
their approach (fields enter like a mathematical aid).

> It is also interesting to note Weinberg's attitude to local gauge
> invariance.
> (he briefly mentions about that in the book, there is more in his
> 1960's papers). He does not present the local gauge invarance
> as a fundamental physical principle (as many other textbooks do).
> Instead, he notices that the quantum field of photons does not transform
> covariantly with respect to boosts. Therefore, in order to get the Lorentz
> invariant S-matrix (which is Weinberg's primary concern) one must build
> interaction in a specific way, i.e., to
> couple the photon field with the "conserved current" built of charged
> particle
> fields. This is known as "minimal coupling".

Yes, but does that 'minimal coupling' rule work for two-particle
states? Note that 'multiparticle' states that Weinberg works are really
ONE-particle states via aplication of the cluster decomposition
principle.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> You are right.
> I found Weinberg's book much closer to my position than any other book.
> Weinberg still speaks about the fundamental character of quantum fields, but
> if
> you read his book carefully, you'll notice that fields are introduced there
> for one and only one reason: to facilitate the construction of the Lorentz
> invariant S-matrix. If one can do that without invoking quantum fields
> (I am pretty sure it is possible), then we can forget about fields
> altogether.

Already Feynman presented his spacetime view of QED based in a
no-fields picture. Hoyle and Narkilar, for instance, have improved that
view. The concept of fields like fundamental elements is unnecesary in
their approach (fields enter like a mathematical aid).

> It is also interesting to note Weinberg's attitude to local gauge
> invariance.
> (he briefly mentions about that in the book, there is more in his
> 1960's papers). He does not present the local gauge invarance
> as a fundamental physical principle (as many other textbooks do).
> Instead, he notices that the quantum field of photons does not transform
> covariantly with respect to boosts. Therefore, in order to get the Lorentz
> invariant S-matrix (which is Weinberg's primary concern) one must build
> interaction in a specific way, i.e., to
> couple the photon field with the "conserved current" built of charged
> particle
> fields. This is known as "minimal coupling".

Yes, but does that 'minimal coupling' rule work for two-particle
states? Note that 'multiparticle' states that Weinberg works are really
ONE-particle states via aplication of the cluster decomposition
principle.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> You are right.
> I found Weinberg's book much closer to my position than any other book.
> Weinberg still speaks about the fundamental character of quantum fields, but
> if
> you read his book carefully, you'll notice that fields are introduced there
> for one and only one reason: to facilitate the construction of the Lorentz
> invariant S-matrix. If one can do that without invoking quantum fields
> (I am pretty sure it is possible), then we can forget about fields
> altogether.

Already Feynman presented his spacetime view of QED based in a
no-fields picture. Hoyle and Narkilar, for instance, have improved that
view. The concept of fields like fundamental elements is unnecesary in
their approach (fields enter like a mathematical aid).

> It is also interesting to note Weinberg's attitude to local gauge
> invariance.
> (he briefly mentions about that in the book, there is more in his
> 1960's papers). He does not present the local gauge invarance
> as a fundamental physical principle (as many other textbooks do).
> Instead, he notices that the quantum field of photons does not transform
> covariantly with respect to boosts. Therefore, in order to get the Lorentz
> invariant S-matrix (which is Weinberg's primary concern) one must build
> interaction in a specific way, i.e., to
> couple the photon field with the "conserved current" built of charged
> particle
> fields. This is known as "minimal coupling".

Yes, but does that 'minimal coupling' rule work for two-particle
states? Note that 'multiparticle' states that Weinberg works are really
ONE-particle states via aplication of the cluster decomposition
principle.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> You are right.
> I found Weinberg's book much closer to my position than any other book.
> Weinberg still speaks about the fundamental character of quantum fields, but
> if
> you read his book carefully, you'll notice that fields are introduced there
> for one and only one reason: to facilitate the construction of the Lorentz
> invariant S-matrix. If one can do that without invoking quantum fields
> (I am pretty sure it is possible), then we can forget about fields
> altogether.

Already Feynman presented his spacetime view of QED based in a
no-fields picture. Hoyle and Narkilar, for instance, have improved that
view. The concept of fields like fundamental elements is unnecesary in
their approach (fields enter like a mathematical aid).

> It is also interesting to note Weinberg's attitude to local gauge
> invariance.
> (he briefly mentions about that in the book, there is more in his
> 1960's papers). He does not present the local gauge invarance
> as a fundamental physical principle (as many other textbooks do).
> Instead, he notices that the quantum field of photons does not transform
> covariantly with respect to boosts. Therefore, in order to get the Lorentz
> invariant S-matrix (which is Weinberg's primary concern) one must build
> interaction in a specific way, i.e., to
> couple the photon field with the "conserved current" built of charged
> particle
> fields. This is known as "minimal coupling".

Yes, but does that 'minimal coupling' rule work for two-particle
states? Note that 'multiparticle' states that Weinberg works are really
ONE-particle states via aplication of the cluster decomposition
principle.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> You are right.
> I found Weinberg's book much closer to my position than any other book.
> Weinberg still speaks about the fundamental character of quantum fields, but
> if
> you read his book carefully, you'll notice that fields are introduced there
> for one and only one reason: to facilitate the construction of the Lorentz
> invariant S-matrix. If one can do that without invoking quantum fields
> (I am pretty sure it is possible), then we can forget about fields
> altogether.

Already Feynman presented his spacetime view of QED based in a
no-fields picture. Hoyle and Narkilar, for instance, have improved that
view. The concept of fields like fundamental elements is unnecesary in
their approach (fields enter like a mathematical aid).

> It is also interesting to note Weinberg's attitude to local gauge
> invariance.
> (he briefly mentions about that in the book, there is more in his
> 1960's papers). He does not present the local gauge invarance
> as a fundamental physical principle (as many other textbooks do).
> Instead, he notices that the quantum field of photons does not transform
> covariantly with respect to boosts. Therefore, in order to get the Lorentz
> invariant S-matrix (which is Weinberg's primary concern) one must build
> interaction in a specific way, i.e., to
> couple the photon field with the "conserved current" built of charged
> particle
> fields. This is known as "minimal coupling".

Yes, but does that 'minimal coupling' rule work for two-particle
states? Note that 'multiparticle' states that Weinberg works are really
ONE-particle states via aplication of the cluster decomposition
principle.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> You are right.
> I found Weinberg's book much closer to my position than any other book.
> Weinberg still speaks about the fundamental character of quantum fields, but
> if
> you read his book carefully, you'll notice that fields are introduced there
> for one and only one reason: to facilitate the construction of the Lorentz
> invariant S-matrix. If one can do that without invoking quantum fields
> (I am pretty sure it is possible), then we can forget about fields
> altogether.

Already Feynman presented his spacetime view of QED based in a
no-fields picture. Hoyle and Narkilar, for instance, have improved that
view. The concept of fields like fundamental elements is unnecesary in
their approach (fields enter like a mathematical aid).

> It is also interesting to note Weinberg's attitude to local gauge
> invariance.
> (he briefly mentions about that in the book, there is more in his
> 1960's papers). He does not present the local gauge invarance
> as a fundamental physical principle (as many other textbooks do).
> Instead, he notices that the quantum field of photons does not transform
> covariantly with respect to boosts. Therefore, in order to get the Lorentz
> invariant S-matrix (which is Weinberg's primary concern) one must build
> interaction in a specific way, i.e., to
> couple the photon field with the "conserved current" built of charged
> particle
> fields. This is known as "minimal coupling".

Yes, but does that 'minimal coupling' rule work for two-particle
states? Note that 'multiparticle' states that Weinberg works are really
ONE-particle states via aplication of the cluster decomposition
principle.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> You are right.
> I found Weinberg's book much closer to my position than any other book.
> Weinberg still speaks about the fundamental character of quantum fields, but
> if
> you read his book carefully, you'll notice that fields are introduced there
> for one and only one reason: to facilitate the construction of the Lorentz
> invariant S-matrix. If one can do that without invoking quantum fields
> (I am pretty sure it is possible), then we can forget about fields
> altogether.

Already Feynman presented his spacetime view of QED based in a
no-fields picture. Hoyle and Narkilar, for instance, have improved that
view. The concept of fields like fundamental elements is unnecesary in
their approach (fields enter like a mathematical aid).

> It is also interesting to note Weinberg's attitude to local gauge
> invariance.
> (he briefly mentions about that in the book, there is more in his
> 1960's papers). He does not present the local gauge invarance
> as a fundamental physical principle (as many other textbooks do).
> Instead, he notices that the quantum field of photons does not transform
> covariantly with respect to boosts. Therefore, in order to get the Lorentz
> invariant S-matrix (which is Weinberg's primary concern) one must build
> interaction in a specific way, i.e., to
> couple the photon field with the "conserved current" built of charged
> particle
> fields. This is known as "minimal coupling".

Yes, but does that 'minimal coupling' rule work for two-particle
states? Note that 'multiparticle' states that Weinberg works are really
ONE-particle states via aplication of the cluster decomposition
principle.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> You are right.
> I found Weinberg's book much closer to my position than any other book.
> Weinberg still speaks about the fundamental character of quantum fields, but
> if
> you read his book carefully, you'll notice that fields are introduced there
> for one and only one reason: to facilitate the construction of the Lorentz
> invariant S-matrix. If one can do that without invoking quantum fields
> (I am pretty sure it is possible), then we can forget about fields
> altogether.

Already Feynman presented his spacetime view of QED based in a
no-fields picture. Hoyle and Narkilar, for instance, have improved that
view. The concept of fields like fundamental elements is unnecesary in
their approach (fields enter like a mathematical aid).

> It is also interesting to note Weinberg's attitude to local gauge
> invariance.
> (he briefly mentions about that in the book, there is more in his
> 1960's papers). He does not present the local gauge invarance
> as a fundamental physical principle (as many other textbooks do).
> Instead, he notices that the quantum field of photons does not transform
> covariantly with respect to boosts. Therefore, in order to get the Lorentz
> invariant S-matrix (which is Weinberg's primary concern) one must build
> interaction in a specific way, i.e., to
> couple the photon field with the "conserved current" built of charged
> particle
> fields. This is known as "minimal coupling".

Yes, but does that 'minimal coupling' rule work for two-particle
states? Note that 'multiparticle' states that Weinberg works are really
ONE-particle states via aplication of the cluster decomposition
principle.

Juan R.

Center for CANONICAL |SCIENCE)

[Igor Khavkine]

Eugene Stefanovich wrote:

> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?

And I don't need to mention "particles" in such a description. The
hydrogen atom is a set of bound states in the spectrum of QED when
restricted to the subspace of its Hilbert space, on which the number
operators for the electron and proton fields both take the value one. A
number operator is defined for quantized fields since every field mode
represents a harmonic oscillator.

In fact, this description can be generalized to include quasi-bound
states (those with a finite lifetime). This opens up the door to the
description of states like positronium at the same time allowing us to
calculate their lifetimes simply by unrestricting the Hilbert space to
full QED. In the same way, the Lamb shift makes an appearance for the
Hydrogen atom.

In this description, all measurable quantities are accessible. Mass,
charge density, position, momentum, etc.

Igor

[Igor Khavkine]

Eugene Stefanovich wrote:

> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?

And I don't need to mention "particles" in such a description. The
hydrogen atom is a set of bound states in the spectrum of QED when
restricted to the subspace of its Hilbert space, on which the number
operators for the electron and proton fields both take the value one. A
number operator is defined for quantized fields since every field mode
represents a harmonic oscillator.

In fact, this description can be generalized to include quasi-bound
states (those with a finite lifetime). This opens up the door to the
description of states like positronium at the same time allowing us to
calculate their lifetimes simply by unrestricting the Hilbert space to
full QED. In the same way, the Lamb shift makes an appearance for the
Hydrogen atom.

In this description, all measurable quantities are accessible. Mass,
charge density, position, momentum, etc.

Igor

[Igor Khavkine]

Eugene Stefanovich wrote:

> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?

And I don't need to mention "particles" in such a description. The
hydrogen atom is a set of bound states in the spectrum of QED when
restricted to the subspace of its Hilbert space, on which the number
operators for the electron and proton fields both take the value one. A
number operator is defined for quantized fields since every field mode
represents a harmonic oscillator.

In fact, this description can be generalized to include quasi-bound
states (those with a finite lifetime). This opens up the door to the
description of states like positronium at the same time allowing us to
calculate their lifetimes simply by unrestricting the Hilbert space to
full QED. In the same way, the Lamb shift makes an appearance for the
Hydrogen atom.

In this description, all measurable quantities are accessible. Mass,
charge density, position, momentum, etc.

Igor

[Igor Khavkine]

Eugene Stefanovich wrote:

> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?

And I don't need to mention "particles" in such a description. The
hydrogen atom is a set of bound states in the spectrum of QED when
restricted to the subspace of its Hilbert space, on which the number
operators for the electron and proton fields both take the value one. A
number operator is defined for quantized fields since every field mode
represents a harmonic oscillator.

In fact, this description can be generalized to include quasi-bound
states (those with a finite lifetime). This opens up the door to the
description of states like positronium at the same time allowing us to
calculate their lifetimes simply by unrestricting the Hilbert space to
full QED. In the same way, the Lamb shift makes an appearance for the
Hydrogen atom.

In this description, all measurable quantities are accessible. Mass,
charge density, position, momentum, etc.

Igor

[Igor Khavkine]

Eugene Stefanovich wrote:

> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?

And I don't need to mention "particles" in such a description. The
hydrogen atom is a set of bound states in the spectrum of QED when
restricted to the subspace of its Hilbert space, on which the number
operators for the electron and proton fields both take the value one. A
number operator is defined for quantized fields since every field mode
represents a harmonic oscillator.

In fact, this description can be generalized to include quasi-bound
states (those with a finite lifetime). This opens up the door to the
description of states like positronium at the same time allowing us to
calculate their lifetimes simply by unrestricting the Hilbert space to
full QED. In the same way, the Lamb shift makes an appearance for the
Hydrogen atom.

In this description, all measurable quantities are accessible. Mass,
charge density, position, momentum, etc.

Igor

[Igor Khavkine]

Eugene Stefanovich wrote:

> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?

And I don't need to mention "particles" in such a description. The
hydrogen atom is a set of bound states in the spectrum of QED when
restricted to the subspace of its Hilbert space, on which the number
operators for the electron and proton fields both take the value one. A
number operator is defined for quantized fields since every field mode
represents a harmonic oscillator.

In fact, this description can be generalized to include quasi-bound
states (those with a finite lifetime). This opens up the door to the
description of states like positronium at the same time allowing us to
calculate their lifetimes simply by unrestricting the Hilbert space to
full QED. In the same way, the Lamb shift makes an appearance for the
Hydrogen atom.

In this description, all measurable quantities are accessible. Mass,
charge density, position, momentum, etc.

Igor

[Igor Khavkine]

Eugene Stefanovich wrote:

> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?

And I don't need to mention "particles" in such a description. The
hydrogen atom is a set of bound states in the spectrum of QED when
restricted to the subspace of its Hilbert space, on which the number
operators for the electron and proton fields both take the value one. A
number operator is defined for quantized fields since every field mode
represents a harmonic oscillator.

In fact, this description can be generalized to include quasi-bound
states (those with a finite lifetime). This opens up the door to the
description of states like positronium at the same time allowing us to
calculate their lifetimes simply by unrestricting the Hilbert space to
full QED. In the same way, the Lamb shift makes an appearance for the
Hydrogen atom.

In this description, all measurable quantities are accessible. Mass,
charge density, position, momentum, etc.

Igor

[Igor Khavkine]

Eugene Stefanovich wrote:

> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?

And I don't need to mention "particles" in such a description. The
hydrogen atom is a set of bound states in the spectrum of QED when
restricted to the subspace of its Hilbert space, on which the number
operators for the electron and proton fields both take the value one. A
number operator is defined for quantized fields since every field mode
represents a harmonic oscillator.

In fact, this description can be generalized to include quasi-bound
states (those with a finite lifetime). This opens up the door to the
description of states like positronium at the same time allowing us to
calculate their lifetimes simply by unrestricting the Hilbert space to
full QED. In the same way, the Lamb shift makes an appearance for the
Hydrogen atom.

In this description, all measurable quantities are accessible. Mass,
charge density, position, momentum, etc.

Igor

[Igor Khavkine]

Eugene Stefanovich wrote:

> The particle approach easily describes the hydrogen atom: this is a
> two-particle state with certain wave-function that determines the
> probabilities of measurements of various observables. I don't need
> to mention "quantum fields" in this description. How would you
> describe the hydrogen atom using quantum fields and not mentioning
> the word "particle"?

And I don't need to mention "particles" in such a description. The
hydrogen atom is a set of bound states in the spectrum of QED when
restricted to the subspace of its Hilbert space, on which the number
operators for the electron and proton fields both take the value one. A
number operator is defined for quantized fields since every field mode
represents a harmonic oscillator.

In fact, this description can be generalized to include quasi-bound
states (those with a finite lifetime). This opens up the door to the
description of states like positronium at the same time allowing us to
calculate their lifetimes simply by unrestricting the Hilbert space to
full QED. In the same way, the Lamb shift makes an appearance for the
Hydrogen atom.

In this description, all measurable quantities are accessible. Mass,
charge density, position, momentum, etc.

Igor

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128183366.819460.154450@g44g2000cwa.googlegr oups.com...

> > A particle is a simple entity without extent
> > and with minimal properties. A field takes a different value at each
> > point in spacetime - in complexity it may be likened to a machine
> > with an infinite number of moving parts. Of course these days the
> > fashion is to say "oh we mustn't think about physical
> > interpretation". So I guess I'm just pig headed, because I think it
> > is the main thing we should think about if we are going to advance
> > our understanding of nature.
>
> [...] Namely, the interpretation must consist of a dictionary to
> translate the properties of objects of a theory into measurable and
> verifiable quantities, and vice versa. For a successful theory, the
> dictionary is required to be as complete as possible going from
> experiment to theory, but there is no such requirement going the
> opposite way. Hence an incompleteness in this second half of the
> dictionary does not have a lot of weight in the discussion. I think the
> point that you bring up, a mechanical interpretation of a particle as
> opposed to a field, belongs to this second half of the dictionary.

I disagree. I insist that our dictionary must have translations from the
physical world to the theory AND back from the theory to the physical
world. The latter part is often neglected, and I think this is a major
source of troubles in modern theoretical physics. It seems that now all
kinds of wild mathematical speculations (extra dimensions, parallel
universes, etc.) are allowed that do not have any relationship to the
observable world.

In my view, doing theoretical physics requires certain discipline, i.e.,
using only objects or concepts that can be (at least in principle)
observed in the physical world. I think we started to lose this
discipline with the arrival of quantum field theory. There are just too
many purely formal ingredients without any connection to observations.
What is quantum field psi(x,t)? Can we measure the value of the field at
point (x,t)? What is the meaning of the gauge transformation of the
field? If two fields related to each other by a gauge transformation are
physically indistinguishable, then probably the gauge field description
is just redundant? What is the meaning of the Lagrangian and action? Can
we measure them, or they are just formal mathematical constructs?

You may argue that theoretical physics was overwhelmed by formal math
much earlier, e.g., in ordinary quantum mechanics the Hilbert space and
the wave function are formal concepts that do not have counterparts in
the observable world. I would disagree with that. Given physical system
in a certain state, we can perform measurements of probability
distributions for position, momentum, and other observables and (at
least in principle) reproduce its complex wave function with desired
degree of precision. So, the wave function is measurable. The formalism
of quantum mechanics may look abstract, but it follows directly from
postulates of quantum logic (Piron's theorem). All these postulates have
(at least in principle) directly verifiable physical meaning.

This direct connection between theoretical concepts and observations is
lacking in the field-based formulation of QFT. There is no proof
(similar to the Piron's theorem) that quantum fields are absolutely
necessary for the description of relativistic systems. It is true that
so far we were not able to figure out any better way to construct
interactions in the Fock space that combines relativistic invariance
with cluster separability and with the possibility of particle creation
and annihilation. But I suggest we keep an open mind. If we convince
ourselves that quantum fields are the only fundamental ingredients of
nature, then we don't have a chance to find any better theory.

As opposed to the field picture, the particle picture (the same as in
ordinary quantum mechanics) has direct connections to observable
quantities, so it has a better chance for survival in the future. It
appears that if we just delete from our QED dictionary some
field-related terms (like "quantum field", "gauge", "lagrangian") we can
still do all calculatons in perfect agreement with experiment. We still
have particle observables, wave functions in the Fock space, the
Hamiltonian, etc. that allow us to do physics. On the other hand, the
theory would be dead if you delete from the dictionary such
particle-related terms as position, momentum, spin, mass, etc.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128183366.819460.154450@g44g2000cwa.googlegr oups.com...

> > A particle is a simple entity without extent
> > and with minimal properties. A field takes a different value at each
> > point in spacetime - in complexity it may be likened to a machine
> > with an infinite number of moving parts. Of course these days the
> > fashion is to say "oh we mustn't think about physical
> > interpretation". So I guess I'm just pig headed, because I think it
> > is the main thing we should think about if we are going to advance
> > our understanding of nature.
>
> [...] Namely, the interpretation must consist of a dictionary to
> translate the properties of objects of a theory into measurable and
> verifiable quantities, and vice versa. For a successful theory, the
> dictionary is required to be as complete as possible going from
> experiment to theory, but there is no such requirement going the
> opposite way. Hence an incompleteness in this second half of the
> dictionary does not have a lot of weight in the discussion. I think the
> point that you bring up, a mechanical interpretation of a particle as
> opposed to a field, belongs to this second half of the dictionary.

I disagree. I insist that our dictionary must have translations from the
physical world to the theory AND back from the theory to the physical
world. The latter part is often neglected, and I think this is a major
source of troubles in modern theoretical physics. It seems that now all
kinds of wild mathematical speculations (extra dimensions, parallel
universes, etc.) are allowed that do not have any relationship to the
observable world.

In my view, doing theoretical physics requires certain discipline, i.e.,
using only objects or concepts that can be (at least in principle)
observed in the physical world. I think we started to lose this
discipline with the arrival of quantum field theory. There are just too
many purely formal ingredients without any connection to observations.
What is quantum field psi(x,t)? Can we measure the value of the field at
point (x,t)? What is the meaning of the gauge transformation of the
field? If two fields related to each other by a gauge transformation are
physically indistinguishable, then probably the gauge field description
is just redundant? What is the meaning of the Lagrangian and action? Can
we measure them, or they are just formal mathematical constructs?

You may argue that theoretical physics was overwhelmed by formal math
much earlier, e.g., in ordinary quantum mechanics the Hilbert space and
the wave function are formal concepts that do not have counterparts in
the observable world. I would disagree with that. Given physical system
in a certain state, we can perform measurements of probability
distributions for position, momentum, and other observables and (at
least in principle) reproduce its complex wave function with desired
degree of precision. So, the wave function is measurable. The formalism
of quantum mechanics may look abstract, but it follows directly from
postulates of quantum logic (Piron's theorem). All these postulates have
(at least in principle) directly verifiable physical meaning.

This direct connection between theoretical concepts and observations is
lacking in the field-based formulation of QFT. There is no proof
(similar to the Piron's theorem) that quantum fields are absolutely
necessary for the description of relativistic systems. It is true that
so far we were not able to figure out any better way to construct
interactions in the Fock space that combines relativistic invariance
with cluster separability and with the possibility of particle creation
and annihilation. But I suggest we keep an open mind. If we convince
ourselves that quantum fields are the only fundamental ingredients of
nature, then we don't have a chance to find any better theory.

As opposed to the field picture, the particle picture (the same as in
ordinary quantum mechanics) has direct connections to observable
quantities, so it has a better chance for survival in the future. It
appears that if we just delete from our QED dictionary some
field-related terms (like "quantum field", "gauge", "lagrangian") we can
still do all calculatons in perfect agreement with experiment. We still
have particle observables, wave functions in the Fock space, the
Hamiltonian, etc. that allow us to do physics. On the other hand, the
theory would be dead if you delete from the dictionary such
particle-related terms as position, momentum, spin, mass, etc.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128183366.819460.154450@g44g2000cwa.googlegr oups.com...

> > A particle is a simple entity without extent
> > and with minimal properties. A field takes a different value at each
> > point in spacetime - in complexity it may be likened to a machine
> > with an infinite number of moving parts. Of course these days the
> > fashion is to say "oh we mustn't think about physical
> > interpretation". So I guess I'm just pig headed, because I think it
> > is the main thing we should think about if we are going to advance
> > our understanding of nature.
>
> [...] Namely, the interpretation must consist of a dictionary to
> translate the properties of objects of a theory into measurable and
> verifiable quantities, and vice versa. For a successful theory, the
> dictionary is required to be as complete as possible going from
> experiment to theory, but there is no such requirement going the
> opposite way. Hence an incompleteness in this second half of the
> dictionary does not have a lot of weight in the discussion. I think the
> point that you bring up, a mechanical interpretation of a particle as
> opposed to a field, belongs to this second half of the dictionary.

I disagree. I insist that our dictionary must have translations from the
physical world to the theory AND back from the theory to the physical
world. The latter part is often neglected, and I think this is a major
source of troubles in modern theoretical physics. It seems that now all
kinds of wild mathematical speculations (extra dimensions, parallel
universes, etc.) are allowed that do not have any relationship to the
observable world.

In my view, doing theoretical physics requires certain discipline, i.e.,
using only objects or concepts that can be (at least in principle)
observed in the physical world. I think we started to lose this
discipline with the arrival of quantum field theory. There are just too
many purely formal ingredients without any connection to observations.
What is quantum field psi(x,t)? Can we measure the value of the field at
point (x,t)? What is the meaning of the gauge transformation of the
field? If two fields related to each other by a gauge transformation are
physically indistinguishable, then probably the gauge field description
is just redundant? What is the meaning of the Lagrangian and action? Can
we measure them, or they are just formal mathematical constructs?

You may argue that theoretical physics was overwhelmed by formal math
much earlier, e.g., in ordinary quantum mechanics the Hilbert space and
the wave function are formal concepts that do not have counterparts in
the observable world. I would disagree with that. Given physical system
in a certain state, we can perform measurements of probability
distributions for position, momentum, and other observables and (at
least in principle) reproduce its complex wave function with desired
degree of precision. So, the wave function is measurable. The formalism
of quantum mechanics may look abstract, but it follows directly from
postulates of quantum logic (Piron's theorem). All these postulates have
(at least in principle) directly verifiable physical meaning.

This direct connection between theoretical concepts and observations is
lacking in the field-based formulation of QFT. There is no proof
(similar to the Piron's theorem) that quantum fields are absolutely
necessary for the description of relativistic systems. It is true that
so far we were not able to figure out any better way to construct
interactions in the Fock space that combines relativistic invariance
with cluster separability and with the possibility of particle creation
and annihilation. But I suggest we keep an open mind. If we convince
ourselves that quantum fields are the only fundamental ingredients of
nature, then we don't have a chance to find any better theory.

As opposed to the field picture, the particle picture (the same as in
ordinary quantum mechanics) has direct connections to observable
quantities, so it has a better chance for survival in the future. It
appears that if we just delete from our QED dictionary some
field-related terms (like "quantum field", "gauge", "lagrangian") we can
still do all calculatons in perfect agreement with experiment. We still
have particle observables, wave functions in the Fock space, the
Hamiltonian, etc. that allow us to do physics. On the other hand, the
theory would be dead if you delete from the dictionary such
particle-related terms as position, momentum, spin, mass, etc.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128183366.819460.154450@g44g2000cwa.googlegr oups.com...

> > A particle is a simple entity without extent
> > and with minimal properties. A field takes a different value at each
> > point in spacetime - in complexity it may be likened to a machine
> > with an infinite number of moving parts. Of course these days the
> > fashion is to say "oh we mustn't think about physical
> > interpretation". So I guess I'm just pig headed, because I think it
> > is the main thing we should think about if we are going to advance
> > our understanding of nature.
>
> [...] Namely, the interpretation must consist of a dictionary to
> translate the properties of objects of a theory into measurable and
> verifiable quantities, and vice versa. For a successful theory, the
> dictionary is required to be as complete as possible going from
> experiment to theory, but there is no such requirement going the
> opposite way. Hence an incompleteness in this second half of the
> dictionary does not have a lot of weight in the discussion. I think the
> point that you bring up, a mechanical interpretation of a particle as
> opposed to a field, belongs to this second half of the dictionary.

I disagree. I insist that our dictionary must have translations from the
physical world to the theory AND back from the theory to the physical
world. The latter part is often neglected, and I think this is a major
source of troubles in modern theoretical physics. It seems that now all
kinds of wild mathematical speculations (extra dimensions, parallel
universes, etc.) are allowed that do not have any relationship to the
observable world.

In my view, doing theoretical physics requires certain discipline, i.e.,
using only objects or concepts that can be (at least in principle)
observed in the physical world. I think we started to lose this
discipline with the arrival of quantum field theory. There are just too
many purely formal ingredients without any connection to observations.
What is quantum field psi(x,t)? Can we measure the value of the field at
point (x,t)? What is the meaning of the gauge transformation of the
field? If two fields related to each other by a gauge transformation are
physically indistinguishable, then probably the gauge field description
is just redundant? What is the meaning of the Lagrangian and action? Can
we measure them, or they are just formal mathematical constructs?

You may argue that theoretical physics was overwhelmed by formal math
much earlier, e.g., in ordinary quantum mechanics the Hilbert space and
the wave function are formal concepts that do not have counterparts in
the observable world. I would disagree with that. Given physical system
in a certain state, we can perform measurements of probability
distributions for position, momentum, and other observables and (at
least in principle) reproduce its complex wave function with desired
degree of precision. So, the wave function is measurable. The formalism
of quantum mechanics may look abstract, but it follows directly from
postulates of quantum logic (Piron's theorem). All these postulates have
(at least in principle) directly verifiable physical meaning.

This direct connection between theoretical concepts and observations is
lacking in the field-based formulation of QFT. There is no proof
(similar to the Piron's theorem) that quantum fields are absolutely
necessary for the description of relativistic systems. It is true that
so far we were not able to figure out any better way to construct
interactions in the Fock space that combines relativistic invariance
with cluster separability and with the possibility of particle creation
and annihilation. But I suggest we keep an open mind. If we convince
ourselves that quantum fields are the only fundamental ingredients of
nature, then we don't have a chance to find any better theory.

As opposed to the field picture, the particle picture (the same as in
ordinary quantum mechanics) has direct connections to observable
quantities, so it has a better chance for survival in the future. It
appears that if we just delete from our QED dictionary some
field-related terms (like "quantum field", "gauge", "lagrangian") we can
still do all calculatons in perfect agreement with experiment. We still
have particle observables, wave functions in the Fock space, the
Hamiltonian, etc. that allow us to do physics. On the other hand, the
theory would be dead if you delete from the dictionary such
particle-related terms as position, momentum, spin, mass, etc.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128186160.894439.227610@o13g2000cwo.googlegr oups.com...

> But I do know that every successful
> physical theory has been generalized to the case of a curved classical
> background. If Eugene's theory is to be successful, why should it be
> any different? There is always the option of picking an alternative
> description of gravity. But then you have to justify the choice and
> explain why that generalization was possible, yet the generalization to
> GR (even in the limit in which it is known to apply) was not.

I don't know what you mean by "success" of QFT on a curved classical
background. As far as I know, there are no experiments probing quantum
effects of gravity. Have anybody seen Hawking radiation? It seems to me
that the question of a consistent quantum gravity theory is still wide
open. Therefore, I don't feel any pressure to formulate my approach on
curved backgrounds. If you read my book, you'll see that I do not accept
the idea of any "background": neither flat nor curved Minkowski
space-time.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128186160.894439.227610@o13g2000cwo.googlegr oups.com...

> But I do know that every successful
> physical theory has been generalized to the case of a curved classical
> background. If Eugene's theory is to be successful, why should it be
> any different? There is always the option of picking an alternative
> description of gravity. But then you have to justify the choice and
> explain why that generalization was possible, yet the generalization to
> GR (even in the limit in which it is known to apply) was not.

I don't know what you mean by "success" of QFT on a curved classical
background. As far as I know, there are no experiments probing quantum
effects of gravity. Have anybody seen Hawking radiation? It seems to me
that the question of a consistent quantum gravity theory is still wide
open. Therefore, I don't feel any pressure to formulate my approach on
curved backgrounds. If you read my book, you'll see that I do not accept
the idea of any "background": neither flat nor curved Minkowski
space-time.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128186160.894439.227610@o13g2000cwo.googlegr oups.com...

> But I do know that every successful
> physical theory has been generalized to the case of a curved classical
> background. If Eugene's theory is to be successful, why should it be
> any different? There is always the option of picking an alternative
> description of gravity. But then you have to justify the choice and
> explain why that generalization was possible, yet the generalization to
> GR (even in the limit in which it is known to apply) was not.

I don't know what you mean by "success" of QFT on a curved classical
background. As far as I know, there are no experiments probing quantum
effects of gravity. Have anybody seen Hawking radiation? It seems to me
that the question of a consistent quantum gravity theory is still wide
open. Therefore, I don't feel any pressure to formulate my approach on
curved backgrounds. If you read my book, you'll see that I do not accept
the idea of any "background": neither flat nor curved Minkowski
space-time.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128186160.894439.227610@o13g2000cwo.googlegr oups.com...

> But I do know that every successful
> physical theory has been generalized to the case of a curved classical
> background. If Eugene's theory is to be successful, why should it be
> any different? There is always the option of picking an alternative
> description of gravity. But then you have to justify the choice and
> explain why that generalization was possible, yet the generalization to
> GR (even in the limit in which it is known to apply) was not.

I don't know what you mean by "success" of QFT on a curved classical
background. As far as I know, there are no experiments probing quantum
effects of gravity. Have anybody seen Hawking radiation? It seems to me
that the question of a consistent quantum gravity theory is still wide
open. Therefore, I don't feel any pressure to formulate my approach on
curved backgrounds. If you read my book, you'll see that I do not accept
the idea of any "background": neither flat nor curved Minkowski
space-time.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128186160.894439.227610@o13g2000cwo.googlegr oups.com...

> But I do know that every successful
> physical theory has been generalized to the case of a curved classical
> background. If Eugene's theory is to be successful, why should it be
> any different? There is always the option of picking an alternative
> description of gravity. But then you have to justify the choice and
> explain why that generalization was possible, yet the generalization to
> GR (even in the limit in which it is known to apply) was not.

I don't know what you mean by "success" of QFT on a curved classical
background. As far as I know, there are no experiments probing quantum
effects of gravity. Have anybody seen Hawking radiation? It seems to me
that the question of a consistent quantum gravity theory is still wide
open. Therefore, I don't feel any pressure to formulate my approach on
curved backgrounds. If you read my book, you'll see that I do not accept
the idea of any "background": neither flat nor curved Minkowski
space-time.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128186160.894439.227610@o13g2000cwo.googlegr oups.com...

> But I do know that every successful
> physical theory has been generalized to the case of a curved classical
> background. If Eugene's theory is to be successful, why should it be
> any different? There is always the option of picking an alternative
> description of gravity. But then you have to justify the choice and
> explain why that generalization was possible, yet the generalization to
> GR (even in the limit in which it is known to apply) was not.

I don't know what you mean by "success" of QFT on a curved classical
background. As far as I know, there are no experiments probing quantum
effects of gravity. Have anybody seen Hawking radiation? It seems to me
that the question of a consistent quantum gravity theory is still wide
open. Therefore, I don't feel any pressure to formulate my approach on
curved backgrounds. If you read my book, you'll see that I do not accept
the idea of any "background": neither flat nor curved Minkowski
space-time.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128186160.894439.227610@o13g2000cwo.googlegr oups.com...

> But I do know that every successful
> physical theory has been generalized to the case of a curved classical
> background. If Eugene's theory is to be successful, why should it be
> any different? There is always the option of picking an alternative
> description of gravity. But then you have to justify the choice and
> explain why that generalization was possible, yet the generalization to
> GR (even in the limit in which it is known to apply) was not.

I don't know what you mean by "success" of QFT on a curved classical
background. As far as I know, there are no experiments probing quantum
effects of gravity. Have anybody seen Hawking radiation? It seems to me
that the question of a consistent quantum gravity theory is still wide
open. Therefore, I don't feel any pressure to formulate my approach on
curved backgrounds. If you read my book, you'll see that I do not accept
the idea of any "background": neither flat nor curved Minkowski
space-time.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128186160.894439.227610@o13g2000cwo.googlegr oups.com...

> But I do know that every successful
> physical theory has been generalized to the case of a curved classical
> background. If Eugene's theory is to be successful, why should it be
> any different? There is always the option of picking an alternative
> description of gravity. But then you have to justify the choice and
> explain why that generalization was possible, yet the generalization to
> GR (even in the limit in which it is known to apply) was not.

I don't know what you mean by "success" of QFT on a curved classical
background. As far as I know, there are no experiments probing quantum
effects of gravity. Have anybody seen Hawking radiation? It seems to me
that the question of a consistent quantum gravity theory is still wide
open. Therefore, I don't feel any pressure to formulate my approach on
curved backgrounds. If you read my book, you'll see that I do not accept
the idea of any "background": neither flat nor curved Minkowski
space-time.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128186160.894439.227610@o13g2000cwo.googlegr oups.com...

> But I do know that every successful
> physical theory has been generalized to the case of a curved classical
> background. If Eugene's theory is to be successful, why should it be
> any different? There is always the option of picking an alternative
> description of gravity. But then you have to justify the choice and
> explain why that generalization was possible, yet the generalization to
> GR (even in the limit in which it is known to apply) was not.

I don't know what you mean by "success" of QFT on a curved classical
background. As far as I know, there are no experiments probing quantum
effects of gravity. Have anybody seen Hawking radiation? It seems to me
that the question of a consistent quantum gravity theory is still wide
open. Therefore, I don't feel any pressure to formulate my approach on
curved backgrounds. If you read my book, you'll see that I do not accept
the idea of any "background": neither flat nor curved Minkowski
space-time.

Eugene.

[Juan R.]

J. Horta wrote:
>
> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.

I would remark again that any field is *by definition* experimentally
unobserved.

AT CLASSICAL LEVEL

That one measures are effects of those *supposed* fields on test
particles. Voltmeters and magnetometers newer measure fields.

This is the reason that one can construct alternative theories without
the use of fields. For example, to first orders in c^2, Weber particle
electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
Physics') offers the *same* results that Maxwell EM except in certain
radiation phenomena (there are some generalization of Weber
electrodynamics dealing with radiation also).

At a more fundamental level, the Wheeler-Feynman like electrodynamics
is totally compatible with standard EM but does not use fields, only
particles (that is we measure in laboratories).

AT QUANTUM LEVEL.

Simply i cite Weinberg chapter 3:

"Rather, the paradigmatic experiment (at least in nuclear or elementary
particle physics) is one in which several particles approach each other
from a macroscopically large distance, and interact in a
microscopically small region, after which the products of the
interaction travel out again to a macroscopically large distance"

"In such an experiment, all that is measured is the probability
distribution, or 'cross-sections', for transitions between the initial
and final states of distant and effectively non-interacting particles."

Quantum fields NEWER are measured, and this is the reason that one can
perfectly to construct relativistic quantum theories without fields.

> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.

That is, i believe, Eugene Stefanovich aim.

Also is the aim of M-theory where the particle (D0-brane) is
fundamental and the fields a low-energy approximation.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

J. Horta wrote:
>
> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.

I would remark again that any field is *by definition* experimentally
unobserved.

AT CLASSICAL LEVEL

That one measures are effects of those *supposed* fields on test
particles. Voltmeters and magnetometers newer measure fields.

This is the reason that one can construct alternative theories without
the use of fields. For example, to first orders in c^2, Weber particle
electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
Physics') offers the *same* results that Maxwell EM except in certain
radiation phenomena (there are some generalization of Weber
electrodynamics dealing with radiation also).

At a more fundamental level, the Wheeler-Feynman like electrodynamics
is totally compatible with standard EM but does not use fields, only
particles (that is we measure in laboratories).

AT QUANTUM LEVEL.

Simply i cite Weinberg chapter 3:

"Rather, the paradigmatic experiment (at least in nuclear or elementary
particle physics) is one in which several particles approach each other
from a macroscopically large distance, and interact in a
microscopically small region, after which the products of the
interaction travel out again to a macroscopically large distance"

"In such an experiment, all that is measured is the probability
distribution, or 'cross-sections', for transitions between the initial
and final states of distant and effectively non-interacting particles."

Quantum fields NEWER are measured, and this is the reason that one can
perfectly to construct relativistic quantum theories without fields.

> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.

That is, i believe, Eugene Stefanovich aim.

Also is the aim of M-theory where the particle (D0-brane) is
fundamental and the fields a low-energy approximation.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

J. Horta wrote:
>
> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.

I would remark again that any field is *by definition* experimentally
unobserved.

AT CLASSICAL LEVEL

That one measures are effects of those *supposed* fields on test
particles. Voltmeters and magnetometers newer measure fields.

This is the reason that one can construct alternative theories without
the use of fields. For example, to first orders in c^2, Weber particle
electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
Physics') offers the *same* results that Maxwell EM except in certain
radiation phenomena (there are some generalization of Weber
electrodynamics dealing with radiation also).

At a more fundamental level, the Wheeler-Feynman like electrodynamics
is totally compatible with standard EM but does not use fields, only
particles (that is we measure in laboratories).

AT QUANTUM LEVEL.

Simply i cite Weinberg chapter 3:

"Rather, the paradigmatic experiment (at least in nuclear or elementary
particle physics) is one in which several particles approach each other
from a macroscopically large distance, and interact in a
microscopically small region, after which the products of the
interaction travel out again to a macroscopically large distance"

"In such an experiment, all that is measured is the probability
distribution, or 'cross-sections', for transitions between the initial
and final states of distant and effectively non-interacting particles."

Quantum fields NEWER are measured, and this is the reason that one can
perfectly to construct relativistic quantum theories without fields.

> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.

That is, i believe, Eugene Stefanovich aim.

Also is the aim of M-theory where the particle (D0-brane) is
fundamental and the fields a low-energy approximation.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

J. Horta wrote:
>
> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.

I would remark again that any field is *by definition* experimentally
unobserved.

AT CLASSICAL LEVEL

That one measures are effects of those *supposed* fields on test
particles. Voltmeters and magnetometers newer measure fields.

This is the reason that one can construct alternative theories without
the use of fields. For example, to first orders in c^2, Weber particle
electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
Physics') offers the *same* results that Maxwell EM except in certain
radiation phenomena (there are some generalization of Weber
electrodynamics dealing with radiation also).

At a more fundamental level, the Wheeler-Feynman like electrodynamics
is totally compatible with standard EM but does not use fields, only
particles (that is we measure in laboratories).

AT QUANTUM LEVEL.

Simply i cite Weinberg chapter 3:

"Rather, the paradigmatic experiment (at least in nuclear or elementary
particle physics) is one in which several particles approach each other
from a macroscopically large distance, and interact in a
microscopically small region, after which the products of the
interaction travel out again to a macroscopically large distance"

"In such an experiment, all that is measured is the probability
distribution, or 'cross-sections', for transitions between the initial
and final states of distant and effectively non-interacting particles."

Quantum fields NEWER are measured, and this is the reason that one can
perfectly to construct relativistic quantum theories without fields.

> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.

That is, i believe, Eugene Stefanovich aim.

Also is the aim of M-theory where the particle (D0-brane) is
fundamental and the fields a low-energy approximation.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

J. Horta wrote:
>
> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.

I would remark again that any field is *by definition* experimentally
unobserved.

AT CLASSICAL LEVEL

That one measures are effects of those *supposed* fields on test
particles. Voltmeters and magnetometers newer measure fields.

This is the reason that one can construct alternative theories without
the use of fields. For example, to first orders in c^2, Weber particle
electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
Physics') offers the *same* results that Maxwell EM except in certain
radiation phenomena (there are some generalization of Weber
electrodynamics dealing with radiation also).

At a more fundamental level, the Wheeler-Feynman like electrodynamics
is totally compatible with standard EM but does not use fields, only
particles (that is we measure in laboratories).

AT QUANTUM LEVEL.

Simply i cite Weinberg chapter 3:

"Rather, the paradigmatic experiment (at least in nuclear or elementary
particle physics) is one in which several particles approach each other
from a macroscopically large distance, and interact in a
microscopically small region, after which the products of the
interaction travel out again to a macroscopically large distance"

"In such an experiment, all that is measured is the probability
distribution, or 'cross-sections', for transitions between the initial
and final states of distant and effectively non-interacting particles."

Quantum fields NEWER are measured, and this is the reason that one can
perfectly to construct relativistic quantum theories without fields.

> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.

That is, i believe, Eugene Stefanovich aim.

Also is the aim of M-theory where the particle (D0-brane) is
fundamental and the fields a low-energy approximation.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

J. Horta wrote:
>
> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.

I would remark again that any field is *by definition* experimentally
unobserved.

AT CLASSICAL LEVEL

That one measures are effects of those *supposed* fields on test
particles. Voltmeters and magnetometers newer measure fields.

This is the reason that one can construct alternative theories without
the use of fields. For example, to first orders in c^2, Weber particle
electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
Physics') offers the *same* results that Maxwell EM except in certain
radiation phenomena (there are some generalization of Weber
electrodynamics dealing with radiation also).

At a more fundamental level, the Wheeler-Feynman like electrodynamics
is totally compatible with standard EM but does not use fields, only
particles (that is we measure in laboratories).

AT QUANTUM LEVEL.

Simply i cite Weinberg chapter 3:

"Rather, the paradigmatic experiment (at least in nuclear or elementary
particle physics) is one in which several particles approach each other
from a macroscopically large distance, and interact in a
microscopically small region, after which the products of the
interaction travel out again to a macroscopically large distance"

"In such an experiment, all that is measured is the probability
distribution, or 'cross-sections', for transitions between the initial
and final states of distant and effectively non-interacting particles."

Quantum fields NEWER are measured, and this is the reason that one can
perfectly to construct relativistic quantum theories without fields.

> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.

That is, i believe, Eugene Stefanovich aim.

Also is the aim of M-theory where the particle (D0-brane) is
fundamental and the fields a low-energy approximation.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

J. Horta wrote:
>
> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.

I would remark again that any field is *by definition* experimentally
unobserved.

AT CLASSICAL LEVEL

That one measures are effects of those *supposed* fields on test
particles. Voltmeters and magnetometers newer measure fields.

This is the reason that one can construct alternative theories without
the use of fields. For example, to first orders in c^2, Weber particle
electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
Physics') offers the *same* results that Maxwell EM except in certain
radiation phenomena (there are some generalization of Weber
electrodynamics dealing with radiation also).

At a more fundamental level, the Wheeler-Feynman like electrodynamics
is totally compatible with standard EM but does not use fields, only
particles (that is we measure in laboratories).

AT QUANTUM LEVEL.

Simply i cite Weinberg chapter 3:

"Rather, the paradigmatic experiment (at least in nuclear or elementary
particle physics) is one in which several particles approach each other
from a macroscopically large distance, and interact in a
microscopically small region, after which the products of the
interaction travel out again to a macroscopically large distance"

"In such an experiment, all that is measured is the probability
distribution, or 'cross-sections', for transitions between the initial
and final states of distant and effectively non-interacting particles."

Quantum fields NEWER are measured, and this is the reason that one can
perfectly to construct relativistic quantum theories without fields.

> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.

That is, i believe, Eugene Stefanovich aim.

Also is the aim of M-theory where the particle (D0-brane) is
fundamental and the fields a low-energy approximation.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

J. Horta wrote:
>
> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.

I would remark again that any field is *by definition* experimentally
unobserved.

AT CLASSICAL LEVEL

That one measures are effects of those *supposed* fields on test
particles. Voltmeters and magnetometers newer measure fields.

This is the reason that one can construct alternative theories without
the use of fields. For example, to first orders in c^2, Weber particle
electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
Physics') offers the *same* results that Maxwell EM except in certain
radiation phenomena (there are some generalization of Weber
electrodynamics dealing with radiation also).

At a more fundamental level, the Wheeler-Feynman like electrodynamics
is totally compatible with standard EM but does not use fields, only
particles (that is we measure in laboratories).

AT QUANTUM LEVEL.

Simply i cite Weinberg chapter 3:

"Rather, the paradigmatic experiment (at least in nuclear or elementary
particle physics) is one in which several particles approach each other
from a macroscopically large distance, and interact in a
microscopically small region, after which the products of the
interaction travel out again to a macroscopically large distance"

"In such an experiment, all that is measured is the probability
distribution, or 'cross-sections', for transitions between the initial
and final states of distant and effectively non-interacting particles."

Quantum fields NEWER are measured, and this is the reason that one can
perfectly to construct relativistic quantum theories without fields.

> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.

That is, i believe, Eugene Stefanovich aim.

Also is the aim of M-theory where the particle (D0-brane) is
fundamental and the fields a low-energy approximation.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

J. Horta wrote:
>
> I very much agree with Igor. What is or is not observable depends on
> the situation. At very low energies electric fields may be measured
> with a volt meter and magnetic fields with a magnetometer. A radio
> picks up em waves. Sure, each one of these may be formulated as
> interaction with quantum fields or, as follows from the formalism,
> absorption and emission of particles. The physics should be the same
> independent of how it is formulated.

I would remark again that any field is *by definition* experimentally
unobserved.

AT CLASSICAL LEVEL

That one measures are effects of those *supposed* fields on test
particles. Voltmeters and magnetometers newer measure fields.

This is the reason that one can construct alternative theories without
the use of fields. For example, to first orders in c^2, Weber particle
electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
Physics') offers the *same* results that Maxwell EM except in certain
radiation phenomena (there are some generalization of Weber
electrodynamics dealing with radiation also).

At a more fundamental level, the Wheeler-Feynman like electrodynamics
is totally compatible with standard EM but does not use fields, only
particles (that is we measure in laboratories).

AT QUANTUM LEVEL.

Simply i cite Weinberg chapter 3:

"Rather, the paradigmatic experiment (at least in nuclear or elementary
particle physics) is one in which several particles approach each other
from a macroscopically large distance, and interact in a
microscopically small region, after which the products of the
interaction travel out again to a macroscopically large distance"

"In such an experiment, all that is measured is the probability
distribution, or 'cross-sections', for transitions between the initial
and final states of distant and effectively non-interacting particles."

Quantum fields NEWER are measured, and this is the reason that one can
perfectly to construct relativistic quantum theories without fields.

> If you should find, however,
> a formulation of the physics using a particle approach which includes
> phenomena which a quantum field approach can't then you could well
> claim a more fundamental theory.

That is, i believe, Eugene Stefanovich aim.

Also is the aim of M-theory where the particle (D0-brane) is
fundamental and the fields a low-energy approximation.

Juan R.

Center for CANONICAL |SCIENCE)

[Eugene Stefanovich]

Juan R. wrote:

>>It is also interesting to note Weinberg's attitude to local gauge
>>invariance.
>>(he briefly mentions about that in the book, there is more in his
>>1960's papers). He does not present the local gauge invarance
>>as a fundamental physical principle (as many other textbooks do).
>>Instead, he notices that the quantum field of photons does not transform
>>covariantly with respect to boosts. Therefore, in order to get the Lorentz
>>invariant S-matrix (which is Weinberg's primary concern) one must build
>>interaction in a specific way, i.e., to
>>couple the photon field with the "conserved current" built of charged
>>particle
>>fields. This is known as "minimal coupling".
>
>
> Yes, but does that 'minimal coupling' rule work for two-particle
> states?

The "minimal coupling" defines how one constructs the interaction
operator V in the Hamiltonian (or Lagrangian). In the Coulomb gauge,
it looks like

V(t) = Integral dx j_u(x,t) A^u(x,t)
+ 1/(8 \pi) Integral dx dy j_0(x,t) j_0(y,t)/|x-y|

This operator acts on the
entire Fock space, therefore it acts on the 2-particle sector as well.

> Note that 'multiparticle' states that Weinberg works are really
> ONE-particle states via aplication of the cluster decomposition
> principle.

The cluster decomposition principle simply states that interaction
between spatially separated clusters tends to zero. I don't know how
this principle can be used for making 1-particle states from
multiparticle states. Where in the Weinberg's book you found this
statement?

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

>>It is also interesting to note Weinberg's attitude to local gauge
>>invariance.
>>(he briefly mentions about that in the book, there is more in his
>>1960's papers). He does not present the local gauge invarance
>>as a fundamental physical principle (as many other textbooks do).
>>Instead, he notices that the quantum field of photons does not transform
>>covariantly with respect to boosts. Therefore, in order to get the Lorentz
>>invariant S-matrix (which is Weinberg's primary concern) one must build
>>interaction in a specific way, i.e., to
>>couple the photon field with the "conserved current" built of charged
>>particle
>>fields. This is known as "minimal coupling".
>
>
> Yes, but does that 'minimal coupling' rule work for two-particle
> states?

The "minimal coupling" defines how one constructs the interaction
operator V in the Hamiltonian (or Lagrangian). In the Coulomb gauge,
it looks like

V(t) = Integral dx j_u(x,t) A^u(x,t)
+ 1/(8 \pi) Integral dx dy j_0(x,t) j_0(y,t)/|x-y|

This operator acts on the
entire Fock space, therefore it acts on the 2-particle sector as well.

> Note that 'multiparticle' states that Weinberg works are really
> ONE-particle states via aplication of the cluster decomposition
> principle.

The cluster decomposition principle simply states that interaction
between spatially separated clusters tends to zero. I don't know how
this principle can be used for making 1-particle states from
multiparticle states. Where in the Weinberg's book you found this
statement?

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

>>It is also interesting to note Weinberg's attitude to local gauge
>>invariance.
>>(he briefly mentions about that in the book, there is more in his
>>1960's papers). He does not present the local gauge invarance
>>as a fundamental physical principle (as many other textbooks do).
>>Instead, he notices that the quantum field of photons does not transform
>>covariantly with respect to boosts. Therefore, in order to get the Lorentz
>>invariant S-matrix (which is Weinberg's primary concern) one must build
>>interaction in a specific way, i.e., to
>>couple the photon field with the "conserved current" built of charged
>>particle
>>fields. This is known as "minimal coupling".
>
>
> Yes, but does that 'minimal coupling' rule work for two-particle
> states?

The "minimal coupling" defines how one constructs the interaction
operator V in the Hamiltonian (or Lagrangian). In the Coulomb gauge,
it looks like

V(t) = Integral dx j_u(x,t) A^u(x,t)
+ 1/(8 \pi) Integral dx dy j_0(x,t) j_0(y,t)/|x-y|

This operator acts on the
entire Fock space, therefore it acts on the 2-particle sector as well.

> Note that 'multiparticle' states that Weinberg works are really
> ONE-particle states via aplication of the cluster decomposition
> principle.

The cluster decomposition principle simply states that interaction
between spatially separated clusters tends to zero. I don't know how
this principle can be used for making 1-particle states from
multiparticle states. Where in the Weinberg's book you found this
statement?

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

>>It is also interesting to note Weinberg's attitude to local gauge
>>invariance.
>>(he briefly mentions about that in the book, there is more in his
>>1960's papers). He does not present the local gauge invarance
>>as a fundamental physical principle (as many other textbooks do).
>>Instead, he notices that the quantum field of photons does not transform
>>covariantly with respect to boosts. Therefore, in order to get the Lorentz
>>invariant S-matrix (which is Weinberg's primary concern) one must build
>>interaction in a specific way, i.e., to
>>couple the photon field with the "conserved current" built of charged
>>particle
>>fields. This is known as "minimal coupling".
>
>
> Yes, but does that 'minimal coupling' rule work for two-particle
> states?

The "minimal coupling" defines how one constructs the interaction
operator V in the Hamiltonian (or Lagrangian). In the Coulomb gauge,
it looks like

V(t) = Integral dx j_u(x,t) A^u(x,t)
+ 1/(8 \pi) Integral dx dy j_0(x,t) j_0(y,t)/|x-y|

This operator acts on the
entire Fock space, therefore it acts on the 2-particle sector as well.

> Note that 'multiparticle' states that Weinberg works are really
> ONE-particle states via aplication of the cluster decomposition
> principle.

The cluster decomposition principle simply states that interaction
between spatially separated clusters tends to zero. I don't know how
this principle can be used for making 1-particle states from
multiparticle states. Where in the Weinberg's book you found this
statement?

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

>>It is also interesting to note Weinberg's attitude to local gauge
>>invariance.
>>(he briefly mentions about that in the book, there is more in his
>>1960's papers). He does not present the local gauge invarance
>>as a fundamental physical principle (as many other textbooks do).
>>Instead, he notices that the quantum field of photons does not transform
>>covariantly with respect to boosts. Therefore, in order to get the Lorentz
>>invariant S-matrix (which is Weinberg's primary concern) one must build
>>interaction in a specific way, i.e., to
>>couple the photon field with the "conserved current" built of charged
>>particle
>>fields. This is known as "minimal coupling".
>
>
> Yes, but does that 'minimal coupling' rule work for two-particle
> states?

The "minimal coupling" defines how one constructs the interaction
operator V in the Hamiltonian (or Lagrangian). In the Coulomb gauge,
it looks like

V(t) = Integral dx j_u(x,t) A^u(x,t)
+ 1/(8 \pi) Integral dx dy j_0(x,t) j_0(y,t)/|x-y|

This operator acts on the
entire Fock space, therefore it acts on the 2-particle sector as well.

> Note that 'multiparticle' states that Weinberg works are really
> ONE-particle states via aplication of the cluster decomposition
> principle.

The cluster decomposition principle simply states that interaction
between spatially separated clusters tends to zero. I don't know how
this principle can be used for making 1-particle states from
multiparticle states. Where in the Weinberg's book you found this
statement?

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

>>It is also interesting to note Weinberg's attitude to local gauge
>>invariance.
>>(he briefly mentions about that in the book, there is more in his
>>1960's papers). He does not present the local gauge invarance
>>as a fundamental physical principle (as many other textbooks do).
>>Instead, he notices that the quantum field of photons does not transform
>>covariantly with respect to boosts. Therefore, in order to get the Lorentz
>>invariant S-matrix (which is Weinberg's primary concern) one must build
>>interaction in a specific way, i.e., to
>>couple the photon field with the "conserved current" built of charged
>>particle
>>fields. This is known as "minimal coupling".
>
>
> Yes, but does that 'minimal coupling' rule work for two-particle
> states?

The "minimal coupling" defines how one constructs the interaction
operator V in the Hamiltonian (or Lagrangian). In the Coulomb gauge,
it looks like

V(t) = Integral dx j_u(x,t) A^u(x,t)
+ 1/(8 \pi) Integral dx dy j_0(x,t) j_0(y,t)/|x-y|

This operator acts on the
entire Fock space, therefore it acts on the 2-particle sector as well.

> Note that 'multiparticle' states that Weinberg works are really
> ONE-particle states via aplication of the cluster decomposition
> principle.

The cluster decomposition principle simply states that interaction
between spatially separated clusters tends to zero. I don't know how
this principle can be used for making 1-particle states from
multiparticle states. Where in the Weinberg's book you found this
statement?

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

>>It is also interesting to note Weinberg's attitude to local gauge
>>invariance.
>>(he briefly mentions about that in the book, there is more in his
>>1960's papers). He does not present the local gauge invarance
>>as a fundamental physical principle (as many other textbooks do).
>>Instead, he notices that the quantum field of photons does not transform
>>covariantly with respect to boosts. Therefore, in order to get the Lorentz
>>invariant S-matrix (which is Weinberg's primary concern) one must build
>>interaction in a specific way, i.e., to
>>couple the photon field with the "conserved current" built of charged
>>particle
>>fields. This is known as "minimal coupling".
>
>
> Yes, but does that 'minimal coupling' rule work for two-particle
> states?

The "minimal coupling" defines how one constructs the interaction
operator V in the Hamiltonian (or Lagrangian). In the Coulomb gauge,
it looks like

V(t) = Integral dx j_u(x,t) A^u(x,t)
+ 1/(8 \pi) Integral dx dy j_0(x,t) j_0(y,t)/|x-y|

This operator acts on the
entire Fock space, therefore it acts on the 2-particle sector as well.

> Note that 'multiparticle' states that Weinberg works are really
> ONE-particle states via aplication of the cluster decomposition
> principle.

The cluster decomposition principle simply states that interaction
between spatially separated clusters tends to zero. I don't know how
this principle can be used for making 1-particle states from
multiparticle states. Where in the Weinberg's book you found this
statement?

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

>>It is also interesting to note Weinberg's attitude to local gauge
>>invariance.
>>(he briefly mentions about that in the book, there is more in his
>>1960's papers). He does not present the local gauge invarance
>>as a fundamental physical principle (as many other textbooks do).
>>Instead, he notices that the quantum field of photons does not transform
>>covariantly with respect to boosts. Therefore, in order to get the Lorentz
>>invariant S-matrix (which is Weinberg's primary concern) one must build
>>interaction in a specific way, i.e., to
>>couple the photon field with the "conserved current" built of charged
>>particle
>>fields. This is known as "minimal coupling".
>
>
> Yes, but does that 'minimal coupling' rule work for two-particle
> states?

The "minimal coupling" defines how one constructs the interaction
operator V in the Hamiltonian (or Lagrangian). In the Coulomb gauge,
it looks like

V(t) = Integral dx j_u(x,t) A^u(x,t)
+ 1/(8 \pi) Integral dx dy j_0(x,t) j_0(y,t)/|x-y|

This operator acts on the
entire Fock space, therefore it acts on the 2-particle sector as well.

> Note that 'multiparticle' states that Weinberg works are really
> ONE-particle states via aplication of the cluster decomposition
> principle.

The cluster decomposition principle simply states that interaction
between spatially separated clusters tends to zero. I don't know how
this principle can be used for making 1-particle states from
multiparticle states. Where in the Weinberg's book you found this
statement?

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

>>It is also interesting to note Weinberg's attitude to local gauge
>>invariance.
>>(he briefly mentions about that in the book, there is more in his
>>1960's papers). He does not present the local gauge invarance
>>as a fundamental physical principle (as many other textbooks do).
>>Instead, he notices that the quantum field of photons does not transform
>>covariantly with respect to boosts. Therefore, in order to get the Lorentz
>>invariant S-matrix (which is Weinberg's primary concern) one must build
>>interaction in a specific way, i.e., to
>>couple the photon field with the "conserved current" built of charged
>>particle
>>fields. This is known as "minimal coupling".
>
>
> Yes, but does that 'minimal coupling' rule work for two-particle
> states?

The "minimal coupling" defines how one constructs the interaction
operator V in the Hamiltonian (or Lagrangian). In the Coulomb gauge,
it looks like

V(t) = Integral dx j_u(x,t) A^u(x,t)
+ 1/(8 \pi) Integral dx dy j_0(x,t) j_0(y,t)/|x-y|

This operator acts on the
entire Fock space, therefore it acts on the 2-particle sector as well.

> Note that 'multiparticle' states that Weinberg works are really
> ONE-particle states via aplication of the cluster decomposition
> principle.

The cluster decomposition principle simply states that interaction
between spatially separated clusters tends to zero. I don't know how
this principle can be used for making 1-particle states from
multiparticle states. Where in the Weinberg's book you found this
statement?

Eugene.

[Eugene Stefanovich]

Juan R. wrote:
> For example, to first orders in c^2, Weber particle
> electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
> Physics') offers the *same* results that Maxwell EM except in certain
> radiation phenomena (there are some generalization of Weber
> electrodynamics dealing with radiation also).

Yes, there is a lot of theoretical
(and experimental, e.g., by P. Graneau) activity
regarding electrodynamics with Weber or Ampere forces.
However, the interparticle force law consistent with the Hamiltonian
of quantum electrodynamics is different (though similar in spirit).
The interaction of charged particles is better described by the Breit
interaction. If spins are neglected, then this interaction is just
Coulomb + Darwin potential. See subsection 12.2.3 in my book.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:
> For example, to first orders in c^2, Weber particle
> electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
> Physics') offers the *same* results that Maxwell EM except in certain
> radiation phenomena (there are some generalization of Weber
> electrodynamics dealing with radiation also).

Yes, there is a lot of theoretical
(and experimental, e.g., by P. Graneau) activity
regarding electrodynamics with Weber or Ampere forces.
However, the interparticle force law consistent with the Hamiltonian
of quantum electrodynamics is different (though similar in spirit).
The interaction of charged particles is better described by the Breit
interaction. If spins are neglected, then this interaction is just
Coulomb + Darwin potential. See subsection 12.2.3 in my book.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:
> For example, to first orders in c^2, Weber particle
> electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
> Physics') offers the *same* results that Maxwell EM except in certain
> radiation phenomena (there are some generalization of Weber
> electrodynamics dealing with radiation also).

Yes, there is a lot of theoretical
(and experimental, e.g., by P. Graneau) activity
regarding electrodynamics with Weber or Ampere forces.
However, the interparticle force law consistent with the Hamiltonian
of quantum electrodynamics is different (though similar in spirit).
The interaction of charged particles is better described by the Breit
interaction. If spins are neglected, then this interaction is just
Coulomb + Darwin potential. See subsection 12.2.3 in my book.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:
> For example, to first orders in c^2, Weber particle
> electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
> Physics') offers the *same* results that Maxwell EM except in certain
> radiation phenomena (there are some generalization of Weber
> electrodynamics dealing with radiation also).

Yes, there is a lot of theoretical
(and experimental, e.g., by P. Graneau) activity
regarding electrodynamics with Weber or Ampere forces.
However, the interparticle force law consistent with the Hamiltonian
of quantum electrodynamics is different (though similar in spirit).
The interaction of charged particles is better described by the Breit
interaction. If spins are neglected, then this interaction is just
Coulomb + Darwin potential. See subsection 12.2.3 in my book.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:
> For example, to first orders in c^2, Weber particle
> electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
> Physics') offers the *same* results that Maxwell EM except in certain
> radiation phenomena (there are some generalization of Weber
> electrodynamics dealing with radiation also).

Yes, there is a lot of theoretical
(and experimental, e.g., by P. Graneau) activity
regarding electrodynamics with Weber or Ampere forces.
However, the interparticle force law consistent with the Hamiltonian
of quantum electrodynamics is different (though similar in spirit).
The interaction of charged particles is better described by the Breit
interaction. If spins are neglected, then this interaction is just
Coulomb + Darwin potential. See subsection 12.2.3 in my book.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:
> For example, to first orders in c^2, Weber particle
> electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
> Physics') offers the *same* results that Maxwell EM except in certain
> radiation phenomena (there are some generalization of Weber
> electrodynamics dealing with radiation also).

Yes, there is a lot of theoretical
(and experimental, e.g., by P. Graneau) activity
regarding electrodynamics with Weber or Ampere forces.
However, the interparticle force law consistent with the Hamiltonian
of quantum electrodynamics is different (though similar in spirit).
The interaction of charged particles is better described by the Breit
interaction. If spins are neglected, then this interaction is just
Coulomb + Darwin potential. See subsection 12.2.3 in my book.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:
> For example, to first orders in c^2, Weber particle
> electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
> Physics') offers the *same* results that Maxwell EM except in certain
> radiation phenomena (there are some generalization of Weber
> electrodynamics dealing with radiation also).

Yes, there is a lot of theoretical
(and experimental, e.g., by P. Graneau) activity
regarding electrodynamics with Weber or Ampere forces.
However, the interparticle force law consistent with the Hamiltonian
of quantum electrodynamics is different (though similar in spirit).
The interaction of charged particles is better described by the Breit
interaction. If spins are neglected, then this interaction is just
Coulomb + Darwin potential. See subsection 12.2.3 in my book.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:
> For example, to first orders in c^2, Weber particle
> electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
> Physics') offers the *same* results that Maxwell EM except in certain
> radiation phenomena (there are some generalization of Weber
> electrodynamics dealing with radiation also).

Yes, there is a lot of theoretical
(and experimental, e.g., by P. Graneau) activity
regarding electrodynamics with Weber or Ampere forces.
However, the interparticle force law consistent with the Hamiltonian
of quantum electrodynamics is different (though similar in spirit).
The interaction of charged particles is better described by the Breit
interaction. If spins are neglected, then this interaction is just
Coulomb + Darwin potential. See subsection 12.2.3 in my book.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:
> For example, to first orders in c^2, Weber particle
> electrodynamics (see e.g. Assis monograph on 'Fundamental Theories of
> Physics') offers the *same* results that Maxwell EM except in certain
> radiation phenomena (there are some generalization of Weber
> electrodynamics dealing with radiation also).

Yes, there is a lot of theoretical
(and experimental, e.g., by P. Graneau) activity
regarding electrodynamics with Weber or Ampere forces.
However, the interparticle force law consistent with the Hamiltonian
of quantum electrodynamics is different (though similar in spirit).
The interaction of charged particles is better described by the Breit
interaction. If spins are neglected, then this interaction is just
Coulomb + Darwin potential. See subsection 12.2.3 in my book.

Eugene.

[Juan R.]

Eugene Stefanovich wrote:
> > Note that 'multiparticle' states that Weinberg works are really
> > ONE-particle states via aplication of the cluster decomposition
> > principle.
>
> The cluster decomposition principle simply states that interaction
> between spatially separated clusters tends to zero. I don't know how
> this principle can be used for making 1-particle states from
> multiparticle states. Where in the Weinberg's book you found this
> statement?

Weinberg begins from we know (and we can measure) that is particles,
QM, and relativity. Anyone who claim that we can measure a field simply
does not know that a field is! As stated by Weinberg in Chapter 3 we
measure particles in particle physics experiments.

Weimberg begins from a N-particle state. But that N-particle state is
not defined in QFT, and therefore, in QFt one focuses only in
asymptotic regimes. Weinberg appeals to the cluster decomposition
principle for shown that in the asymptotic regime the N-particle state
is transformed into a N-product of 1-particle states. Then next he
introduces fields.

The fields, obviously, are not defined for multiparticle states and
this is the reason that QFT can only deal with asymptotic regimes.

In reaction A + B = C + D

QFT is only defined for the asymptotic A, B and C, D (as was proven by
Landau). The interaction regime cannot be defined in QFT because there
is no fields therein. This is the reason that QFT cannot be applied
into condensed-matter chemistry. And this is the reason that Nobel
Prize for chemistry Prigogine developed a generalization of QFT some
years ago. QFT is recovered in the asymptotic regime.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> > Note that 'multiparticle' states that Weinberg works are really
> > ONE-particle states via aplication of the cluster decomposition
> > principle.
>
> The cluster decomposition principle simply states that interaction
> between spatially separated clusters tends to zero. I don't know how
> this principle can be used for making 1-particle states from
> multiparticle states. Where in the Weinberg's book you found this
> statement?

Weinberg begins from we know (and we can measure) that is particles,
QM, and relativity. Anyone who claim that we can measure a field simply
does not know that a field is! As stated by Weinberg in Chapter 3 we
measure particles in particle physics experiments.

Weimberg begins from a N-particle state. But that N-particle state is
not defined in QFT, and therefore, in QFt one focuses only in
asymptotic regimes. Weinberg appeals to the cluster decomposition
principle for shown that in the asymptotic regime the N-particle state
is transformed into a N-product of 1-particle states. Then next he
introduces fields.

The fields, obviously, are not defined for multiparticle states and
this is the reason that QFT can only deal with asymptotic regimes.

In reaction A + B = C + D

QFT is only defined for the asymptotic A, B and C, D (as was proven by
Landau). The interaction regime cannot be defined in QFT because there
is no fields therein. This is the reason that QFT cannot be applied
into condensed-matter chemistry. And this is the reason that Nobel
Prize for chemistry Prigogine developed a generalization of QFT some
years ago. QFT is recovered in the asymptotic regime.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> > Note that 'multiparticle' states that Weinberg works are really
> > ONE-particle states via aplication of the cluster decomposition
> > principle.
>
> The cluster decomposition principle simply states that interaction
> between spatially separated clusters tends to zero. I don't know how
> this principle can be used for making 1-particle states from
> multiparticle states. Where in the Weinberg's book you found this
> statement?

Weinberg begins from we know (and we can measure) that is particles,
QM, and relativity. Anyone who claim that we can measure a field simply
does not know that a field is! As stated by Weinberg in Chapter 3 we
measure particles in particle physics experiments.

Weimberg begins from a N-particle state. But that N-particle state is
not defined in QFT, and therefore, in QFt one focuses only in
asymptotic regimes. Weinberg appeals to the cluster decomposition
principle for shown that in the asymptotic regime the N-particle state
is transformed into a N-product of 1-particle states. Then next he
introduces fields.

The fields, obviously, are not defined for multiparticle states and
this is the reason that QFT can only deal with asymptotic regimes.

In reaction A + B = C + D

QFT is only defined for the asymptotic A, B and C, D (as was proven by
Landau). The interaction regime cannot be defined in QFT because there
is no fields therein. This is the reason that QFT cannot be applied
into condensed-matter chemistry. And this is the reason that Nobel
Prize for chemistry Prigogine developed a generalization of QFT some
years ago. QFT is recovered in the asymptotic regime.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> > Note that 'multiparticle' states that Weinberg works are really
> > ONE-particle states via aplication of the cluster decomposition
> > principle.
>
> The cluster decomposition principle simply states that interaction
> between spatially separated clusters tends to zero. I don't know how
> this principle can be used for making 1-particle states from
> multiparticle states. Where in the Weinberg's book you found this
> statement?

Weinberg begins from we know (and we can measure) that is particles,
QM, and relativity. Anyone who claim that we can measure a field simply
does not know that a field is! As stated by Weinberg in Chapter 3 we
measure particles in particle physics experiments.

Weimberg begins from a N-particle state. But that N-particle state is
not defined in QFT, and therefore, in QFt one focuses only in
asymptotic regimes. Weinberg appeals to the cluster decomposition
principle for shown that in the asymptotic regime the N-particle state
is transformed into a N-product of 1-particle states. Then next he
introduces fields.

The fields, obviously, are not defined for multiparticle states and
this is the reason that QFT can only deal with asymptotic regimes.

In reaction A + B = C + D

QFT is only defined for the asymptotic A, B and C, D (as was proven by
Landau). The interaction regime cannot be defined in QFT because there
is no fields therein. This is the reason that QFT cannot be applied
into condensed-matter chemistry. And this is the reason that Nobel
Prize for chemistry Prigogine developed a generalization of QFT some
years ago. QFT is recovered in the asymptotic regime.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> > Note that 'multiparticle' states that Weinberg works are really
> > ONE-particle states via aplication of the cluster decomposition
> > principle.
>
> The cluster decomposition principle simply states that interaction
> between spatially separated clusters tends to zero. I don't know how
> this principle can be used for making 1-particle states from
> multiparticle states. Where in the Weinberg's book you found this
> statement?

Weinberg begins from we know (and we can measure) that is particles,
QM, and relativity. Anyone who claim that we can measure a field simply
does not know that a field is! As stated by Weinberg in Chapter 3 we
measure particles in particle physics experiments.

Weimberg begins from a N-particle state. But that N-particle state is
not defined in QFT, and therefore, in QFt one focuses only in
asymptotic regimes. Weinberg appeals to the cluster decomposition
principle for shown that in the asymptotic regime the N-particle state
is transformed into a N-product of 1-particle states. Then next he
introduces fields.

The fields, obviously, are not defined for multiparticle states and
this is the reason that QFT can only deal with asymptotic regimes.

In reaction A + B = C + D

QFT is only defined for the asymptotic A, B and C, D (as was proven by
Landau). The interaction regime cannot be defined in QFT because there
is no fields therein. This is the reason that QFT cannot be applied
into condensed-matter chemistry. And this is the reason that Nobel
Prize for chemistry Prigogine developed a generalization of QFT some
years ago. QFT is recovered in the asymptotic regime.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> > Note that 'multiparticle' states that Weinberg works are really
> > ONE-particle states via aplication of the cluster decomposition
> > principle.
>
> The cluster decomposition principle simply states that interaction
> between spatially separated clusters tends to zero. I don't know how
> this principle can be used for making 1-particle states from
> multiparticle states. Where in the Weinberg's book you found this
> statement?

Weinberg begins from we know (and we can measure) that is particles,
QM, and relativity. Anyone who claim that we can measure a field simply
does not know that a field is! As stated by Weinberg in Chapter 3 we
measure particles in particle physics experiments.

Weimberg begins from a N-particle state. But that N-particle state is
not defined in QFT, and therefore, in QFt one focuses only in
asymptotic regimes. Weinberg appeals to the cluster decomposition
principle for shown that in the asymptotic regime the N-particle state
is transformed into a N-product of 1-particle states. Then next he
introduces fields.

The fields, obviously, are not defined for multiparticle states and
this is the reason that QFT can only deal with asymptotic regimes.

In reaction A + B = C + D

QFT is only defined for the asymptotic A, B and C, D (as was proven by
Landau). The interaction regime cannot be defined in QFT because there
is no fields therein. This is the reason that QFT cannot be applied
into condensed-matter chemistry. And this is the reason that Nobel
Prize for chemistry Prigogine developed a generalization of QFT some
years ago. QFT is recovered in the asymptotic regime.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> > Note that 'multiparticle' states that Weinberg works are really
> > ONE-particle states via aplication of the cluster decomposition
> > principle.
>
> The cluster decomposition principle simply states that interaction
> between spatially separated clusters tends to zero. I don't know how
> this principle can be used for making 1-particle states from
> multiparticle states. Where in the Weinberg's book you found this
> statement?

Weinberg begins from we know (and we can measure) that is particles,
QM, and relativity. Anyone who claim that we can measure a field simply
does not know that a field is! As stated by Weinberg in Chapter 3 we
measure particles in particle physics experiments.

Weimberg begins from a N-particle state. But that N-particle state is
not defined in QFT, and therefore, in QFt one focuses only in
asymptotic regimes. Weinberg appeals to the cluster decomposition
principle for shown that in the asymptotic regime the N-particle state
is transformed into a N-product of 1-particle states. Then next he
introduces fields.

The fields, obviously, are not defined for multiparticle states and
this is the reason that QFT can only deal with asymptotic regimes.

In reaction A + B = C + D

QFT is only defined for the asymptotic A, B and C, D (as was proven by
Landau). The interaction regime cannot be defined in QFT because there
is no fields therein. This is the reason that QFT cannot be applied
into condensed-matter chemistry. And this is the reason that Nobel
Prize for chemistry Prigogine developed a generalization of QFT some
years ago. QFT is recovered in the asymptotic regime.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> > Note that 'multiparticle' states that Weinberg works are really
> > ONE-particle states via aplication of the cluster decomposition
> > principle.
>
> The cluster decomposition principle simply states that interaction
> between spatially separated clusters tends to zero. I don't know how
> this principle can be used for making 1-particle states from
> multiparticle states. Where in the Weinberg's book you found this
> statement?

Weinberg begins from we know (and we can measure) that is particles,
QM, and relativity. Anyone who claim that we can measure a field simply
does not know that a field is! As stated by Weinberg in Chapter 3 we
measure particles in particle physics experiments.

Weimberg begins from a N-particle state. But that N-particle state is
not defined in QFT, and therefore, in QFt one focuses only in
asymptotic regimes. Weinberg appeals to the cluster decomposition
principle for shown that in the asymptotic regime the N-particle state
is transformed into a N-product of 1-particle states. Then next he
introduces fields.

The fields, obviously, are not defined for multiparticle states and
this is the reason that QFT can only deal with asymptotic regimes.

In reaction A + B = C + D

QFT is only defined for the asymptotic A, B and C, D (as was proven by
Landau). The interaction regime cannot be defined in QFT because there
is no fields therein. This is the reason that QFT cannot be applied
into condensed-matter chemistry. And this is the reason that Nobel
Prize for chemistry Prigogine developed a generalization of QFT some
years ago. QFT is recovered in the asymptotic regime.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> > Note that 'multiparticle' states that Weinberg works are really
> > ONE-particle states via aplication of the cluster decomposition
> > principle.
>
> The cluster decomposition principle simply states that interaction
> between spatially separated clusters tends to zero. I don't know how
> this principle can be used for making 1-particle states from
> multiparticle states. Where in the Weinberg's book you found this
> statement?

Weinberg begins from we know (and we can measure) that is particles,
QM, and relativity. Anyone who claim that we can measure a field simply
does not know that a field is! As stated by Weinberg in Chapter 3 we
measure particles in particle physics experiments.

Weimberg begins from a N-particle state. But that N-particle state is
not defined in QFT, and therefore, in QFt one focuses only in
asymptotic regimes. Weinberg appeals to the cluster decomposition
principle for shown that in the asymptotic regime the N-particle state
is transformed into a N-product of 1-particle states. Then next he
introduces fields.

The fields, obviously, are not defined for multiparticle states and
this is the reason that QFT can only deal with asymptotic regimes.

In reaction A + B = C + D

QFT is only defined for the asymptotic A, B and C, D (as was proven by
Landau). The interaction regime cannot be defined in QFT because there
is no fields therein. This is the reason that QFT cannot be applied
into condensed-matter chemistry. And this is the reason that Nobel
Prize for chemistry Prigogine developed a generalization of QFT some
years ago. QFT is recovered in the asymptotic regime.

Juan R.

Center for CANONICAL |SCIENCE)

[J. Horta]

On Wed, 05 Oct 2005 21:06:20 +0000, Juan R. wrote:

> Eugene Stefanovich wrote:
>> > Note that 'multiparticle' states that Weinberg works are really
>> > ONE-particle states via aplication of the cluster decomposition
>> > principle.
>>
>> The cluster decomposition principle simply states that interaction
>> between spatially separated clusters tends to zero. I don't know how
>> this principle can be used for making 1-particle states from
>> multiparticle states. Where in the Weinberg's book you found this
>> statement?
>
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.
>

Then help me out. What particles (photons) am I measuring if I use
an oscilloscope to measure the time varying voltage for a 60 cycle
120v wall outlet? As far as I am aware this is a measurement of a
line integral of electric *field*. Clearly a development ala Weinberg
is pretty far removed from the observed phenomena in this limit
just as an oscilloscope would be from measuring gamma ray scattering
from deuterium. For me this is *why* the multi-leveled treatment one
gets with QFT is so impressive.

Regards
Paul C.

[J. Horta]

On Wed, 05 Oct 2005 21:06:20 +0000, Juan R. wrote:

> Eugene Stefanovich wrote:
>> > Note that 'multiparticle' states that Weinberg works are really
>> > ONE-particle states via aplication of the cluster decomposition
>> > principle.
>>
>> The cluster decomposition principle simply states that interaction
>> between spatially separated clusters tends to zero. I don't know how
>> this principle can be used for making 1-particle states from
>> multiparticle states. Where in the Weinberg's book you found this
>> statement?
>
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.
>

Then help me out. What particles (photons) am I measuring if I use
an oscilloscope to measure the time varying voltage for a 60 cycle
120v wall outlet? As far as I am aware this is a measurement of a
line integral of electric *field*. Clearly a development ala Weinberg
is pretty far removed from the observed phenomena in this limit
just as an oscilloscope would be from measuring gamma ray scattering
from deuterium. For me this is *why* the multi-leveled treatment one
gets with QFT is so impressive.

Regards
Paul C.

[J. Horta]

On Wed, 05 Oct 2005 21:06:20 +0000, Juan R. wrote:

> Eugene Stefanovich wrote:
>> > Note that 'multiparticle' states that Weinberg works are really
>> > ONE-particle states via aplication of the cluster decomposition
>> > principle.
>>
>> The cluster decomposition principle simply states that interaction
>> between spatially separated clusters tends to zero. I don't know how
>> this principle can be used for making 1-particle states from
>> multiparticle states. Where in the Weinberg's book you found this
>> statement?
>
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.
>

Then help me out. What particles (photons) am I measuring if I use
an oscilloscope to measure the time varying voltage for a 60 cycle
120v wall outlet? As far as I am aware this is a measurement of a
line integral of electric *field*. Clearly a development ala Weinberg
is pretty far removed from the observed phenomena in this limit
just as an oscilloscope would be from measuring gamma ray scattering
from deuterium. For me this is *why* the multi-leveled treatment one
gets with QFT is so impressive.

Regards
Paul C.

[J. Horta]

On Wed, 05 Oct 2005 21:06:20 +0000, Juan R. wrote:

> Eugene Stefanovich wrote:
>> > Note that 'multiparticle' states that Weinberg works are really
>> > ONE-particle states via aplication of the cluster decomposition
>> > principle.
>>
>> The cluster decomposition principle simply states that interaction
>> between spatially separated clusters tends to zero. I don't know how
>> this principle can be used for making 1-particle states from
>> multiparticle states. Where in the Weinberg's book you found this
>> statement?
>
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.
>

Then help me out. What particles (photons) am I measuring if I use
an oscilloscope to measure the time varying voltage for a 60 cycle
120v wall outlet? As far as I am aware this is a measurement of a
line integral of electric *field*. Clearly a development ala Weinberg
is pretty far removed from the observed phenomena in this limit
just as an oscilloscope would be from measuring gamma ray scattering
from deuterium. For me this is *why* the multi-leveled treatment one
gets with QFT is so impressive.

Regards
Paul C.

[J. Horta]

On Wed, 05 Oct 2005 21:06:20 +0000, Juan R. wrote:

> Eugene Stefanovich wrote:
>> > Note that 'multiparticle' states that Weinberg works are really
>> > ONE-particle states via aplication of the cluster decomposition
>> > principle.
>>
>> The cluster decomposition principle simply states that interaction
>> between spatially separated clusters tends to zero. I don't know how
>> this principle can be used for making 1-particle states from
>> multiparticle states. Where in the Weinberg's book you found this
>> statement?
>
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.
>

Then help me out. What particles (photons) am I measuring if I use
an oscilloscope to measure the time varying voltage for a 60 cycle
120v wall outlet? As far as I am aware this is a measurement of a
line integral of electric *field*. Clearly a development ala Weinberg
is pretty far removed from the observed phenomena in this limit
just as an oscilloscope would be from measuring gamma ray scattering
from deuterium. For me this is *why* the multi-leveled treatment one
gets with QFT is so impressive.

Regards
Paul C.

[J. Horta]

On Wed, 05 Oct 2005 21:06:20 +0000, Juan R. wrote:

> Eugene Stefanovich wrote:
>> > Note that 'multiparticle' states that Weinberg works are really
>> > ONE-particle states via aplication of the cluster decomposition
>> > principle.
>>
>> The cluster decomposition principle simply states that interaction
>> between spatially separated clusters tends to zero. I don't know how
>> this principle can be used for making 1-particle states from
>> multiparticle states. Where in the Weinberg's book you found this
>> statement?
>
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.
>

Then help me out. What particles (photons) am I measuring if I use
an oscilloscope to measure the time varying voltage for a 60 cycle
120v wall outlet? As far as I am aware this is a measurement of a
line integral of electric *field*. Clearly a development ala Weinberg
is pretty far removed from the observed phenomena in this limit
just as an oscilloscope would be from measuring gamma ray scattering
from deuterium. For me this is *why* the multi-leveled treatment one
gets with QFT is so impressive.

Regards
Paul C.

[J. Horta]

On Wed, 05 Oct 2005 21:06:20 +0000, Juan R. wrote:

> Eugene Stefanovich wrote:
>> > Note that 'multiparticle' states that Weinberg works are really
>> > ONE-particle states via aplication of the cluster decomposition
>> > principle.
>>
>> The cluster decomposition principle simply states that interaction
>> between spatially separated clusters tends to zero. I don't know how
>> this principle can be used for making 1-particle states from
>> multiparticle states. Where in the Weinberg's book you found this
>> statement?
>
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.
>

Then help me out. What particles (photons) am I measuring if I use
an oscilloscope to measure the time varying voltage for a 60 cycle
120v wall outlet? As far as I am aware this is a measurement of a
line integral of electric *field*. Clearly a development ala Weinberg
is pretty far removed from the observed phenomena in this limit
just as an oscilloscope would be from measuring gamma ray scattering
from deuterium. For me this is *why* the multi-leveled treatment one
gets with QFT is so impressive.

Regards
Paul C.

[Juan R.]

Juan R. wrote:
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.

Weimberg (p200):

"Traditionally in quantum field theory one begins with such field
equations, or with the Lagrangian from which they are derived, and then
uses them to derive the expansion of the fields in terms of
one-particle annihilation and creation operators. In the approach
followed here, we start with the particles, and derive the fields
according to the dictates of Lorentz invariance, /with the field
equations arising almost incidentally as a byproduct of this
construction/."

Emphasis mine. I will remark again, that Weimberg begins from
ONE-particle states and *derives* fields. But if one begins from FULL
N-particle states (do NOT factorization into N one-particle states,
which is ONLY valid on asymptotic regimes of scattering matrix) there
is not posibility for a rigorous EXACT derivation of fields. The field
vanishes Remember that both Dirac and KG equations are valid only for
ONE-particle systems. There is nothing like a 'two-particle Dirac
equation' (IN RIGOR) for relativistic particles.

Previously, i cited Weinberg comments on bounded states but i cited
incorrectly the page, p560:

"It must be said that the theory of relativistic effects and radiative
corrections in bound states is not yet in entirely satisfactory shape".

Moreover, in curved spacetimes, the particles theories can be done
rigorous using Synge parallel propagators. Wald 'and company' are wrong
about particle theories cannot 'work' in curved spacetimes.
Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Juan R. wrote:
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.

Weimberg (p200):

"Traditionally in quantum field theory one begins with such field
equations, or with the Lagrangian from which they are derived, and then
uses them to derive the expansion of the fields in terms of
one-particle annihilation and creation operators. In the approach
followed here, we start with the particles, and derive the fields
according to the dictates of Lorentz invariance, /with the field
equations arising almost incidentally as a byproduct of this
construction/."

Emphasis mine. I will remark again, that Weimberg begins from
ONE-particle states and *derives* fields. But if one begins from FULL
N-particle states (do NOT factorization into N one-particle states,
which is ONLY valid on asymptotic regimes of scattering matrix) there
is not posibility for a rigorous EXACT derivation of fields. The field
vanishes Remember that both Dirac and KG equations are valid only for
ONE-particle systems. There is nothing like a 'two-particle Dirac
equation' (IN RIGOR) for relativistic particles.

Previously, i cited Weinberg comments on bounded states but i cited
incorrectly the page, p560:

"It must be said that the theory of relativistic effects and radiative
corrections in bound states is not yet in entirely satisfactory shape".

Moreover, in curved spacetimes, the particles theories can be done
rigorous using Synge parallel propagators. Wald 'and company' are wrong
about particle theories cannot 'work' in curved spacetimes.
Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Juan R. wrote:
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.

Weimberg (p200):

"Traditionally in quantum field theory one begins with such field
equations, or with the Lagrangian from which they are derived, and then
uses them to derive the expansion of the fields in terms of
one-particle annihilation and creation operators. In the approach
followed here, we start with the particles, and derive the fields
according to the dictates of Lorentz invariance, /with the field
equations arising almost incidentally as a byproduct of this
construction/."

Emphasis mine. I will remark again, that Weimberg begins from
ONE-particle states and *derives* fields. But if one begins from FULL
N-particle states (do NOT factorization into N one-particle states,
which is ONLY valid on asymptotic regimes of scattering matrix) there
is not posibility for a rigorous EXACT derivation of fields. The field
vanishes Remember that both Dirac and KG equations are valid only for
ONE-particle systems. There is nothing like a 'two-particle Dirac
equation' (IN RIGOR) for relativistic particles.

Previously, i cited Weinberg comments on bounded states but i cited
incorrectly the page, p560:

"It must be said that the theory of relativistic effects and radiative
corrections in bound states is not yet in entirely satisfactory shape".

Moreover, in curved spacetimes, the particles theories can be done
rigorous using Synge parallel propagators. Wald 'and company' are wrong
about particle theories cannot 'work' in curved spacetimes.
Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Juan R. wrote:
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.

Weimberg (p200):

"Traditionally in quantum field theory one begins with such field
equations, or with the Lagrangian from which they are derived, and then
uses them to derive the expansion of the fields in terms of
one-particle annihilation and creation operators. In the approach
followed here, we start with the particles, and derive the fields
according to the dictates of Lorentz invariance, /with the field
equations arising almost incidentally as a byproduct of this
construction/."

Emphasis mine. I will remark again, that Weimberg begins from
ONE-particle states and *derives* fields. But if one begins from FULL
N-particle states (do NOT factorization into N one-particle states,
which is ONLY valid on asymptotic regimes of scattering matrix) there
is not posibility for a rigorous EXACT derivation of fields. The field
vanishes Remember that both Dirac and KG equations are valid only for
ONE-particle systems. There is nothing like a 'two-particle Dirac
equation' (IN RIGOR) for relativistic particles.

Previously, i cited Weinberg comments on bounded states but i cited
incorrectly the page, p560:

"It must be said that the theory of relativistic effects and radiative
corrections in bound states is not yet in entirely satisfactory shape".

Moreover, in curved spacetimes, the particles theories can be done
rigorous using Synge parallel propagators. Wald 'and company' are wrong
about particle theories cannot 'work' in curved spacetimes.
Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Juan R. wrote:
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.

Weimberg (p200):

"Traditionally in quantum field theory one begins with such field
equations, or with the Lagrangian from which they are derived, and then
uses them to derive the expansion of the fields in terms of
one-particle annihilation and creation operators. In the approach
followed here, we start with the particles, and derive the fields
according to the dictates of Lorentz invariance, /with the field
equations arising almost incidentally as a byproduct of this
construction/."

Emphasis mine. I will remark again, that Weimberg begins from
ONE-particle states and *derives* fields. But if one begins from FULL
N-particle states (do NOT factorization into N one-particle states,
which is ONLY valid on asymptotic regimes of scattering matrix) there
is not posibility for a rigorous EXACT derivation of fields. The field
vanishes Remember that both Dirac and KG equations are valid only for
ONE-particle systems. There is nothing like a 'two-particle Dirac
equation' (IN RIGOR) for relativistic particles.

Previously, i cited Weinberg comments on bounded states but i cited
incorrectly the page, p560:

"It must be said that the theory of relativistic effects and radiative
corrections in bound states is not yet in entirely satisfactory shape".

Moreover, in curved spacetimes, the particles theories can be done
rigorous using Synge parallel propagators. Wald 'and company' are wrong
about particle theories cannot 'work' in curved spacetimes.
Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Juan R.

Center for CANONICAL |SCIENCE)

[J. Horta]

On Wed, 05 Oct 2005 21:06:20 +0000, Juan R. wrote:

> Eugene Stefanovich wrote:
>> > Note that 'multiparticle' states that Weinberg works are really
>> > ONE-particle states via aplication of the cluster decomposition
>> > principle.
>>
>> The cluster decomposition principle simply states that interaction
>> between spatially separated clusters tends to zero. I don't know how
>> this principle can be used for making 1-particle states from
>> multiparticle states. Where in the Weinberg's book you found this
>> statement?
>
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.
>

Then help me out. What particles (photons) am I measuring if I use
an oscilloscope to measure the time varying voltage for a 60 cycle
120v wall outlet? As far as I am aware this is a measurement of a
line integral of electric *field*. Clearly a development ala Weinberg
is pretty far removed from the observed phenomena in this limit
just as an oscilloscope would be from measuring gamma ray scattering
from deuterium. For me this is *why* the multi-leveled treatment one
gets with QFT is so impressive.

Regards
Paul C.

[Juan R.]

Juan R. wrote:
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.

Weimberg (p200):

"Traditionally in quantum field theory one begins with such field
equations, or with the Lagrangian from which they are derived, and then
uses them to derive the expansion of the fields in terms of
one-particle annihilation and creation operators. In the approach
followed here, we start with the particles, and derive the fields
according to the dictates of Lorentz invariance, /with the field
equations arising almost incidentally as a byproduct of this
construction/."

Emphasis mine. I will remark again, that Weimberg begins from
ONE-particle states and *derives* fields. But if one begins from FULL
N-particle states (do NOT factorization into N one-particle states,
which is ONLY valid on asymptotic regimes of scattering matrix) there
is not posibility for a rigorous EXACT derivation of fields. The field
vanishes Remember that both Dirac and KG equations are valid only for
ONE-particle systems. There is nothing like a 'two-particle Dirac
equation' (IN RIGOR) for relativistic particles.

Previously, i cited Weinberg comments on bounded states but i cited
incorrectly the page, p560:

"It must be said that the theory of relativistic effects and radiative
corrections in bound states is not yet in entirely satisfactory shape".

Moreover, in curved spacetimes, the particles theories can be done
rigorous using Synge parallel propagators. Wald 'and company' are wrong
about particle theories cannot 'work' in curved spacetimes.
Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Juan R.

Center for CANONICAL |SCIENCE)

[J. Horta]

On Wed, 05 Oct 2005 21:06:20 +0000, Juan R. wrote:

> Eugene Stefanovich wrote:
>> > Note that 'multiparticle' states that Weinberg works are really
>> > ONE-particle states via aplication of the cluster decomposition
>> > principle.
>>
>> The cluster decomposition principle simply states that interaction
>> between spatially separated clusters tends to zero. I don't know how
>> this principle can be used for making 1-particle states from
>> multiparticle states. Where in the Weinberg's book you found this
>> statement?
>
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.
>

Then help me out. What particles (photons) am I measuring if I use
an oscilloscope to measure the time varying voltage for a 60 cycle
120v wall outlet? As far as I am aware this is a measurement of a
line integral of electric *field*. Clearly a development ala Weinberg
is pretty far removed from the observed phenomena in this limit
just as an oscilloscope would be from measuring gamma ray scattering
from deuterium. For me this is *why* the multi-leveled treatment one
gets with QFT is so impressive.

Regards
Paul C.

[Juan R.]

Juan R. wrote:
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.

Weimberg (p200):

"Traditionally in quantum field theory one begins with such field
equations, or with the Lagrangian from which they are derived, and then
uses them to derive the expansion of the fields in terms of
one-particle annihilation and creation operators. In the approach
followed here, we start with the particles, and derive the fields
according to the dictates of Lorentz invariance, /with the field
equations arising almost incidentally as a byproduct of this
construction/."

Emphasis mine. I will remark again, that Weimberg begins from
ONE-particle states and *derives* fields. But if one begins from FULL
N-particle states (do NOT factorization into N one-particle states,
which is ONLY valid on asymptotic regimes of scattering matrix) there
is not posibility for a rigorous EXACT derivation of fields. The field
vanishes Remember that both Dirac and KG equations are valid only for
ONE-particle systems. There is nothing like a 'two-particle Dirac
equation' (IN RIGOR) for relativistic particles.

Previously, i cited Weinberg comments on bounded states but i cited
incorrectly the page, p560:

"It must be said that the theory of relativistic effects and radiative
corrections in bound states is not yet in entirely satisfactory shape".

Moreover, in curved spacetimes, the particles theories can be done
rigorous using Synge parallel propagators. Wald 'and company' are wrong
about particle theories cannot 'work' in curved spacetimes.
Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Juan R. wrote:
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.

Weimberg (p200):

"Traditionally in quantum field theory one begins with such field
equations, or with the Lagrangian from which they are derived, and then
uses them to derive the expansion of the fields in terms of
one-particle annihilation and creation operators. In the approach
followed here, we start with the particles, and derive the fields
according to the dictates of Lorentz invariance, /with the field
equations arising almost incidentally as a byproduct of this
construction/."

Emphasis mine. I will remark again, that Weimberg begins from
ONE-particle states and *derives* fields. But if one begins from FULL
N-particle states (do NOT factorization into N one-particle states,
which is ONLY valid on asymptotic regimes of scattering matrix) there
is not posibility for a rigorous EXACT derivation of fields. The field
vanishes Remember that both Dirac and KG equations are valid only for
ONE-particle systems. There is nothing like a 'two-particle Dirac
equation' (IN RIGOR) for relativistic particles.

Previously, i cited Weinberg comments on bounded states but i cited
incorrectly the page, p560:

"It must be said that the theory of relativistic effects and radiative
corrections in bound states is not yet in entirely satisfactory shape".

Moreover, in curved spacetimes, the particles theories can be done
rigorous using Synge parallel propagators. Wald 'and company' are wrong
about particle theories cannot 'work' in curved spacetimes.
Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Juan R. wrote:
> Weinberg begins from we know (and we can measure) that is particles,
> QM, and relativity. Anyone who claim that we can measure a field simply
> does not know that a field is! As stated by Weinberg in Chapter 3 we
> measure particles in particle physics experiments.

Weimberg (p200):

"Traditionally in quantum field theory one begins with such field
equations, or with the Lagrangian from which they are derived, and then
uses them to derive the expansion of the fields in terms of
one-particle annihilation and creation operators. In the approach
followed here, we start with the particles, and derive the fields
according to the dictates of Lorentz invariance, /with the field
equations arising almost incidentally as a byproduct of this
construction/."

Emphasis mine. I will remark again, that Weimberg begins from
ONE-particle states and *derives* fields. But if one begins from FULL
N-particle states (do NOT factorization into N one-particle states,
which is ONLY valid on asymptotic regimes of scattering matrix) there
is not posibility for a rigorous EXACT derivation of fields. The field
vanishes Remember that both Dirac and KG equations are valid only for
ONE-particle systems. There is nothing like a 'two-particle Dirac
equation' (IN RIGOR) for relativistic particles.

Previously, i cited Weinberg comments on bounded states but i cited
incorrectly the page, p560:

"It must be said that the theory of relativistic effects and radiative
corrections in bound states is not yet in entirely satisfactory shape".

Moreover, in curved spacetimes, the particles theories can be done
rigorous using Synge parallel propagators. Wald 'and company' are wrong
about particle theories cannot 'work' in curved spacetimes.
Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Juan R.

Center for CANONICAL |SCIENCE)

[Charles Francis]

In message <1128183366.819460.154450@g44g2000cwa.googlegroups. com>,
Igor Khavkine <igor.kh@gmail.com> writes
>Charles Francis wrote:

>> Of course these days the
>> fashion is to say "oh we mustn't think about physical
>> interpretation". So I guess I'm just pig headed, because I think it
>> is the main thing we should think about if we are going to advance
>> our understanding of nature.
>
>I completely agree that physical interpretation must be considered.
>However, since we are dealing with a scientific theory, the
>interpretation must be held to as high a standard as any other part of
>the theory. Namely, the interpretation must consist of a dictionary to
>translate the properties of objects of a theory into measurable and
>verifiable quantities, and vice versa. For a successful theory, the
>dictionary is required to be as complete as possible going from
>experiment to theory, but there is no such requirement going the
>opposite way. Hence an incompleteness in this second half of the
>dictionary does not have a lot of weight in the discussion. I think the
>point that you bring up, a mechanical interpretation of a particle as
>opposed to a field, belongs to this second half of the dictionary.

I can't agree with that. For me the purpose of physics is not just to
make good experimental predictions, but to have as true and accurate a
view of nature as possible. I do agree that interpretation must be held
to as high a standard as any other part of the theory, and since I find
that this is not currently the case it seems to me that this is the very
area of research which will most likely yield our next advances in the
understanding of nature.

>As you can see, Wald has much less kind words about the formulation of
>quantum particles on curved spacetime.

Perhaps. But as I come from a relationist perspective which does not
believe in the existence of prior space time, I tend to feel the
discussion has diverged somewhat at that point and start losing
interest. As I have it, space-time is just a mathematical construction,
part of an ordering principle, and does not physically exist between
initial and final measurements in quantum theory. The purpose of the
teleconnection was to formulate quantum theory in flat space-time, even
in an expanding universe.
>
>
>A Fock space is merely a Hilbert space with some extra structure
>(it's closed under tensor products of states). A Fock space can arise
>in two ways. The Hilbert space on which a theory of quantum fields is
>formulated can be given a Fock space structure, in which case matrix
>elements of the field operators give wave functions for free.
>A quantum theory of an indefinite number of identitcal particles can
>also be constructed on a Hilbert space with a Fock structure, in which
>case single particle wave functions yield quantum field operators,
>again for free. The equivalence of these two constructions is the main
>theorem of second quantization. In other words, quantum fields are
>unavoidable, even if you forget about the quantization of classical
>fields.

True.

>Now, Haag's theorem says nothing about Fock spaces. It only talks of
>Hilbert spaces and quantum fields. It says nothing about the
>impossibility of constructing interacting fields. It only says that if
>a free theory and an interacting theory are constructed, they cannot be
>related by a unitary transformation (where both theories are Poincare
>invariant). Compare to ordinary quantum mechanics where knowing the
>matrix elements of the x and p operators between the momentum states of
>a free particle and between the bound states of the harmonic
>oscillator, we can connect one set of matrix elements to the other
>through a unitary transformation whose own matrix elements are given by
>the Fourier transforms of weighted Hermite polynomials.
>
>Note that the paragraph on second quantization specified nothing of
>interaction, while the paragraph on Haag's theorem specified nothing of
>Fock spaces. Which means that you can put them together and draw your
>own conclusions about the fictitiousness of interacting quantum fields
>and the applicability of Haag's theorem to a particle description.
>
>
I simply don't see a problem at this point. It is natural to model an
interaction as a map from the space of non-interacting particles to the
space of non interacting particles. It is natural to do that using the
field operators which we got for free with Fock space. These are the so
called "non-interacting" fields, and there is no reason to introduce
"interacting fields". The problem comes a little way down the road when
the Landau pole appears. That may make the model inconsistent, but it
does not mean that it is entirely wrong. The natural solution is to
modify the theory at short range. If we can find a way to do that the
Landau pole will go away.

Regards

--
Charles Francis

[Charles Francis]

In message <1128183366.819460.154450@g44g2000cwa.googlegroups. com>,
Igor Khavkine <igor.kh@gmail.com> writes
>Charles Francis wrote:

>> Of course these days the
>> fashion is to say "oh we mustn't think about physical
>> interpretation". So I guess I'm just pig headed, because I think it
>> is the main thing we should think about if we are going to advance
>> our understanding of nature.
>
>I completely agree that physical interpretation must be considered.
>However, since we are dealing with a scientific theory, the
>interpretation must be held to as high a standard as any other part of
>the theory. Namely, the interpretation must consist of a dictionary to
>translate the properties of objects of a theory into measurable and
>verifiable quantities, and vice versa. For a successful theory, the
>dictionary is required to be as complete as possible going from
>experiment to theory, but there is no such requirement going the
>opposite way. Hence an incompleteness in this second half of the
>dictionary does not have a lot of weight in the discussion. I think the
>point that you bring up, a mechanical interpretation of a particle as
>opposed to a field, belongs to this second half of the dictionary.

I can't agree with that. For me the purpose of physics is not just to
make good experimental predictions, but to have as true and accurate a
view of nature as possible. I do agree that interpretation must be held
to as high a standard as any other part of the theory, and since I find
that this is not currently the case it seems to me that this is the very
area of research which will most likely yield our next advances in the
understanding of nature.

>As you can see, Wald has much less kind words about the formulation of
>quantum particles on curved spacetime.

Perhaps. But as I come from a relationist perspective which does not
believe in the existence of prior space time, I tend to feel the
discussion has diverged somewhat at that point and start losing
interest. As I have it, space-time is just a mathematical construction,
part of an ordering principle, and does not physically exist between
initial and final measurements in quantum theory. The purpose of the
teleconnection was to formulate quantum theory in flat space-time, even
in an expanding universe.
>
>
>A Fock space is merely a Hilbert space with some extra structure
>(it's closed under tensor products of states). A Fock space can arise
>in two ways. The Hilbert space on which a theory of quantum fields is
>formulated can be given a Fock space structure, in which case matrix
>elements of the field operators give wave functions for free.
>A quantum theory of an indefinite number of identitcal particles can
>also be constructed on a Hilbert space with a Fock structure, in which
>case single particle wave functions yield quantum field operators,
>again for free. The equivalence of these two constructions is the main
>theorem of second quantization. In other words, quantum fields are
>unavoidable, even if you forget about the quantization of classical
>fields.

True.

>Now, Haag's theorem says nothing about Fock spaces. It only talks of
>Hilbert spaces and quantum fields. It says nothing about the
>impossibility of constructing interacting fields. It only says that if
>a free theory and an interacting theory are constructed, they cannot be
>related by a unitary transformation (where both theories are Poincare
>invariant). Compare to ordinary quantum mechanics where knowing the
>matrix elements of the x and p operators between the momentum states of
>a free particle and between the bound states of the harmonic
>oscillator, we can connect one set of matrix elements to the other
>through a unitary transformation whose own matrix elements are given by
>the Fourier transforms of weighted Hermite polynomials.
>
>Note that the paragraph on second quantization specified nothing of
>interaction, while the paragraph on Haag's theorem specified nothing of
>Fock spaces. Which means that you can put them together and draw your
>own conclusions about the fictitiousness of interacting quantum fields
>and the applicability of Haag's theorem to a particle description.
>
>
I simply don't see a problem at this point. It is natural to model an
interaction as a map from the space of non-interacting particles to the
space of non interacting particles. It is natural to do that using the
field operators which we got for free with Fock space. These are the so
called "non-interacting" fields, and there is no reason to introduce
"interacting fields". The problem comes a little way down the road when
the Landau pole appears. That may make the model inconsistent, but it
does not mean that it is entirely wrong. The natural solution is to
modify the theory at short range. If we can find a way to do that the
Landau pole will go away.

Regards

--
Charles Francis

[Charles Francis]

In message <1128183366.819460.154450@g44g2000cwa.googlegroups. com>,
Igor Khavkine <igor.kh@gmail.com> writes
>Charles Francis wrote:

>> Of course these days the
>> fashion is to say "oh we mustn't think about physical
>> interpretation". So I guess I'm just pig headed, because I think it
>> is the main thing we should think about if we are going to advance
>> our understanding of nature.
>
>I completely agree that physical interpretation must be considered.
>However, since we are dealing with a scientific theory, the
>interpretation must be held to as high a standard as any other part of
>the theory. Namely, the interpretation must consist of a dictionary to
>translate the properties of objects of a theory into measurable and
>verifiable quantities, and vice versa. For a successful theory, the
>dictionary is required to be as complete as possible going from
>experiment to theory, but there is no such requirement going the
>opposite way. Hence an incompleteness in this second half of the
>dictionary does not have a lot of weight in the discussion. I think the
>point that you bring up, a mechanical interpretation of a particle as
>opposed to a field, belongs to this second half of the dictionary.

I can't agree with that. For me the purpose of physics is not just to
make good experimental predictions, but to have as true and accurate a
view of nature as possible. I do agree that interpretation must be held
to as high a standard as any other part of the theory, and since I find
that this is not currently the case it seems to me that this is the very
area of research which will most likely yield our next advances in the
understanding of nature.

>As you can see, Wald has much less kind words about the formulation of
>quantum particles on curved spacetime.

Perhaps. But as I come from a relationist perspective which does not
believe in the existence of prior space time, I tend to feel the
discussion has diverged somewhat at that point and start losing
interest. As I have it, space-time is just a mathematical construction,
part of an ordering principle, and does not physically exist between
initial and final measurements in quantum theory. The purpose of the
teleconnection was to formulate quantum theory in flat space-time, even
in an expanding universe.
>
>
>A Fock space is merely a Hilbert space with some extra structure
>(it's closed under tensor products of states). A Fock space can arise
>in two ways. The Hilbert space on which a theory of quantum fields is
>formulated can be given a Fock space structure, in which case matrix
>elements of the field operators give wave functions for free.
>A quantum theory of an indefinite number of identitcal particles can
>also be constructed on a Hilbert space with a Fock structure, in which
>case single particle wave functions yield quantum field operators,
>again for free. The equivalence of these two constructions is the main
>theorem of second quantization. In other words, quantum fields are
>unavoidable, even if you forget about the quantization of classical
>fields.

True.

>Now, Haag's theorem says nothing about Fock spaces. It only talks of
>Hilbert spaces and quantum fields. It says nothing about the
>impossibility of constructing interacting fields. It only says that if
>a free theory and an interacting theory are constructed, they cannot be
>related by a unitary transformation (where both theories are Poincare
>invariant). Compare to ordinary quantum mechanics where knowing the
>matrix elements of the x and p operators between the momentum states of
>a free particle and between the bound states of the harmonic
>oscillator, we can connect one set of matrix elements to the other
>through a unitary transformation whose own matrix elements are given by
>the Fourier transforms of weighted Hermite polynomials.
>
>Note that the paragraph on second quantization specified nothing of
>interaction, while the paragraph on Haag's theorem specified nothing of
>Fock spaces. Which means that you can put them together and draw your
>own conclusions about the fictitiousness of interacting quantum fields
>and the applicability of Haag's theorem to a particle description.
>
>
I simply don't see a problem at this point. It is natural to model an
interaction as a map from the space of non-interacting particles to the
space of non interacting particles. It is natural to do that using the
field operators which we got for free with Fock space. These are the so
called "non-interacting" fields, and there is no reason to introduce
"interacting fields". The problem comes a little way down the road when
the Landau pole appears. That may make the model inconsistent, but it
does not mean that it is entirely wrong. The natural solution is to
modify the theory at short range. If we can find a way to do that the
Landau pole will go away.

Regards

--
Charles Francis

[Charles Francis]

In message <1128183366.819460.154450@g44g2000cwa.googlegroups. com>,
Igor Khavkine <igor.kh@gmail.com> writes
>Charles Francis wrote:

>> Of course these days the
>> fashion is to say "oh we mustn't think about physical
>> interpretation". So I guess I'm just pig headed, because I think it
>> is the main thing we should think about if we are going to advance
>> our understanding of nature.
>
>I completely agree that physical interpretation must be considered.
>However, since we are dealing with a scientific theory, the
>interpretation must be held to as high a standard as any other part of
>the theory. Namely, the interpretation must consist of a dictionary to
>translate the properties of objects of a theory into measurable and
>verifiable quantities, and vice versa. For a successful theory, the
>dictionary is required to be as complete as possible going from
>experiment to theory, but there is no such requirement going the
>opposite way. Hence an incompleteness in this second half of the
>dictionary does not have a lot of weight in the discussion. I think the
>point that you bring up, a mechanical interpretation of a particle as
>opposed to a field, belongs to this second half of the dictionary.

I can't agree with that. For me the purpose of physics is not just to
make good experimental predictions, but to have as true and accurate a
view of nature as possible. I do agree that interpretation must be held
to as high a standard as any other part of the theory, and since I find
that this is not currently the case it seems to me that this is the very
area of research which will most likely yield our next advances in the
understanding of nature.

>As you can see, Wald has much less kind words about the formulation of
>quantum particles on curved spacetime.

Perhaps. But as I come from a relationist perspective which does not
believe in the existence of prior space time, I tend to feel the
discussion has diverged somewhat at that point and start losing
interest. As I have it, space-time is just a mathematical construction,
part of an ordering principle, and does not physically exist between
initial and final measurements in quantum theory. The purpose of the
teleconnection was to formulate quantum theory in flat space-time, even
in an expanding universe.
>
>
>A Fock space is merely a Hilbert space with some extra structure
>(it's closed under tensor products of states). A Fock space can arise
>in two ways. The Hilbert space on which a theory of quantum fields is
>formulated can be given a Fock space structure, in which case matrix
>elements of the field operators give wave functions for free.
>A quantum theory of an indefinite number of identitcal particles can
>also be constructed on a Hilbert space with a Fock structure, in which
>case single particle wave functions yield quantum field operators,
>again for free. The equivalence of these two constructions is the main
>theorem of second quantization. In other words, quantum fields are
>unavoidable, even if you forget about the quantization of classical
>fields.

True.

>Now, Haag's theorem says nothing about Fock spaces. It only talks of
>Hilbert spaces and quantum fields. It says nothing about the
>impossibility of constructing interacting fields. It only says that if
>a free theory and an interacting theory are constructed, they cannot be
>related by a unitary transformation (where both theories are Poincare
>invariant). Compare to ordinary quantum mechanics where knowing the
>matrix elements of the x and p operators between the momentum states of
>a free particle and between the bound states of the harmonic
>oscillator, we can connect one set of matrix elements to the other
>through a unitary transformation whose own matrix elements are given by
>the Fourier transforms of weighted Hermite polynomials.
>
>Note that the paragraph on second quantization specified nothing of
>interaction, while the paragraph on Haag's theorem specified nothing of
>Fock spaces. Which means that you can put them together and draw your
>own conclusions about the fictitiousness of interacting quantum fields
>and the applicability of Haag's theorem to a particle description.
>
>
I simply don't see a problem at this point. It is natural to model an
interaction as a map from the space of non-interacting particles to the
space of non interacting particles. It is natural to do that using the
field operators which we got for free with Fock space. These are the so
called "non-interacting" fields, and there is no reason to introduce
"interacting fields". The problem comes a little way down the road when
the Landau pole appears. That may make the model inconsistent, but it
does not mean that it is entirely wrong. The natural solution is to
modify the theory at short range. If we can find a way to do that the
Landau pole will go away.

Regards

--
Charles Francis

[Charles Francis]

In message <1128183366.819460.154450@g44g2000cwa.googlegroups. com>,
Igor Khavkine <igor.kh@gmail.com> writes
>Charles Francis wrote:

>> Of course these days the
>> fashion is to say "oh we mustn't think about physical
>> interpretation". So I guess I'm just pig headed, because I think it
>> is the main thing we should think about if we are going to advance
>> our understanding of nature.
>
>I completely agree that physical interpretation must be considered.
>However, since we are dealing with a scientific theory, the
>interpretation must be held to as high a standard as any other part of
>the theory. Namely, the interpretation must consist of a dictionary to
>translate the properties of objects of a theory into measurable and
>verifiable quantities, and vice versa. For a successful theory, the
>dictionary is required to be as complete as possible going from
>experiment to theory, but there is no such requirement going the
>opposite way. Hence an incompleteness in this second half of the
>dictionary does not have a lot of weight in the discussion. I think the
>point that you bring up, a mechanical interpretation of a particle as
>opposed to a field, belongs to this second half of the dictionary.

I can't agree with that. For me the purpose of physics is not just to
make good experimental predictions, but to have as true and accurate a
view of nature as possible. I do agree that interpretation must be held
to as high a standard as any other part of the theory, and since I find
that this is not currently the case it seems to me that this is the very
area of research which will most likely yield our next advances in the
understanding of nature.

>As you can see, Wald has much less kind words about the formulation of
>quantum particles on curved spacetime.

Perhaps. But as I come from a relationist perspective which does not
believe in the existence of prior space time, I tend to feel the
discussion has diverged somewhat at that point and start losing
interest. As I have it, space-time is just a mathematical construction,
part of an ordering principle, and does not physically exist between
initial and final measurements in quantum theory. The purpose of the
teleconnection was to formulate quantum theory in flat space-time, even
in an expanding universe.
>
>
>A Fock space is merely a Hilbert space with some extra structure
>(it's closed under tensor products of states). A Fock space can arise
>in two ways. The Hilbert space on which a theory of quantum fields is
>formulated can be given a Fock space structure, in which case matrix
>elements of the field operators give wave functions for free.
>A quantum theory of an indefinite number of identitcal particles can
>also be constructed on a Hilbert space with a Fock structure, in which
>case single particle wave functions yield quantum field operators,
>again for free. The equivalence of these two constructions is the main
>theorem of second quantization. In other words, quantum fields are
>unavoidable, even if you forget about the quantization of classical
>fields.

True.

>Now, Haag's theorem says nothing about Fock spaces. It only talks of
>Hilbert spaces and quantum fields. It says nothing about the
>impossibility of constructing interacting fields. It only says that if
>a free theory and an interacting theory are constructed, they cannot be
>related by a unitary transformation (where both theories are Poincare
>invariant). Compare to ordinary quantum mechanics where knowing the
>matrix elements of the x and p operators between the momentum states of
>a free particle and between the bound states of the harmonic
>oscillator, we can connect one set of matrix elements to the other
>through a unitary transformation whose own matrix elements are given by
>the Fourier transforms of weighted Hermite polynomials.
>
>Note that the paragraph on second quantization specified nothing of
>interaction, while the paragraph on Haag's theorem specified nothing of
>Fock spaces. Which means that you can put them together and draw your
>own conclusions about the fictitiousness of interacting quantum fields
>and the applicability of Haag's theorem to a particle description.
>
>
I simply don't see a problem at this point. It is natural to model an
interaction as a map from the space of non-interacting particles to the
space of non interacting particles. It is natural to do that using the
field operators which we got for free with Fock space. These are the so
called "non-interacting" fields, and there is no reason to introduce
"interacting fields". The problem comes a little way down the road when
the Landau pole appears. That may make the model inconsistent, but it
does not mean that it is entirely wrong. The natural solution is to
modify the theory at short range. If we can find a way to do that the
Landau pole will go away.

Regards

--
Charles Francis

[Charles Francis]

In message <1128183366.819460.154450@g44g2000cwa.googlegroups. com>,
Igor Khavkine <igor.kh@gmail.com> writes
>Charles Francis wrote:

>> Of course these days the
>> fashion is to say "oh we mustn't think about physical
>> interpretation". So I guess I'm just pig headed, because I think it
>> is the main thing we should think about if we are going to advance
>> our understanding of nature.
>
>I completely agree that physical interpretation must be considered.
>However, since we are dealing with a scientific theory, the
>interpretation must be held to as high a standard as any other part of
>the theory. Namely, the interpretation must consist of a dictionary to
>translate the properties of objects of a theory into measurable and
>verifiable quantities, and vice versa. For a successful theory, the
>dictionary is required to be as complete as possible going from
>experiment to theory, but there is no such requirement going the
>opposite way. Hence an incompleteness in this second half of the
>dictionary does not have a lot of weight in the discussion. I think the
>point that you bring up, a mechanical interpretation of a particle as
>opposed to a field, belongs to this second half of the dictionary.

I can't agree with that. For me the purpose of physics is not just to
make good experimental predictions, but to have as true and accurate a
view of nature as possible. I do agree that interpretation must be held
to as high a standard as any other part of the theory, and since I find
that this is not currently the case it seems to me that this is the very
area of research which will most likely yield our next advances in the
understanding of nature.

>As you can see, Wald has much less kind words about the formulation of
>quantum particles on curved spacetime.

Perhaps. But as I come from a relationist perspective which does not
believe in the existence of prior space time, I tend to feel the
discussion has diverged somewhat at that point and start losing
interest. As I have it, space-time is just a mathematical construction,
part of an ordering principle, and does not physically exist between
initial and final measurements in quantum theory. The purpose of the
teleconnection was to formulate quantum theory in flat space-time, even
in an expanding universe.
>
>
>A Fock space is merely a Hilbert space with some extra structure
>(it's closed under tensor products of states). A Fock space can arise
>in two ways. The Hilbert space on which a theory of quantum fields is
>formulated can be given a Fock space structure, in which case matrix
>elements of the field operators give wave functions for free.
>A quantum theory of an indefinite number of identitcal particles can
>also be constructed on a Hilbert space with a Fock structure, in which
>case single particle wave functions yield quantum field operators,
>again for free. The equivalence of these two constructions is the main
>theorem of second quantization. In other words, quantum fields are
>unavoidable, even if you forget about the quantization of classical
>fields.

True.

>Now, Haag's theorem says nothing about Fock spaces. It only talks of
>Hilbert spaces and quantum fields. It says nothing about the
>impossibility of constructing interacting fields. It only says that if
>a free theory and an interacting theory are constructed, they cannot be
>related by a unitary transformation (where both theories are Poincare
>invariant). Compare to ordinary quantum mechanics where knowing the
>matrix elements of the x and p operators between the momentum states of
>a free particle and between the bound states of the harmonic
>oscillator, we can connect one set of matrix elements to the other
>through a unitary transformation whose own matrix elements are given by
>the Fourier transforms of weighted Hermite polynomials.
>
>Note that the paragraph on second quantization specified nothing of
>interaction, while the paragraph on Haag's theorem specified nothing of
>Fock spaces. Which means that you can put them together and draw your
>own conclusions about the fictitiousness of interacting quantum fields
>and the applicability of Haag's theorem to a particle description.
>
>
I simply don't see a problem at this point. It is natural to model an
interaction as a map from the space of non-interacting particles to the
space of non interacting particles. It is natural to do that using the
field operators which we got for free with Fock space. These are the so
called "non-interacting" fields, and there is no reason to introduce
"interacting fields". The problem comes a little way down the road when
the Landau pole appears. That may make the model inconsistent, but it
does not mean that it is entirely wrong. The natural solution is to
modify the theory at short range. If we can find a way to do that the
Landau pole will go away.

Regards

--
Charles Francis

[Charles Francis]

In message <1128183366.819460.154450@g44g2000cwa.googlegroups. com>,
Igor Khavkine <igor.kh@gmail.com> writes
>Charles Francis wrote:

>> Of course these days the
>> fashion is to say "oh we mustn't think about physical
>> interpretation". So I guess I'm just pig headed, because I think it
>> is the main thing we should think about if we are going to advance
>> our understanding of nature.
>
>I completely agree that physical interpretation must be considered.
>However, since we are dealing with a scientific theory, the
>interpretation must be held to as high a standard as any other part of
>the theory. Namely, the interpretation must consist of a dictionary to
>translate the properties of objects of a theory into measurable and
>verifiable quantities, and vice versa. For a successful theory, the
>dictionary is required to be as complete as possible going from
>experiment to theory, but there is no such requirement going the
>opposite way. Hence an incompleteness in this second half of the
>dictionary does not have a lot of weight in the discussion. I think the
>point that you bring up, a mechanical interpretation of a particle as
>opposed to a field, belongs to this second half of the dictionary.

I can't agree with that. For me the purpose of physics is not just to
make good experimental predictions, but to have as true and accurate a
view of nature as possible. I do agree that interpretation must be held
to as high a standard as any other part of the theory, and since I find
that this is not currently the case it seems to me that this is the very
area of research which will most likely yield our next advances in the
understanding of nature.

>As you can see, Wald has much less kind words about the formulation of
>quantum particles on curved spacetime.

Perhaps. But as I come from a relationist perspective which does not
believe in the existence of prior space time, I tend to feel the
discussion has diverged somewhat at that point and start losing
interest. As I have it, space-time is just a mathematical construction,
part of an ordering principle, and does not physically exist between
initial and final measurements in quantum theory. The purpose of the
teleconnection was to formulate quantum theory in flat space-time, even
in an expanding universe.
>
>
>A Fock space is merely a Hilbert space with some extra structure
>(it's closed under tensor products of states). A Fock space can arise
>in two ways. The Hilbert space on which a theory of quantum fields is
>formulated can be given a Fock space structure, in which case matrix
>elements of the field operators give wave functions for free.
>A quantum theory of an indefinite number of identitcal particles can
>also be constructed on a Hilbert space with a Fock structure, in which
>case single particle wave functions yield quantum field operators,
>again for free. The equivalence of these two constructions is the main
>theorem of second quantization. In other words, quantum fields are
>unavoidable, even if you forget about the quantization of classical
>fields.

True.

>Now, Haag's theorem says nothing about Fock spaces. It only talks of
>Hilbert spaces and quantum fields. It says nothing about the
>impossibility of constructing interacting fields. It only says that if
>a free theory and an interacting theory are constructed, they cannot be
>related by a unitary transformation (where both theories are Poincare
>invariant). Compare to ordinary quantum mechanics where knowing the
>matrix elements of the x and p operators between the momentum states of
>a free particle and between the bound states of the harmonic
>oscillator, we can connect one set of matrix elements to the other
>through a unitary transformation whose own matrix elements are given by
>the Fourier transforms of weighted Hermite polynomials.
>
>Note that the paragraph on second quantization specified nothing of
>interaction, while the paragraph on Haag's theorem specified nothing of
>Fock spaces. Which means that you can put them together and draw your
>own conclusions about the fictitiousness of interacting quantum fields
>and the applicability of Haag's theorem to a particle description.
>
>
I simply don't see a problem at this point. It is natural to model an
interaction as a map from the space of non-interacting particles to the
space of non interacting particles. It is natural to do that using the
field operators which we got for free with Fock space. These are the so
called "non-interacting" fields, and there is no reason to introduce
"interacting fields". The problem comes a little way down the road when
the Landau pole appears. That may make the model inconsistent, but it
does not mean that it is entirely wrong. The natural solution is to
modify the theory at short range. If we can find a way to do that the
Landau pole will go away.

Regards

--
Charles Francis

[Charles Francis]

In message <1128183366.819460.154450@g44g2000cwa.googlegroups. com>,
Igor Khavkine <igor.kh@gmail.com> writes
>Charles Francis wrote:

>> Of course these days the
>> fashion is to say "oh we mustn't think about physical
>> interpretation". So I guess I'm just pig headed, because I think it
>> is the main thing we should think about if we are going to advance
>> our understanding of nature.
>
>I completely agree that physical interpretation must be considered.
>However, since we are dealing with a scientific theory, the
>interpretation must be held to as high a standard as any other part of
>the theory. Namely, the interpretation must consist of a dictionary to
>translate the properties of objects of a theory into measurable and
>verifiable quantities, and vice versa. For a successful theory, the
>dictionary is required to be as complete as possible going from
>experiment to theory, but there is no such requirement going the
>opposite way. Hence an incompleteness in this second half of the
>dictionary does not have a lot of weight in the discussion. I think the
>point that you bring up, a mechanical interpretation of a particle as
>opposed to a field, belongs to this second half of the dictionary.

I can't agree with that. For me the purpose of physics is not just to
make good experimental predictions, but to have as true and accurate a
view of nature as possible. I do agree that interpretation must be held
to as high a standard as any other part of the theory, and since I find
that this is not currently the case it seems to me that this is the very
area of research which will most likely yield our next advances in the
understanding of nature.

>As you can see, Wald has much less kind words about the formulation of
>quantum particles on curved spacetime.

Perhaps. But as I come from a relationist perspective which does not
believe in the existence of prior space time, I tend to feel the
discussion has diverged somewhat at that point and start losing
interest. As I have it, space-time is just a mathematical construction,
part of an ordering principle, and does not physically exist between
initial and final measurements in quantum theory. The purpose of the
teleconnection was to formulate quantum theory in flat space-time, even
in an expanding universe.
>
>
>A Fock space is merely a Hilbert space with some extra structure
>(it's closed under tensor products of states). A Fock space can arise
>in two ways. The Hilbert space on which a theory of quantum fields is
>formulated can be given a Fock space structure, in which case matrix
>elements of the field operators give wave functions for free.
>A quantum theory of an indefinite number of identitcal particles can
>also be constructed on a Hilbert space with a Fock structure, in which
>case single particle wave functions yield quantum field operators,
>again for free. The equivalence of these two constructions is the main
>theorem of second quantization. In other words, quantum fields are
>unavoidable, even if you forget about the quantization of classical
>fields.

True.

>Now, Haag's theorem says nothing about Fock spaces. It only talks of
>Hilbert spaces and quantum fields. It says nothing about the
>impossibility of constructing interacting fields. It only says that if
>a free theory and an interacting theory are constructed, they cannot be
>related by a unitary transformation (where both theories are Poincare
>invariant). Compare to ordinary quantum mechanics where knowing the
>matrix elements of the x and p operators between the momentum states of
>a free particle and between the bound states of the harmonic
>oscillator, we can connect one set of matrix elements to the other
>through a unitary transformation whose own matrix elements are given by
>the Fourier transforms of weighted Hermite polynomials.
>
>Note that the paragraph on second quantization specified nothing of
>interaction, while the paragraph on Haag's theorem specified nothing of
>Fock spaces. Which means that you can put them together and draw your
>own conclusions about the fictitiousness of interacting quantum fields
>and the applicability of Haag's theorem to a particle description.
>
>
I simply don't see a problem at this point. It is natural to model an
interaction as a map from the space of non-interacting particles to the
space of non interacting particles. It is natural to do that using the
field operators which we got for free with Fock space. These are the so
called "non-interacting" fields, and there is no reason to introduce
"interacting fields". The problem comes a little way down the road when
the Landau pole appears. That may make the model inconsistent, but it
does not mean that it is entirely wrong. The natural solution is to
modify the theory at short range. If we can find a way to do that the
Landau pole will go away.

Regards

--
Charles Francis

[Charles Francis]

In message <1128183366.819460.154450@g44g2000cwa.googlegroups. com>,
Igor Khavkine <igor.kh@gmail.com> writes
>Charles Francis wrote:

>> Of course these days the
>> fashion is to say "oh we mustn't think about physical
>> interpretation". So I guess I'm just pig headed, because I think it
>> is the main thing we should think about if we are going to advance
>> our understanding of nature.
>
>I completely agree that physical interpretation must be considered.
>However, since we are dealing with a scientific theory, the
>interpretation must be held to as high a standard as any other part of
>the theory. Namely, the interpretation must consist of a dictionary to
>translate the properties of objects of a theory into measurable and
>verifiable quantities, and vice versa. For a successful theory, the
>dictionary is required to be as complete as possible going from
>experiment to theory, but there is no such requirement going the
>opposite way. Hence an incompleteness in this second half of the
>dictionary does not have a lot of weight in the discussion. I think the
>point that you bring up, a mechanical interpretation of a particle as
>opposed to a field, belongs to this second half of the dictionary.

I can't agree with that. For me the purpose of physics is not just to
make good experimental predictions, but to have as true and accurate a
view of nature as possible. I do agree that interpretation must be held
to as high a standard as any other part of the theory, and since I find
that this is not currently the case it seems to me that this is the very
area of research which will most likely yield our next advances in the
understanding of nature.

>As you can see, Wald has much less kind words about the formulation of
>quantum particles on curved spacetime.

Perhaps. But as I come from a relationist perspective which does not
believe in the existence of prior space time, I tend to feel the
discussion has diverged somewhat at that point and start losing
interest. As I have it, space-time is just a mathematical construction,
part of an ordering principle, and does not physically exist between
initial and final measurements in quantum theory. The purpose of the
teleconnection was to formulate quantum theory in flat space-time, even
in an expanding universe.
>
>
>A Fock space is merely a Hilbert space with some extra structure
>(it's closed under tensor products of states). A Fock space can arise
>in two ways. The Hilbert space on which a theory of quantum fields is
>formulated can be given a Fock space structure, in which case matrix
>elements of the field operators give wave functions for free.
>A quantum theory of an indefinite number of identitcal particles can
>also be constructed on a Hilbert space with a Fock structure, in which
>case single particle wave functions yield quantum field operators,
>again for free. The equivalence of these two constructions is the main
>theorem of second quantization. In other words, quantum fields are
>unavoidable, even if you forget about the quantization of classical
>fields.

True.

>Now, Haag's theorem says nothing about Fock spaces. It only talks of
>Hilbert spaces and quantum fields. It says nothing about the
>impossibility of constructing interacting fields. It only says that if
>a free theory and an interacting theory are constructed, they cannot be
>related by a unitary transformation (where both theories are Poincare
>invariant). Compare to ordinary quantum mechanics where knowing the
>matrix elements of the x and p operators between the momentum states of
>a free particle and between the bound states of the harmonic
>oscillator, we can connect one set of matrix elements to the other
>through a unitary transformation whose own matrix elements are given by
>the Fourier transforms of weighted Hermite polynomials.
>
>Note that the paragraph on second quantization specified nothing of
>interaction, while the paragraph on Haag's theorem specified nothing of
>Fock spaces. Which means that you can put them together and draw your
>own conclusions about the fictitiousness of interacting quantum fields
>and the applicability of Haag's theorem to a particle description.
>
>
I simply don't see a problem at this point. It is natural to model an
interaction as a map from the space of non-interacting particles to the
space of non interacting particles. It is natural to do that using the
field operators which we got for free with Fock space. These are the so
called "non-interacting" fields, and there is no reason to introduce
"interacting fields". The problem comes a little way down the road when
the Landau pole appears. That may make the model inconsistent, but it
does not mean that it is entirely wrong. The natural solution is to
modify the theory at short range. If we can find a way to do that the
Landau pole will go away.

Regards

--
Charles Francis

[Charles Francis]

In message <pan.2005.10.01.13.44.07.528839@me.spam>, J. Horta
<bite@me.spam> writes
>On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:
>
>> Igor Khavkine wrote:
>>>
>>> Regarding your original post. Your argument falls through when
>>> you declare that particles are "fundamental" in the way you define that
>>> term. One can take fields as a starting point and never mention
>>> particles at all, they will fall out automatically. Just as fields will
>>> fall out automatically when you start with particles. Both formulations
>>> are equally "fundamental" or equally not so, however you want to
>>> consider them.
>>
>> Particles are more fundamental that fields because fields are ALWAYS
>> -by definition- unobserved, one measures in scattering experiments are
>> particles, newer fields. As clearly stated by Weinberg in his volume 1,
>> we know more about particles that about fields.
>>
>
>I very much agree with Igor. What is or is not observable depends on
>the situation. At very low energies electric fields may be measured
>with a volt meter and magnetic fields with a magnetometer.

The supposedly fundamental fields are not e.m. But quantum fields, which
may not be measured.

>A radio
>picks up em waves. Sure, each one of these may be formulated as
>interaction with quantum fields or, as follows from the formalism,
>absorption and emission of particles. The physics should be the same
>independent of how it is formulated. If you should find, however,
>a formulation of the physics using a particle approach which includes
>phenomena which a quantum field approach can't then you could well
>claim a more fundamental theory.
>
I do. See gr-qc/0508077

Regards

--
Charles Francis

[Charles Francis]

In message <pan.2005.10.01.13.44.07.528839@me.spam>, J. Horta
<bite@me.spam> writes
>On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:
>
>> Igor Khavkine wrote:
>>>
>>> Regarding your original post. Your argument falls through when
>>> you declare that particles are "fundamental" in the way you define that
>>> term. One can take fields as a starting point and never mention
>>> particles at all, they will fall out automatically. Just as fields will
>>> fall out automatically when you start with particles. Both formulations
>>> are equally "fundamental" or equally not so, however you want to
>>> consider them.
>>
>> Particles are more fundamental that fields because fields are ALWAYS
>> -by definition- unobserved, one measures in scattering experiments are
>> particles, newer fields. As clearly stated by Weinberg in his volume 1,
>> we know more about particles that about fields.
>>
>
>I very much agree with Igor. What is or is not observable depends on
>the situation. At very low energies electric fields may be measured
>with a volt meter and magnetic fields with a magnetometer.

The supposedly fundamental fields are not e.m. But quantum fields, which
may not be measured.

>A radio
>picks up em waves. Sure, each one of these may be formulated as
>interaction with quantum fields or, as follows from the formalism,
>absorption and emission of particles. The physics should be the same
>independent of how it is formulated. If you should find, however,
>a formulation of the physics using a particle approach which includes
>phenomena which a quantum field approach can't then you could well
>claim a more fundamental theory.
>
I do. See gr-qc/0508077

Regards

--
Charles Francis

[Charles Francis]

In message <pan.2005.10.01.13.44.07.528839@me.spam>, J. Horta
<bite@me.spam> writes
>On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:
>
>> Igor Khavkine wrote:
>>>
>>> Regarding your original post. Your argument falls through when
>>> you declare that particles are "fundamental" in the way you define that
>>> term. One can take fields as a starting point and never mention
>>> particles at all, they will fall out automatically. Just as fields will
>>> fall out automatically when you start with particles. Both formulations
>>> are equally "fundamental" or equally not so, however you want to
>>> consider them.
>>
>> Particles are more fundamental that fields because fields are ALWAYS
>> -by definition- unobserved, one measures in scattering experiments are
>> particles, newer fields. As clearly stated by Weinberg in his volume 1,
>> we know more about particles that about fields.
>>
>
>I very much agree with Igor. What is or is not observable depends on
>the situation. At very low energies electric fields may be measured
>with a volt meter and magnetic fields with a magnetometer.

The supposedly fundamental fields are not e.m. But quantum fields, which
may not be measured.

>A radio
>picks up em waves. Sure, each one of these may be formulated as
>interaction with quantum fields or, as follows from the formalism,
>absorption and emission of particles. The physics should be the same
>independent of how it is formulated. If you should find, however,
>a formulation of the physics using a particle approach which includes
>phenomena which a quantum field approach can't then you could well
>claim a more fundamental theory.
>
I do. See gr-qc/0508077

Regards

--
Charles Francis

[Charles Francis]

In message <pan.2005.10.01.13.44.07.528839@me.spam>, J. Horta
<bite@me.spam> writes
>On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:
>
>> Igor Khavkine wrote:
>>>
>>> Regarding your original post. Your argument falls through when
>>> you declare that particles are "fundamental" in the way you define that
>>> term. One can take fields as a starting point and never mention
>>> particles at all, they will fall out automatically. Just as fields will
>>> fall out automatically when you start with particles. Both formulations
>>> are equally "fundamental" or equally not so, however you want to
>>> consider them.
>>
>> Particles are more fundamental that fields because fields are ALWAYS
>> -by definition- unobserved, one measures in scattering experiments are
>> particles, newer fields. As clearly stated by Weinberg in his volume 1,
>> we know more about particles that about fields.
>>
>
>I very much agree with Igor. What is or is not observable depends on
>the situation. At very low energies electric fields may be measured
>with a volt meter and magnetic fields with a magnetometer.

The supposedly fundamental fields are not e.m. But quantum fields, which
may not be measured.

>A radio
>picks up em waves. Sure, each one of these may be formulated as
>interaction with quantum fields or, as follows from the formalism,
>absorption and emission of particles. The physics should be the same
>independent of how it is formulated. If you should find, however,
>a formulation of the physics using a particle approach which includes
>phenomena which a quantum field approach can't then you could well
>claim a more fundamental theory.
>
I do. See gr-qc/0508077

Regards

--
Charles Francis

[Charles Francis]

In message <pan.2005.10.01.13.44.07.528839@me.spam>, J. Horta
<bite@me.spam> writes
>On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:
>
>> Igor Khavkine wrote:
>>>
>>> Regarding your original post. Your argument falls through when
>>> you declare that particles are "fundamental" in the way you define that
>>> term. One can take fields as a starting point and never mention
>>> particles at all, they will fall out automatically. Just as fields will
>>> fall out automatically when you start with particles. Both formulations
>>> are equally "fundamental" or equally not so, however you want to
>>> consider them.
>>
>> Particles are more fundamental that fields because fields are ALWAYS
>> -by definition- unobserved, one measures in scattering experiments are
>> particles, newer fields. As clearly stated by Weinberg in his volume 1,
>> we know more about particles that about fields.
>>
>
>I very much agree with Igor. What is or is not observable depends on
>the situation. At very low energies electric fields may be measured
>with a volt meter and magnetic fields with a magnetometer.

The supposedly fundamental fields are not e.m. But quantum fields, which
may not be measured.

>A radio
>picks up em waves. Sure, each one of these may be formulated as
>interaction with quantum fields or, as follows from the formalism,
>absorption and emission of particles. The physics should be the same
>independent of how it is formulated. If you should find, however,
>a formulation of the physics using a particle approach which includes
>phenomena which a quantum field approach can't then you could well
>claim a more fundamental theory.
>
I do. See gr-qc/0508077

Regards

--
Charles Francis

[Charles Francis]

In message <pan.2005.10.01.13.44.07.528839@me.spam>, J. Horta
<bite@me.spam> writes
>On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:
>
>> Igor Khavkine wrote:
>>>
>>> Regarding your original post. Your argument falls through when
>>> you declare that particles are "fundamental" in the way you define that
>>> term. One can take fields as a starting point and never mention
>>> particles at all, they will fall out automatically. Just as fields will
>>> fall out automatically when you start with particles. Both formulations
>>> are equally "fundamental" or equally not so, however you want to
>>> consider them.
>>
>> Particles are more fundamental that fields because fields are ALWAYS
>> -by definition- unobserved, one measures in scattering experiments are
>> particles, newer fields. As clearly stated by Weinberg in his volume 1,
>> we know more about particles that about fields.
>>
>
>I very much agree with Igor. What is or is not observable depends on
>the situation. At very low energies electric fields may be measured
>with a volt meter and magnetic fields with a magnetometer.

The supposedly fundamental fields are not e.m. But quantum fields, which
may not be measured.

>A radio
>picks up em waves. Sure, each one of these may be formulated as
>interaction with quantum fields or, as follows from the formalism,
>absorption and emission of particles. The physics should be the same
>independent of how it is formulated. If you should find, however,
>a formulation of the physics using a particle approach which includes
>phenomena which a quantum field approach can't then you could well
>claim a more fundamental theory.
>
I do. See gr-qc/0508077

Regards

--
Charles Francis

[Charles Francis]

In message <pan.2005.10.01.13.44.07.528839@me.spam>, J. Horta
<bite@me.spam> writes
>On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:
>
>> Igor Khavkine wrote:
>>>
>>> Regarding your original post. Your argument falls through when
>>> you declare that particles are "fundamental" in the way you define that
>>> term. One can take fields as a starting point and never mention
>>> particles at all, they will fall out automatically. Just as fields will
>>> fall out automatically when you start with particles. Both formulations
>>> are equally "fundamental" or equally not so, however you want to
>>> consider them.
>>
>> Particles are more fundamental that fields because fields are ALWAYS
>> -by definition- unobserved, one measures in scattering experiments are
>> particles, newer fields. As clearly stated by Weinberg in his volume 1,
>> we know more about particles that about fields.
>>
>
>I very much agree with Igor. What is or is not observable depends on
>the situation. At very low energies electric fields may be measured
>with a volt meter and magnetic fields with a magnetometer.

The supposedly fundamental fields are not e.m. But quantum fields, which
may not be measured.

>A radio
>picks up em waves. Sure, each one of these may be formulated as
>interaction with quantum fields or, as follows from the formalism,
>absorption and emission of particles. The physics should be the same
>independent of how it is formulated. If you should find, however,
>a formulation of the physics using a particle approach which includes
>phenomena which a quantum field approach can't then you could well
>claim a more fundamental theory.
>
I do. See gr-qc/0508077

Regards

--
Charles Francis

[Charles Francis]

In message <pan.2005.10.01.13.44.07.528839@me.spam>, J. Horta
<bite@me.spam> writes
>On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:
>
>> Igor Khavkine wrote:
>>>
>>> Regarding your original post. Your argument falls through when
>>> you declare that particles are "fundamental" in the way you define that
>>> term. One can take fields as a starting point and never mention
>>> particles at all, they will fall out automatically. Just as fields will
>>> fall out automatically when you start with particles. Both formulations
>>> are equally "fundamental" or equally not so, however you want to
>>> consider them.
>>
>> Particles are more fundamental that fields because fields are ALWAYS
>> -by definition- unobserved, one measures in scattering experiments are
>> particles, newer fields. As clearly stated by Weinberg in his volume 1,
>> we know more about particles that about fields.
>>
>
>I very much agree with Igor. What is or is not observable depends on
>the situation. At very low energies electric fields may be measured
>with a volt meter and magnetic fields with a magnetometer.

The supposedly fundamental fields are not e.m. But quantum fields, which
may not be measured.

>A radio
>picks up em waves. Sure, each one of these may be formulated as
>interaction with quantum fields or, as follows from the formalism,
>absorption and emission of particles. The physics should be the same
>independent of how it is formulated. If you should find, however,
>a formulation of the physics using a particle approach which includes
>phenomena which a quantum field approach can't then you could well
>claim a more fundamental theory.
>
I do. See gr-qc/0508077

Regards

--
Charles Francis

[Charles Francis]

In message <pan.2005.10.01.13.44.07.528839@me.spam>, J. Horta
<bite@me.spam> writes
>On Sat, 01 Oct 2005 08:31:47 +0000, Juan R. wrote:
>
>> Igor Khavkine wrote:
>>>
>>> Regarding your original post. Your argument falls through when
>>> you declare that particles are "fundamental" in the way you define that
>>> term. One can take fields as a starting point and never mention
>>> particles at all, they will fall out automatically. Just as fields will
>>> fall out automatically when you start with particles. Both formulations
>>> are equally "fundamental" or equally not so, however you want to
>>> consider them.
>>
>> Particles are more fundamental that fields because fields are ALWAYS
>> -by definition- unobserved, one measures in scattering experiments are
>> particles, newer fields. As clearly stated by Weinberg in his volume 1,
>> we know more about particles that about fields.
>>
>
>I very much agree with Igor. What is or is not observable depends on
>the situation. At very low energies electric fields may be measured
>with a volt meter and magnetic fields with a magnetometer.

The supposedly fundamental fields are not e.m. But quantum fields, which
may not be measured.

>A radio
>picks up em waves. Sure, each one of these may be formulated as
>interaction with quantum fields or, as follows from the formalism,
>absorption and emission of particles. The physics should be the same
>independent of how it is formulated. If you should find, however,
>a formulation of the physics using a particle approach which includes
>phenomena which a quantum field approach can't then you could well
>claim a more fundamental theory.
>
I do. See gr-qc/0508077

Regards

--
Charles Francis

[Igor Khavkine]

Juan R. wrote:
> Juan R. wrote:
> > Weinberg begins from we know (and we can measure) that is particles,
> > QM, and relativity. Anyone who claim that we can measure a field simply
> > does not know that a field is! As stated by Weinberg in Chapter 3 we
> > measure particles in particle physics experiments.
>
> Weimberg (p200):
>
> "Traditionally in quantum field theory one begins with such field
> equations, or with the Lagrangian from which they are derived, and then
> uses them to derive the expansion of the fields in terms of
> one-particle annihilation and creation operators. In the approach
> followed here, we start with the particles, and derive the fields
> according to the dictates of Lorentz invariance, /with the field
> equations arising almost incidentally as a byproduct of this
> construction/."
>
> Emphasis mine. I will remark again, that Weimberg begins from
> ONE-particle states and *derives* fields. But if one begins from FULL
> N-particle states (do NOT factorization into N one-particle states,
> which is ONLY valid on asymptotic regimes of scattering matrix) there
> is not posibility for a rigorous EXACT derivation of fields. The field
> vanishes Remember that both Dirac and KG equations are valid only for
> ONE-particle systems. There is nothing like a 'two-particle Dirac
> equation' (IN RIGOR) for relativistic particles.

The fact that one can derive fields starting with multiparticle states
is the main theorem of second quantization and has been known since
Dirac, Pauli, and others. I don't understand the point you are trying
to make about states that factor and those that don't. Multiparticle
states that factor form a basis for all other ones. Whether you are
dealing with only the basis vectors of the whole Hilbert space is
irrelevant with respect to the construction of field operators.

You also seem to be confused about the KG and Dirac field equations.
There are several distinct objects that satisfy them. Classical fields
(which should be Grassmann valued for fermions), the quantum field
operators, and the (single particle) wave functions. Multiparticle wave
functions are also possible. However, the equations of motion that they
satisfy are not closed (unlike the single particle case) unless all the
E&M field modes are coupled in as well. This is a consequence of the no
go theorem for the existence of a relativistically invariant
two-particle potential. However, in approximations, the infinite
dimensional system of equations including the E&M field can be
truncated to a finite one, like for example the two-particle Dirac
equation with an effective interaction given to order 1/c^2.

> Previously, i cited Weinberg comments on bounded states but i cited
> incorrectly the page, p560:
>
> "It must be said that the theory of relativistic effects and radiative
> corrections in bound states is not yet in entirely satisfactory shape".

On the same page Weinberg talks about the difficulty in finding the
right approximation to use to reduce the full QED theory to an
approximation in which bound state calculations are possible. The same
situation is seen in all of theoretical physics whenever a simple
explanation is sought for complex phenomena, which range from weather
prediction, to high temperature superconductors, the two-body problem
in general relativity.

> Moreover, in curved spacetimes, the particles theories can be done
> rigorous using Synge parallel propagators. Wald 'and company' are wrong
> about particle theories cannot 'work' in curved spacetimes.
> Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Juan R. wrote:
> > Weinberg begins from we know (and we can measure) that is particles,
> > QM, and relativity. Anyone who claim that we can measure a field simply
> > does not know that a field is! As stated by Weinberg in Chapter 3 we
> > measure particles in particle physics experiments.
>
> Weimberg (p200):
>
> "Traditionally in quantum field theory one begins with such field
> equations, or with the Lagrangian from which they are derived, and then
> uses them to derive the expansion of the fields in terms of
> one-particle annihilation and creation operators. In the approach
> followed here, we start with the particles, and derive the fields
> according to the dictates of Lorentz invariance, /with the field
> equations arising almost incidentally as a byproduct of this
> construction/."
>
> Emphasis mine. I will remark again, that Weimberg begins from
> ONE-particle states and *derives* fields. But if one begins from FULL
> N-particle states (do NOT factorization into N one-particle states,
> which is ONLY valid on asymptotic regimes of scattering matrix) there
> is not posibility for a rigorous EXACT derivation of fields. The field
> vanishes Remember that both Dirac and KG equations are valid only for
> ONE-particle systems. There is nothing like a 'two-particle Dirac
> equation' (IN RIGOR) for relativistic particles.

The fact that one can derive fields starting with multiparticle states
is the main theorem of second quantization and has been known since
Dirac, Pauli, and others. I don't understand the point you are trying
to make about states that factor and those that don't. Multiparticle
states that factor form a basis for all other ones. Whether you are
dealing with only the basis vectors of the whole Hilbert space is
irrelevant with respect to the construction of field operators.

You also seem to be confused about the KG and Dirac field equations.
There are several distinct objects that satisfy them. Classical fields
(which should be Grassmann valued for fermions), the quantum field
operators, and the (single particle) wave functions. Multiparticle wave
functions are also possible. However, the equations of motion that they
satisfy are not closed (unlike the single particle case) unless all the
E&M field modes are coupled in as well. This is a consequence of the no
go theorem for the existence of a relativistically invariant
two-particle potential. However, in approximations, the infinite
dimensional system of equations including the E&M field can be
truncated to a finite one, like for example the two-particle Dirac
equation with an effective interaction given to order 1/c^2.

> Previously, i cited Weinberg comments on bounded states but i cited
> incorrectly the page, p560:
>
> "It must be said that the theory of relativistic effects and radiative
> corrections in bound states is not yet in entirely satisfactory shape".

On the same page Weinberg talks about the difficulty in finding the
right approximation to use to reduce the full QED theory to an
approximation in which bound state calculations are possible. The same
situation is seen in all of theoretical physics whenever a simple
explanation is sought for complex phenomena, which range from weather
prediction, to high temperature superconductors, the two-body problem
in general relativity.

> Moreover, in curved spacetimes, the particles theories can be done
> rigorous using Synge parallel propagators. Wald 'and company' are wrong
> about particle theories cannot 'work' in curved spacetimes.
> Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Juan R. wrote:
> > Weinberg begins from we know (and we can measure) that is particles,
> > QM, and relativity. Anyone who claim that we can measure a field simply
> > does not know that a field is! As stated by Weinberg in Chapter 3 we
> > measure particles in particle physics experiments.
>
> Weimberg (p200):
>
> "Traditionally in quantum field theory one begins with such field
> equations, or with the Lagrangian from which they are derived, and then
> uses them to derive the expansion of the fields in terms of
> one-particle annihilation and creation operators. In the approach
> followed here, we start with the particles, and derive the fields
> according to the dictates of Lorentz invariance, /with the field
> equations arising almost incidentally as a byproduct of this
> construction/."
>
> Emphasis mine. I will remark again, that Weimberg begins from
> ONE-particle states and *derives* fields. But if one begins from FULL
> N-particle states (do NOT factorization into N one-particle states,
> which is ONLY valid on asymptotic regimes of scattering matrix) there
> is not posibility for a rigorous EXACT derivation of fields. The field
> vanishes Remember that both Dirac and KG equations are valid only for
> ONE-particle systems. There is nothing like a 'two-particle Dirac
> equation' (IN RIGOR) for relativistic particles.

The fact that one can derive fields starting with multiparticle states
is the main theorem of second quantization and has been known since
Dirac, Pauli, and others. I don't understand the point you are trying
to make about states that factor and those that don't. Multiparticle
states that factor form a basis for all other ones. Whether you are
dealing with only the basis vectors of the whole Hilbert space is
irrelevant with respect to the construction of field operators.

You also seem to be confused about the KG and Dirac field equations.
There are several distinct objects that satisfy them. Classical fields
(which should be Grassmann valued for fermions), the quantum field
operators, and the (single particle) wave functions. Multiparticle wave
functions are also possible. However, the equations of motion that they
satisfy are not closed (unlike the single particle case) unless all the
E&M field modes are coupled in as well. This is a consequence of the no
go theorem for the existence of a relativistically invariant
two-particle potential. However, in approximations, the infinite
dimensional system of equations including the E&M field can be
truncated to a finite one, like for example the two-particle Dirac
equation with an effective interaction given to order 1/c^2.

> Previously, i cited Weinberg comments on bounded states but i cited
> incorrectly the page, p560:
>
> "It must be said that the theory of relativistic effects and radiative
> corrections in bound states is not yet in entirely satisfactory shape".

On the same page Weinberg talks about the difficulty in finding the
right approximation to use to reduce the full QED theory to an
approximation in which bound state calculations are possible. The same
situation is seen in all of theoretical physics whenever a simple
explanation is sought for complex phenomena, which range from weather
prediction, to high temperature superconductors, the two-body problem
in general relativity.

> Moreover, in curved spacetimes, the particles theories can be done
> rigorous using Synge parallel propagators. Wald 'and company' are wrong
> about particle theories cannot 'work' in curved spacetimes.
> Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Juan R. wrote:
> > Weinberg begins from we know (and we can measure) that is particles,
> > QM, and relativity. Anyone who claim that we can measure a field simply
> > does not know that a field is! As stated by Weinberg in Chapter 3 we
> > measure particles in particle physics experiments.
>
> Weimberg (p200):
>
> "Traditionally in quantum field theory one begins with such field
> equations, or with the Lagrangian from which they are derived, and then
> uses them to derive the expansion of the fields in terms of
> one-particle annihilation and creation operators. In the approach
> followed here, we start with the particles, and derive the fields
> according to the dictates of Lorentz invariance, /with the field
> equations arising almost incidentally as a byproduct of this
> construction/."
>
> Emphasis mine. I will remark again, that Weimberg begins from
> ONE-particle states and *derives* fields. But if one begins from FULL
> N-particle states (do NOT factorization into N one-particle states,
> which is ONLY valid on asymptotic regimes of scattering matrix) there
> is not posibility for a rigorous EXACT derivation of fields. The field
> vanishes Remember that both Dirac and KG equations are valid only for
> ONE-particle systems. There is nothing like a 'two-particle Dirac
> equation' (IN RIGOR) for relativistic particles.

The fact that one can derive fields starting with multiparticle states
is the main theorem of second quantization and has been known since
Dirac, Pauli, and others. I don't understand the point you are trying
to make about states that factor and those that don't. Multiparticle
states that factor form a basis for all other ones. Whether you are
dealing with only the basis vectors of the whole Hilbert space is
irrelevant with respect to the construction of field operators.

You also seem to be confused about the KG and Dirac field equations.
There are several distinct objects that satisfy them. Classical fields
(which should be Grassmann valued for fermions), the quantum field
operators, and the (single particle) wave functions. Multiparticle wave
functions are also possible. However, the equations of motion that they
satisfy are not closed (unlike the single particle case) unless all the
E&M field modes are coupled in as well. This is a consequence of the no
go theorem for the existence of a relativistically invariant
two-particle potential. However, in approximations, the infinite
dimensional system of equations including the E&M field can be
truncated to a finite one, like for example the two-particle Dirac
equation with an effective interaction given to order 1/c^2.

> Previously, i cited Weinberg comments on bounded states but i cited
> incorrectly the page, p560:
>
> "It must be said that the theory of relativistic effects and radiative
> corrections in bound states is not yet in entirely satisfactory shape".

On the same page Weinberg talks about the difficulty in finding the
right approximation to use to reduce the full QED theory to an
approximation in which bound state calculations are possible. The same
situation is seen in all of theoretical physics whenever a simple
explanation is sought for complex phenomena, which range from weather
prediction, to high temperature superconductors, the two-body problem
in general relativity.

> Moreover, in curved spacetimes, the particles theories can be done
> rigorous using Synge parallel propagators. Wald 'and company' are wrong
> about particle theories cannot 'work' in curved spacetimes.
> Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Juan R. wrote:
> > Weinberg begins from we know (and we can measure) that is particles,
> > QM, and relativity. Anyone who claim that we can measure a field simply
> > does not know that a field is! As stated by Weinberg in Chapter 3 we
> > measure particles in particle physics experiments.
>
> Weimberg (p200):
>
> "Traditionally in quantum field theory one begins with such field
> equations, or with the Lagrangian from which they are derived, and then
> uses them to derive the expansion of the fields in terms of
> one-particle annihilation and creation operators. In the approach
> followed here, we start with the particles, and derive the fields
> according to the dictates of Lorentz invariance, /with the field
> equations arising almost incidentally as a byproduct of this
> construction/."
>
> Emphasis mine. I will remark again, that Weimberg begins from
> ONE-particle states and *derives* fields. But if one begins from FULL
> N-particle states (do NOT factorization into N one-particle states,
> which is ONLY valid on asymptotic regimes of scattering matrix) there
> is not posibility for a rigorous EXACT derivation of fields. The field
> vanishes Remember that both Dirac and KG equations are valid only for
> ONE-particle systems. There is nothing like a 'two-particle Dirac
> equation' (IN RIGOR) for relativistic particles.

The fact that one can derive fields starting with multiparticle states
is the main theorem of second quantization and has been known since
Dirac, Pauli, and others. I don't understand the point you are trying
to make about states that factor and those that don't. Multiparticle
states that factor form a basis for all other ones. Whether you are
dealing with only the basis vectors of the whole Hilbert space is
irrelevant with respect to the construction of field operators.

You also seem to be confused about the KG and Dirac field equations.
There are several distinct objects that satisfy them. Classical fields
(which should be Grassmann valued for fermions), the quantum field
operators, and the (single particle) wave functions. Multiparticle wave
functions are also possible. However, the equations of motion that they
satisfy are not closed (unlike the single particle case) unless all the
E&M field modes are coupled in as well. This is a consequence of the no
go theorem for the existence of a relativistically invariant
two-particle potential. However, in approximations, the infinite
dimensional system of equations including the E&M field can be
truncated to a finite one, like for example the two-particle Dirac
equation with an effective interaction given to order 1/c^2.

> Previously, i cited Weinberg comments on bounded states but i cited
> incorrectly the page, p560:
>
> "It must be said that the theory of relativistic effects and radiative
> corrections in bound states is not yet in entirely satisfactory shape".

On the same page Weinberg talks about the difficulty in finding the
right approximation to use to reduce the full QED theory to an
approximation in which bound state calculations are possible. The same
situation is seen in all of theoretical physics whenever a simple
explanation is sought for complex phenomena, which range from weather
prediction, to high temperature superconductors, the two-body problem
in general relativity.

> Moreover, in curved spacetimes, the particles theories can be done
> rigorous using Synge parallel propagators. Wald 'and company' are wrong
> about particle theories cannot 'work' in curved spacetimes.
> Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Juan R. wrote:
> > Weinberg begins from we know (and we can measure) that is particles,
> > QM, and relativity. Anyone who claim that we can measure a field simply
> > does not know that a field is! As stated by Weinberg in Chapter 3 we
> > measure particles in particle physics experiments.
>
> Weimberg (p200):
>
> "Traditionally in quantum field theory one begins with such field
> equations, or with the Lagrangian from which they are derived, and then
> uses them to derive the expansion of the fields in terms of
> one-particle annihilation and creation operators. In the approach
> followed here, we start with the particles, and derive the fields
> according to the dictates of Lorentz invariance, /with the field
> equations arising almost incidentally as a byproduct of this
> construction/."
>
> Emphasis mine. I will remark again, that Weimberg begins from
> ONE-particle states and *derives* fields. But if one begins from FULL
> N-particle states (do NOT factorization into N one-particle states,
> which is ONLY valid on asymptotic regimes of scattering matrix) there
> is not posibility for a rigorous EXACT derivation of fields. The field
> vanishes Remember that both Dirac and KG equations are valid only for
> ONE-particle systems. There is nothing like a 'two-particle Dirac
> equation' (IN RIGOR) for relativistic particles.

The fact that one can derive fields starting with multiparticle states
is the main theorem of second quantization and has been known since
Dirac, Pauli, and others. I don't understand the point you are trying
to make about states that factor and those that don't. Multiparticle
states that factor form a basis for all other ones. Whether you are
dealing with only the basis vectors of the whole Hilbert space is
irrelevant with respect to the construction of field operators.

You also seem to be confused about the KG and Dirac field equations.
There are several distinct objects that satisfy them. Classical fields
(which should be Grassmann valued for fermions), the quantum field
operators, and the (single particle) wave functions. Multiparticle wave
functions are also possible. However, the equations of motion that they
satisfy are not closed (unlike the single particle case) unless all the
E&M field modes are coupled in as well. This is a consequence of the no
go theorem for the existence of a relativistically invariant
two-particle potential. However, in approximations, the infinite
dimensional system of equations including the E&M field can be
truncated to a finite one, like for example the two-particle Dirac
equation with an effective interaction given to order 1/c^2.

> Previously, i cited Weinberg comments on bounded states but i cited
> incorrectly the page, p560:
>
> "It must be said that the theory of relativistic effects and radiative
> corrections in bound states is not yet in entirely satisfactory shape".

On the same page Weinberg talks about the difficulty in finding the
right approximation to use to reduce the full QED theory to an
approximation in which bound state calculations are possible. The same
situation is seen in all of theoretical physics whenever a simple
explanation is sought for complex phenomena, which range from weather
prediction, to high temperature superconductors, the two-body problem
in general relativity.

> Moreover, in curved spacetimes, the particles theories can be done
> rigorous using Synge parallel propagators. Wald 'and company' are wrong
> about particle theories cannot 'work' in curved spacetimes.
> Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Juan R. wrote:
> > Weinberg begins from we know (and we can measure) that is particles,
> > QM, and relativity. Anyone who claim that we can measure a field simply
> > does not know that a field is! As stated by Weinberg in Chapter 3 we
> > measure particles in particle physics experiments.
>
> Weimberg (p200):
>
> "Traditionally in quantum field theory one begins with such field
> equations, or with the Lagrangian from which they are derived, and then
> uses them to derive the expansion of the fields in terms of
> one-particle annihilation and creation operators. In the approach
> followed here, we start with the particles, and derive the fields
> according to the dictates of Lorentz invariance, /with the field
> equations arising almost incidentally as a byproduct of this
> construction/."
>
> Emphasis mine. I will remark again, that Weimberg begins from
> ONE-particle states and *derives* fields. But if one begins from FULL
> N-particle states (do NOT factorization into N one-particle states,
> which is ONLY valid on asymptotic regimes of scattering matrix) there
> is not posibility for a rigorous EXACT derivation of fields. The field
> vanishes Remember that both Dirac and KG equations are valid only for
> ONE-particle systems. There is nothing like a 'two-particle Dirac
> equation' (IN RIGOR) for relativistic particles.

The fact that one can derive fields starting with multiparticle states
is the main theorem of second quantization and has been known since
Dirac, Pauli, and others. I don't understand the point you are trying
to make about states that factor and those that don't. Multiparticle
states that factor form a basis for all other ones. Whether you are
dealing with only the basis vectors of the whole Hilbert space is
irrelevant with respect to the construction of field operators.

You also seem to be confused about the KG and Dirac field equations.
There are several distinct objects that satisfy them. Classical fields
(which should be Grassmann valued for fermions), the quantum field
operators, and the (single particle) wave functions. Multiparticle wave
functions are also possible. However, the equations of motion that they
satisfy are not closed (unlike the single particle case) unless all the
E&M field modes are coupled in as well. This is a consequence of the no
go theorem for the existence of a relativistically invariant
two-particle potential. However, in approximations, the infinite
dimensional system of equations including the E&M field can be
truncated to a finite one, like for example the two-particle Dirac
equation with an effective interaction given to order 1/c^2.

> Previously, i cited Weinberg comments on bounded states but i cited
> incorrectly the page, p560:
>
> "It must be said that the theory of relativistic effects and radiative
> corrections in bound states is not yet in entirely satisfactory shape".

On the same page Weinberg talks about the difficulty in finding the
right approximation to use to reduce the full QED theory to an
approximation in which bound state calculations are possible. The same
situation is seen in all of theoretical physics whenever a simple
explanation is sought for complex phenomena, which range from weather
prediction, to high temperature superconductors, the two-body problem
in general relativity.

> Moreover, in curved spacetimes, the particles theories can be done
> rigorous using Synge parallel propagators. Wald 'and company' are wrong
> about particle theories cannot 'work' in curved spacetimes.
> Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Juan R. wrote:
> > Weinberg begins from we know (and we can measure) that is particles,
> > QM, and relativity. Anyone who claim that we can measure a field simply
> > does not know that a field is! As stated by Weinberg in Chapter 3 we
> > measure particles in particle physics experiments.
>
> Weimberg (p200):
>
> "Traditionally in quantum field theory one begins with such field
> equations, or with the Lagrangian from which they are derived, and then
> uses them to derive the expansion of the fields in terms of
> one-particle annihilation and creation operators. In the approach
> followed here, we start with the particles, and derive the fields
> according to the dictates of Lorentz invariance, /with the field
> equations arising almost incidentally as a byproduct of this
> construction/."
>
> Emphasis mine. I will remark again, that Weimberg begins from
> ONE-particle states and *derives* fields. But if one begins from FULL
> N-particle states (do NOT factorization into N one-particle states,
> which is ONLY valid on asymptotic regimes of scattering matrix) there
> is not posibility for a rigorous EXACT derivation of fields. The field
> vanishes Remember that both Dirac and KG equations are valid only for
> ONE-particle systems. There is nothing like a 'two-particle Dirac
> equation' (IN RIGOR) for relativistic particles.

The fact that one can derive fields starting with multiparticle states
is the main theorem of second quantization and has been known since
Dirac, Pauli, and others. I don't understand the point you are trying
to make about states that factor and those that don't. Multiparticle
states that factor form a basis for all other ones. Whether you are
dealing with only the basis vectors of the whole Hilbert space is
irrelevant with respect to the construction of field operators.

You also seem to be confused about the KG and Dirac field equations.
There are several distinct objects that satisfy them. Classical fields
(which should be Grassmann valued for fermions), the quantum field
operators, and the (single particle) wave functions. Multiparticle wave
functions are also possible. However, the equations of motion that they
satisfy are not closed (unlike the single particle case) unless all the
E&M field modes are coupled in as well. This is a consequence of the no
go theorem for the existence of a relativistically invariant
two-particle potential. However, in approximations, the infinite
dimensional system of equations including the E&M field can be
truncated to a finite one, like for example the two-particle Dirac
equation with an effective interaction given to order 1/c^2.

> Previously, i cited Weinberg comments on bounded states but i cited
> incorrectly the page, p560:
>
> "It must be said that the theory of relativistic effects and radiative
> corrections in bound states is not yet in entirely satisfactory shape".

On the same page Weinberg talks about the difficulty in finding the
right approximation to use to reduce the full QED theory to an
approximation in which bound state calculations are possible. The same
situation is seen in all of theoretical physics whenever a simple
explanation is sought for complex phenomena, which range from weather
prediction, to high temperature superconductors, the two-body problem
in general relativity.

> Moreover, in curved spacetimes, the particles theories can be done
> rigorous using Synge parallel propagators. Wald 'and company' are wrong
> about particle theories cannot 'work' in curved spacetimes.
> Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Reference please.

Igor

[Igor Khavkine]

Juan R. wrote:
> Juan R. wrote:
> > Weinberg begins from we know (and we can measure) that is particles,
> > QM, and relativity. Anyone who claim that we can measure a field simply
> > does not know that a field is! As stated by Weinberg in Chapter 3 we
> > measure particles in particle physics experiments.
>
> Weimberg (p200):
>
> "Traditionally in quantum field theory one begins with such field
> equations, or with the Lagrangian from which they are derived, and then
> uses them to derive the expansion of the fields in terms of
> one-particle annihilation and creation operators. In the approach
> followed here, we start with the particles, and derive the fields
> according to the dictates of Lorentz invariance, /with the field
> equations arising almost incidentally as a byproduct of this
> construction/."
>
> Emphasis mine. I will remark again, that Weimberg begins from
> ONE-particle states and *derives* fields. But if one begins from FULL
> N-particle states (do NOT factorization into N one-particle states,
> which is ONLY valid on asymptotic regimes of scattering matrix) there
> is not posibility for a rigorous EXACT derivation of fields. The field
> vanishes Remember that both Dirac and KG equations are valid only for
> ONE-particle systems. There is nothing like a 'two-particle Dirac
> equation' (IN RIGOR) for relativistic particles.

The fact that one can derive fields starting with multiparticle states
is the main theorem of second quantization and has been known since
Dirac, Pauli, and others. I don't understand the point you are trying
to make about states that factor and those that don't. Multiparticle
states that factor form a basis for all other ones. Whether you are
dealing with only the basis vectors of the whole Hilbert space is
irrelevant with respect to the construction of field operators.

You also seem to be confused about the KG and Dirac field equations.
There are several distinct objects that satisfy them. Classical fields
(which should be Grassmann valued for fermions), the quantum field
operators, and the (single particle) wave functions. Multiparticle wave
functions are also possible. However, the equations of motion that they
satisfy are not closed (unlike the single particle case) unless all the
E&M field modes are coupled in as well. This is a consequence of the no
go theorem for the existence of a relativistically invariant
two-particle potential. However, in approximations, the infinite
dimensional system of equations including the E&M field can be
truncated to a finite one, like for example the two-particle Dirac
equation with an effective interaction given to order 1/c^2.

> Previously, i cited Weinberg comments on bounded states but i cited
> incorrectly the page, p560:
>
> "It must be said that the theory of relativistic effects and radiative
> corrections in bound states is not yet in entirely satisfactory shape".

On the same page Weinberg talks about the difficulty in finding the
right approximation to use to reduce the full QED theory to an
approximation in which bound state calculations are possible. The same
situation is seen in all of theoretical physics whenever a simple
explanation is sought for complex phenomena, which range from weather
prediction, to high temperature superconductors, the two-body problem
in general relativity.

> Moreover, in curved spacetimes, the particles theories can be done
> rigorous using Synge parallel propagators. Wald 'and company' are wrong
> about particle theories cannot 'work' in curved spacetimes.
> Hoyle/Narkilar theory in curved spacetime is a beatiful example...

Reference please.

Igor

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1127528209.775506.155930@o13g2000cwo.googlegr oups.com...

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.
> Therefore, any consequence of Haag's theorem will apply to such a
> theory equally well. And, unlike you claim, Haag's theorem is not a
> major obstacle. The non-trivial representations of the operator algebra
> do exist and they are implicitly (perturbatively) constructed during
> any QFT calculation involving interactions.

Regarding Haag's theorem, I found an interesting reference in which all
relevant constructions are performed explicitly:

H. Kita, "A non-trivial example of a relativistic quantum theory of
particles without divergence difficulties", Prog. Theor. Phys. 35
(1966), 934.

Kita builds a relativistically invariant theory in which all ten
generators of the Poincare group are given as explicit functions of
creation and annihilation operators. Then he constructs the interacting
Heisenberg field and explicitly calculates how this field transforms
with respect to boost transformations with interacting generators. This
interacting field does not transform as scalar, in full agreement with
the Haag's theorem.

Does this mean that Kita's theory is wrong or unacceptable? Not at all.
The non-scalar transformation law of the interacting field does not
change the fact that the theory is fully relativistic.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1127528209.775506.155930@o13g2000cwo.googlegr oups.com...

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.
> Therefore, any consequence of Haag's theorem will apply to such a
> theory equally well. And, unlike you claim, Haag's theorem is not a
> major obstacle. The non-trivial representations of the operator algebra
> do exist and they are implicitly (perturbatively) constructed during
> any QFT calculation involving interactions.

Regarding Haag's theorem, I found an interesting reference in which all
relevant constructions are performed explicitly:

H. Kita, "A non-trivial example of a relativistic quantum theory of
particles without divergence difficulties", Prog. Theor. Phys. 35
(1966), 934.

Kita builds a relativistically invariant theory in which all ten
generators of the Poincare group are given as explicit functions of
creation and annihilation operators. Then he constructs the interacting
Heisenberg field and explicitly calculates how this field transforms
with respect to boost transformations with interacting generators. This
interacting field does not transform as scalar, in full agreement with
the Haag's theorem.

Does this mean that Kita's theory is wrong or unacceptable? Not at all.
The non-scalar transformation law of the interacting field does not
change the fact that the theory is fully relativistic.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1127528209.775506.155930@o13g2000cwo.googlegr oups.com...

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.
> Therefore, any consequence of Haag's theorem will apply to such a
> theory equally well. And, unlike you claim, Haag's theorem is not a
> major obstacle. The non-trivial representations of the operator algebra
> do exist and they are implicitly (perturbatively) constructed during
> any QFT calculation involving interactions.

Regarding Haag's theorem, I found an interesting reference in which all
relevant constructions are performed explicitly:

H. Kita, "A non-trivial example of a relativistic quantum theory of
particles without divergence difficulties", Prog. Theor. Phys. 35
(1966), 934.

Kita builds a relativistically invariant theory in which all ten
generators of the Poincare group are given as explicit functions of
creation and annihilation operators. Then he constructs the interacting
Heisenberg field and explicitly calculates how this field transforms
with respect to boost transformations with interacting generators. This
interacting field does not transform as scalar, in full agreement with
the Haag's theorem.

Does this mean that Kita's theory is wrong or unacceptable? Not at all.
The non-scalar transformation law of the interacting field does not
change the fact that the theory is fully relativistic.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1127528209.775506.155930@o13g2000cwo.googlegr oups.com...

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.
> Therefore, any consequence of Haag's theorem will apply to such a
> theory equally well. And, unlike you claim, Haag's theorem is not a
> major obstacle. The non-trivial representations of the operator algebra
> do exist and they are implicitly (perturbatively) constructed during
> any QFT calculation involving interactions.

Regarding Haag's theorem, I found an interesting reference in which all
relevant constructions are performed explicitly:

H. Kita, "A non-trivial example of a relativistic quantum theory of
particles without divergence difficulties", Prog. Theor. Phys. 35
(1966), 934.

Kita builds a relativistically invariant theory in which all ten
generators of the Poincare group are given as explicit functions of
creation and annihilation operators. Then he constructs the interacting
Heisenberg field and explicitly calculates how this field transforms
with respect to boost transformations with interacting generators. This
interacting field does not transform as scalar, in full agreement with
the Haag's theorem.

Does this mean that Kita's theory is wrong or unacceptable? Not at all.
The non-scalar transformation law of the interacting field does not
change the fact that the theory is fully relativistic.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1127528209.775506.155930@o13g2000cwo.googlegr oups.com...

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.
> Therefore, any consequence of Haag's theorem will apply to such a
> theory equally well. And, unlike you claim, Haag's theorem is not a
> major obstacle. The non-trivial representations of the operator algebra
> do exist and they are implicitly (perturbatively) constructed during
> any QFT calculation involving interactions.

Regarding Haag's theorem, I found an interesting reference in which all
relevant constructions are performed explicitly:

H. Kita, "A non-trivial example of a relativistic quantum theory of
particles without divergence difficulties", Prog. Theor. Phys. 35
(1966), 934.

Kita builds a relativistically invariant theory in which all ten
generators of the Poincare group are given as explicit functions of
creation and annihilation operators. Then he constructs the interacting
Heisenberg field and explicitly calculates how this field transforms
with respect to boost transformations with interacting generators. This
interacting field does not transform as scalar, in full agreement with
the Haag's theorem.

Does this mean that Kita's theory is wrong or unacceptable? Not at all.
The non-scalar transformation law of the interacting field does not
change the fact that the theory is fully relativistic.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1127528209.775506.155930@o13g2000cwo.googlegr oups.com...

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.
> Therefore, any consequence of Haag's theorem will apply to such a
> theory equally well. And, unlike you claim, Haag's theorem is not a
> major obstacle. The non-trivial representations of the operator algebra
> do exist and they are implicitly (perturbatively) constructed during
> any QFT calculation involving interactions.

Regarding Haag's theorem, I found an interesting reference in which all
relevant constructions are performed explicitly:

H. Kita, "A non-trivial example of a relativistic quantum theory of
particles without divergence difficulties", Prog. Theor. Phys. 35
(1966), 934.

Kita builds a relativistically invariant theory in which all ten
generators of the Poincare group are given as explicit functions of
creation and annihilation operators. Then he constructs the interacting
Heisenberg field and explicitly calculates how this field transforms
with respect to boost transformations with interacting generators. This
interacting field does not transform as scalar, in full agreement with
the Haag's theorem.

Does this mean that Kita's theory is wrong or unacceptable? Not at all.
The non-scalar transformation law of the interacting field does not
change the fact that the theory is fully relativistic.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1127528209.775506.155930@o13g2000cwo.googlegr oups.com...

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.
> Therefore, any consequence of Haag's theorem will apply to such a
> theory equally well. And, unlike you claim, Haag's theorem is not a
> major obstacle. The non-trivial representations of the operator algebra
> do exist and they are implicitly (perturbatively) constructed during
> any QFT calculation involving interactions.

Regarding Haag's theorem, I found an interesting reference in which all
relevant constructions are performed explicitly:

H. Kita, "A non-trivial example of a relativistic quantum theory of
particles without divergence difficulties", Prog. Theor. Phys. 35
(1966), 934.

Kita builds a relativistically invariant theory in which all ten
generators of the Poincare group are given as explicit functions of
creation and annihilation operators. Then he constructs the interacting
Heisenberg field and explicitly calculates how this field transforms
with respect to boost transformations with interacting generators. This
interacting field does not transform as scalar, in full agreement with
the Haag's theorem.

Does this mean that Kita's theory is wrong or unacceptable? Not at all.
The non-scalar transformation law of the interacting field does not
change the fact that the theory is fully relativistic.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1127528209.775506.155930@o13g2000cwo.googlegr oups.com...

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.
> Therefore, any consequence of Haag's theorem will apply to such a
> theory equally well. And, unlike you claim, Haag's theorem is not a
> major obstacle. The non-trivial representations of the operator algebra
> do exist and they are implicitly (perturbatively) constructed during
> any QFT calculation involving interactions.

Regarding Haag's theorem, I found an interesting reference in which all
relevant constructions are performed explicitly:

H. Kita, "A non-trivial example of a relativistic quantum theory of
particles without divergence difficulties", Prog. Theor. Phys. 35
(1966), 934.

Kita builds a relativistically invariant theory in which all ten
generators of the Poincare group are given as explicit functions of
creation and annihilation operators. Then he constructs the interacting
Heisenberg field and explicitly calculates how this field transforms
with respect to boost transformations with interacting generators. This
interacting field does not transform as scalar, in full agreement with
the Haag's theorem.

Does this mean that Kita's theory is wrong or unacceptable? Not at all.
The non-scalar transformation law of the interacting field does not
change the fact that the theory is fully relativistic.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1127528209.775506.155930@o13g2000cwo.googlegr oups.com...

> > The statement of [Haag's] the theorem is that interacting quantum field
> > (i.e., the one whose time evolution is described by the full interacting
> > Hamiltonian) cannot have usual tensor transformations with respect to
> > Lorentz boosts. If you insist on the field-based description of nature,
> > then this theorem is a big obstacle. However, this theorem can be safely
> > ignored in the particle-based description. I do not assign any physical
> > meaning to free fields and to interacting fields. So, I do not care
> > whether interacting field is "Lorentz invariant" or not.
>
> As soon as you have a Fock space, you have a field theory. This is a
> mathematical fact, no matter how you constructed the Fock space.
> Therefore, any consequence of Haag's theorem will apply to such a
> theory equally well. And, unlike you claim, Haag's theorem is not a
> major obstacle. The non-trivial representations of the operator algebra
> do exist and they are implicitly (perturbatively) constructed during
> any QFT calculation involving interactions.

Regarding Haag's theorem, I found an interesting reference in which all
relevant constructions are performed explicitly:

H. Kita, "A non-trivial example of a relativistic quantum theory of
particles without divergence difficulties", Prog. Theor. Phys. 35
(1966), 934.

Kita builds a relativistically invariant theory in which all ten
generators of the Poincare group are given as explicit functions of
creation and annihilation operators. Then he constructs the interacting
Heisenberg field and explicitly calculates how this field transforms
with respect to boost transformations with interacting generators. This
interacting field does not transform as scalar, in full agreement with
the Haag's theorem.

Does this mean that Kita's theory is wrong or unacceptable? Not at all.
The non-scalar transformation law of the interacting field does not
change the fact that the theory is fully relativistic.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128800211.416846.280170@o13g2000cwo.googlegr oups.com...

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

No objections about classical and quantum fields. However,
single particle wave functions do not satisfy KG or Dirac equations.
Single particle wave functions are (by definition) probability amplitudes.
Their transformations with respect to the Poincare group must preserve
the probabilities in all reference frames. Thus, single particle wave
functions
must form a Hilbert space of the irreducible unitary representation of the
Poincare group. Such representations were constructed and analyzed by
E.P. Wigner in 1939. Based on Wigner's work one can wride down
the relativistic analog of the Schroedinger equation for free particles

i hbar d psi(p,t)/dt = sqrt(p^2 + m^2) psi(p,t)

which is valid for particles of any spin. KG is not a valid analog of
the Schroedinger equation for spin-0 particles, because it involves the
2nd time derivative. Dirac equation is not a valid analog of
the Schroedinger equation for spin-1/2 particles, because it is formulated
for 4-component functions, while according to Wigner (and to experiment)
the wave functions of spin-1/2 particles have only 2 components.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128800211.416846.280170@o13g2000cwo.googlegr oups.com...

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

No objections about classical and quantum fields. However,
single particle wave functions do not satisfy KG or Dirac equations.
Single particle wave functions are (by definition) probability amplitudes.
Their transformations with respect to the Poincare group must preserve
the probabilities in all reference frames. Thus, single particle wave
functions
must form a Hilbert space of the irreducible unitary representation of the
Poincare group. Such representations were constructed and analyzed by
E.P. Wigner in 1939. Based on Wigner's work one can wride down
the relativistic analog of the Schroedinger equation for free particles

i hbar d psi(p,t)/dt = sqrt(p^2 + m^2) psi(p,t)

which is valid for particles of any spin. KG is not a valid analog of
the Schroedinger equation for spin-0 particles, because it involves the
2nd time derivative. Dirac equation is not a valid analog of
the Schroedinger equation for spin-1/2 particles, because it is formulated
for 4-component functions, while according to Wigner (and to experiment)
the wave functions of spin-1/2 particles have only 2 components.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128800211.416846.280170@o13g2000cwo.googlegr oups.com...

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

No objections about classical and quantum fields. However,
single particle wave functions do not satisfy KG or Dirac equations.
Single particle wave functions are (by definition) probability amplitudes.
Their transformations with respect to the Poincare group must preserve
the probabilities in all reference frames. Thus, single particle wave
functions
must form a Hilbert space of the irreducible unitary representation of the
Poincare group. Such representations were constructed and analyzed by
E.P. Wigner in 1939. Based on Wigner's work one can wride down
the relativistic analog of the Schroedinger equation for free particles

i hbar d psi(p,t)/dt = sqrt(p^2 + m^2) psi(p,t)

which is valid for particles of any spin. KG is not a valid analog of
the Schroedinger equation for spin-0 particles, because it involves the
2nd time derivative. Dirac equation is not a valid analog of
the Schroedinger equation for spin-1/2 particles, because it is formulated
for 4-component functions, while according to Wigner (and to experiment)
the wave functions of spin-1/2 particles have only 2 components.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128800211.416846.280170@o13g2000cwo.googlegr oups.com...

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

No objections about classical and quantum fields. However,
single particle wave functions do not satisfy KG or Dirac equations.
Single particle wave functions are (by definition) probability amplitudes.
Their transformations with respect to the Poincare group must preserve
the probabilities in all reference frames. Thus, single particle wave
functions
must form a Hilbert space of the irreducible unitary representation of the
Poincare group. Such representations were constructed and analyzed by
E.P. Wigner in 1939. Based on Wigner's work one can wride down
the relativistic analog of the Schroedinger equation for free particles

i hbar d psi(p,t)/dt = sqrt(p^2 + m^2) psi(p,t)

which is valid for particles of any spin. KG is not a valid analog of
the Schroedinger equation for spin-0 particles, because it involves the
2nd time derivative. Dirac equation is not a valid analog of
the Schroedinger equation for spin-1/2 particles, because it is formulated
for 4-component functions, while according to Wigner (and to experiment)
the wave functions of spin-1/2 particles have only 2 components.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128800211.416846.280170@o13g2000cwo.googlegr oups.com...

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

No objections about classical and quantum fields. However,
single particle wave functions do not satisfy KG or Dirac equations.
Single particle wave functions are (by definition) probability amplitudes.
Their transformations with respect to the Poincare group must preserve
the probabilities in all reference frames. Thus, single particle wave
functions
must form a Hilbert space of the irreducible unitary representation of the
Poincare group. Such representations were constructed and analyzed by
E.P. Wigner in 1939. Based on Wigner's work one can wride down
the relativistic analog of the Schroedinger equation for free particles

i hbar d psi(p,t)/dt = sqrt(p^2 + m^2) psi(p,t)

which is valid for particles of any spin. KG is not a valid analog of
the Schroedinger equation for spin-0 particles, because it involves the
2nd time derivative. Dirac equation is not a valid analog of
the Schroedinger equation for spin-1/2 particles, because it is formulated
for 4-component functions, while according to Wigner (and to experiment)
the wave functions of spin-1/2 particles have only 2 components.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128800211.416846.280170@o13g2000cwo.googlegr oups.com...

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

No objections about classical and quantum fields. However,
single particle wave functions do not satisfy KG or Dirac equations.
Single particle wave functions are (by definition) probability amplitudes.
Their transformations with respect to the Poincare group must preserve
the probabilities in all reference frames. Thus, single particle wave
functions
must form a Hilbert space of the irreducible unitary representation of the
Poincare group. Such representations were constructed and analyzed by
E.P. Wigner in 1939. Based on Wigner's work one can wride down
the relativistic analog of the Schroedinger equation for free particles

i hbar d psi(p,t)/dt = sqrt(p^2 + m^2) psi(p,t)

which is valid for particles of any spin. KG is not a valid analog of
the Schroedinger equation for spin-0 particles, because it involves the
2nd time derivative. Dirac equation is not a valid analog of
the Schroedinger equation for spin-1/2 particles, because it is formulated
for 4-component functions, while according to Wigner (and to experiment)
the wave functions of spin-1/2 particles have only 2 components.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128800211.416846.280170@o13g2000cwo.googlegr oups.com...

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

No objections about classical and quantum fields. However,
single particle wave functions do not satisfy KG or Dirac equations.
Single particle wave functions are (by definition) probability amplitudes.
Their transformations with respect to the Poincare group must preserve
the probabilities in all reference frames. Thus, single particle wave
functions
must form a Hilbert space of the irreducible unitary representation of the
Poincare group. Such representations were constructed and analyzed by
E.P. Wigner in 1939. Based on Wigner's work one can wride down
the relativistic analog of the Schroedinger equation for free particles

i hbar d psi(p,t)/dt = sqrt(p^2 + m^2) psi(p,t)

which is valid for particles of any spin. KG is not a valid analog of
the Schroedinger equation for spin-0 particles, because it involves the
2nd time derivative. Dirac equation is not a valid analog of
the Schroedinger equation for spin-1/2 particles, because it is formulated
for 4-component functions, while according to Wigner (and to experiment)
the wave functions of spin-1/2 particles have only 2 components.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128800211.416846.280170@o13g2000cwo.googlegr oups.com...

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

No objections about classical and quantum fields. However,
single particle wave functions do not satisfy KG or Dirac equations.
Single particle wave functions are (by definition) probability amplitudes.
Their transformations with respect to the Poincare group must preserve
the probabilities in all reference frames. Thus, single particle wave
functions
must form a Hilbert space of the irreducible unitary representation of the
Poincare group. Such representations were constructed and analyzed by
E.P. Wigner in 1939. Based on Wigner's work one can wride down
the relativistic analog of the Schroedinger equation for free particles

i hbar d psi(p,t)/dt = sqrt(p^2 + m^2) psi(p,t)

which is valid for particles of any spin. KG is not a valid analog of
the Schroedinger equation for spin-0 particles, because it involves the
2nd time derivative. Dirac equation is not a valid analog of
the Schroedinger equation for spin-1/2 particles, because it is formulated
for 4-component functions, while according to Wigner (and to experiment)
the wave functions of spin-1/2 particles have only 2 components.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1128800211.416846.280170@o13g2000cwo.googlegr oups.com...

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

No objections about classical and quantum fields. However,
single particle wave functions do not satisfy KG or Dirac equations.
Single particle wave functions are (by definition) probability amplitudes.
Their transformations with respect to the Poincare group must preserve
the probabilities in all reference frames. Thus, single particle wave
functions
must form a Hilbert space of the irreducible unitary representation of the
Poincare group. Such representations were constructed and analyzed by
E.P. Wigner in 1939. Based on Wigner's work one can wride down
the relativistic analog of the Schroedinger equation for free particles

i hbar d psi(p,t)/dt = sqrt(p^2 + m^2) psi(p,t)

which is valid for particles of any spin. KG is not a valid analog of
the Schroedinger equation for spin-0 particles, because it involves the
2nd time derivative. Dirac equation is not a valid analog of
the Schroedinger equation for spin-1/2 particles, because it is formulated
for 4-component functions, while according to Wigner (and to experiment)
the wave functions of spin-1/2 particles have only 2 components.

Eugene.

[Juan R.]

Igor Khavkine wrote:
> The fact that one can derive fields starting with multiparticle states
> is the main theorem of second quantization and has been known since
> Dirac, Pauli, and others. I don't understand the point you are trying
> to make about states that factor and those that don't. Multiparticle
> states that factor form a basis for all other ones. Whether you are
> dealing with only the basis vectors of the whole Hilbert space is
> irrelevant with respect to the construction of field operators.

One can derive fields begining from 'impure' multiparticle states that
can be factorized into one-body states. I was all time talking of pure
multiparticle states that does not factorize. The field description can
be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

You claim that you do not understand the point about states that
factorize and those that do not. Have you studied relativistic QFT? Why
may one study only scattering states in RQFT?

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

You are wrong. The (single particle) wave functions do not satisfy the
KG and Dirac field equations, ***in RIGOR***, and this was the reason
that both are abandoned in relativistic quantum field theory. In fact,
the evolution equation in R-QFT is a Schrödinger equation for
functionals. In RQFT, both KG and Dirac equations are field operator
identities, *newer* wavefunction equations.

> Multiparticle wave
> functions are also possible. However, the equations of motion that they
> satisfy are not closed (unlike the single particle case) unless all the
> E&M field modes are coupled in as well. This is a consequence of the no
> go theorem for the existence of a relativistically invariant
> two-particle potential. However, in approximations, the infinite
> dimensional system of equations including the E&M field can be
> truncated to a finite one, like for example the two-particle Dirac
> equation with an effective interaction given to order 1/c^2.

In rigor, multiparticle wave functions are NOT possible in relativistic
QFT, because there is not dynamical variables for those vawefunctions.
This is the reason that R-QFT can only deal with scattering states,
where N-particle states factorize and via conservation of momentum *FOR
FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
the ONE-body wavefunctions.

The so-called 'two-body wavefunctions' are not rigorous, and only in
stationary states could be partially interpreted if one is not very
precise and moreover one is only interested in some observables like
energy.

You claim that the equations of motion that they satisfy are not
closed, which is misleading. The equations of motion are *not* defined,
and all i know are mixed -totally inconsistent- approaches where, at
one hand, one defines /ad hoc/ two-body wavefunctions (for example with
16 components that are not wavefunctions of relativistic field theory)
and, at the other hand, one defines /ad hoc/ equations of 'motion' for
that two-body system (the dynamical equation in quantum field theory
*is* Schrödinger-like one).

But the equation of 'motion' is formed by single particle terms
inspired in Dirac equation (which is *wrong* in relativistic regimes,
because in RQFT the Dirac equation is an indentity for field operators
and has not wavefunction interpretation) more an interacting term.

Taking interacting term zero, one obtains two Dirac equations, which
all of us know are not correct, and this is the reason that were
abandoned in R-QFT by their field theoretic counterparts.

The interacting term is obtained from R-QFT. The term is mathematically
divergent and physically only explains scattering of particles in the
infinite past and future. What is the interaction term when particles
are not separated infinitely?

All that RQFT can do is to obtain a kind of *effective* interactions,
NEWER the real interaction for the particles in bound states. Do those
*effective* potentials derived from RQFT work? of course! but ONLY in
stationary states, where the detailed mechanism of interaction can be
ignored.

Of course, even ignoring that Coulomb-Breit U_CB or Coulomb-Gaunt U_CB
'effective' operators obtained from RQFT at second order in c^2 and e^2
are not well-founded even at stationary states. As it is well-known in
quantum chemistry, one may correct both via positive energy Casimir
type projection operators for the leptons

V_CB = Lambda_{++} U_CB Lambda_{++}

V_CG = Lambda_{++} U_CG Lambda_{++}

> > Previously, i cited Weinberg comments on bounded states but i cited
> > incorrectly the page, p560:
> >
> > "It must be said that the theory of relativistic effects and radiative
> > corrections in bound states is not yet in entirely satisfactory shape".
>
> On the same page Weinberg talks about the difficulty in finding the
> right approximation to use to reduce the full QED theory to an
> approximation in which bound state calculations are possible. The same
> situation is seen in all of theoretical physics whenever a simple
> explanation is sought for complex phenomena, which range from weather
> prediction, to high temperature superconductors, the two-body problem
> in general relativity.

There is no such one thing as full QED in bound states, because only
free particle states can be defined in QED. Moreover Weinberg said
'theory of' not 'application of'.

> > Moreover, in curved spacetimes, the particles theories can be done
> > rigorous using Synge parallel propagators. Wald 'and company' are wrong
> > about particle theories cannot 'work' in curved spacetimes.
> > Hoyle/Narkilar theory in curved spacetime is a beatiful example...
>
> Reference please.

For instance,

(1963) Proc R Soc London A 277, 1.

(1964) Proc R Soc London A 282, 184.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Igor Khavkine wrote:
> The fact that one can derive fields starting with multiparticle states
> is the main theorem of second quantization and has been known since
> Dirac, Pauli, and others. I don't understand the point you are trying
> to make about states that factor and those that don't. Multiparticle
> states that factor form a basis for all other ones. Whether you are
> dealing with only the basis vectors of the whole Hilbert space is
> irrelevant with respect to the construction of field operators.

One can derive fields begining from 'impure' multiparticle states that
can be factorized into one-body states. I was all time talking of pure
multiparticle states that does not factorize. The field description can
be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

You claim that you do not understand the point about states that
factorize and those that do not. Have you studied relativistic QFT? Why
may one study only scattering states in RQFT?

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

You are wrong. The (single particle) wave functions do not satisfy the
KG and Dirac field equations, ***in RIGOR***, and this was the reason
that both are abandoned in relativistic quantum field theory. In fact,
the evolution equation in R-QFT is a Schrödinger equation for
functionals. In RQFT, both KG and Dirac equations are field operator
identities, *newer* wavefunction equations.

> Multiparticle wave
> functions are also possible. However, the equations of motion that they
> satisfy are not closed (unlike the single particle case) unless all the
> E&M field modes are coupled in as well. This is a consequence of the no
> go theorem for the existence of a relativistically invariant
> two-particle potential. However, in approximations, the infinite
> dimensional system of equations including the E&M field can be
> truncated to a finite one, like for example the two-particle Dirac
> equation with an effective interaction given to order 1/c^2.

In rigor, multiparticle wave functions are NOT possible in relativistic
QFT, because there is not dynamical variables for those vawefunctions.
This is the reason that R-QFT can only deal with scattering states,
where N-particle states factorize and via conservation of momentum *FOR
FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
the ONE-body wavefunctions.

The so-called 'two-body wavefunctions' are not rigorous, and only in
stationary states could be partially interpreted if one is not very
precise and moreover one is only interested in some observables like
energy.

You claim that the equations of motion that they satisfy are not
closed, which is misleading. The equations of motion are *not* defined,
and all i know are mixed -totally inconsistent- approaches where, at
one hand, one defines /ad hoc/ two-body wavefunctions (for example with
16 components that are not wavefunctions of relativistic field theory)
and, at the other hand, one defines /ad hoc/ equations of 'motion' for
that two-body system (the dynamical equation in quantum field theory
*is* Schrödinger-like one).

But the equation of 'motion' is formed by single particle terms
inspired in Dirac equation (which is *wrong* in relativistic regimes,
because in RQFT the Dirac equation is an indentity for field operators
and has not wavefunction interpretation) more an interacting term.

Taking interacting term zero, one obtains two Dirac equations, which
all of us know are not correct, and this is the reason that were
abandoned in R-QFT by their field theoretic counterparts.

The interacting term is obtained from R-QFT. The term is mathematically
divergent and physically only explains scattering of particles in the
infinite past and future. What is the interaction term when particles
are not separated infinitely?

All that RQFT can do is to obtain a kind of *effective* interactions,
NEWER the real interaction for the particles in bound states. Do those
*effective* potentials derived from RQFT work? of course! but ONLY in
stationary states, where the detailed mechanism of interaction can be
ignored.

Of course, even ignoring that Coulomb-Breit U_CB or Coulomb-Gaunt U_CB
'effective' operators obtained from RQFT at second order in c^2 and e^2
are not well-founded even at stationary states. As it is well-known in
quantum chemistry, one may correct both via positive energy Casimir
type projection operators for the leptons

V_CB = Lambda_{++} U_CB Lambda_{++}

V_CG = Lambda_{++} U_CG Lambda_{++}

> > Previously, i cited Weinberg comments on bounded states but i cited
> > incorrectly the page, p560:
> >
> > "It must be said that the theory of relativistic effects and radiative
> > corrections in bound states is not yet in entirely satisfactory shape".
>
> On the same page Weinberg talks about the difficulty in finding the
> right approximation to use to reduce the full QED theory to an
> approximation in which bound state calculations are possible. The same
> situation is seen in all of theoretical physics whenever a simple
> explanation is sought for complex phenomena, which range from weather
> prediction, to high temperature superconductors, the two-body problem
> in general relativity.

There is no such one thing as full QED in bound states, because only
free particle states can be defined in QED. Moreover Weinberg said
'theory of' not 'application of'.

> > Moreover, in curved spacetimes, the particles theories can be done
> > rigorous using Synge parallel propagators. Wald 'and company' are wrong
> > about particle theories cannot 'work' in curved spacetimes.
> > Hoyle/Narkilar theory in curved spacetime is a beatiful example...
>
> Reference please.

For instance,

(1963) Proc R Soc London A 277, 1.

(1964) Proc R Soc London A 282, 184.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Igor Khavkine wrote:
> The fact that one can derive fields starting with multiparticle states
> is the main theorem of second quantization and has been known since
> Dirac, Pauli, and others. I don't understand the point you are trying
> to make about states that factor and those that don't. Multiparticle
> states that factor form a basis for all other ones. Whether you are
> dealing with only the basis vectors of the whole Hilbert space is
> irrelevant with respect to the construction of field operators.

One can derive fields begining from 'impure' multiparticle states that
can be factorized into one-body states. I was all time talking of pure
multiparticle states that does not factorize. The field description can
be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

You claim that you do not understand the point about states that
factorize and those that do not. Have you studied relativistic QFT? Why
may one study only scattering states in RQFT?

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

You are wrong. The (single particle) wave functions do not satisfy the
KG and Dirac field equations, ***in RIGOR***, and this was the reason
that both are abandoned in relativistic quantum field theory. In fact,
the evolution equation in R-QFT is a Schrödinger equation for
functionals. In RQFT, both KG and Dirac equations are field operator
identities, *newer* wavefunction equations.

> Multiparticle wave
> functions are also possible. However, the equations of motion that they
> satisfy are not closed (unlike the single particle case) unless all the
> E&M field modes are coupled in as well. This is a consequence of the no
> go theorem for the existence of a relativistically invariant
> two-particle potential. However, in approximations, the infinite
> dimensional system of equations including the E&M field can be
> truncated to a finite one, like for example the two-particle Dirac
> equation with an effective interaction given to order 1/c^2.

In rigor, multiparticle wave functions are NOT possible in relativistic
QFT, because there is not dynamical variables for those vawefunctions.
This is the reason that R-QFT can only deal with scattering states,
where N-particle states factorize and via conservation of momentum *FOR
FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
the ONE-body wavefunctions.

The so-called 'two-body wavefunctions' are not rigorous, and only in
stationary states could be partially interpreted if one is not very
precise and moreover one is only interested in some observables like
energy.

You claim that the equations of motion that they satisfy are not
closed, which is misleading. The equations of motion are *not* defined,
and all i know are mixed -totally inconsistent- approaches where, at
one hand, one defines /ad hoc/ two-body wavefunctions (for example with
16 components that are not wavefunctions of relativistic field theory)
and, at the other hand, one defines /ad hoc/ equations of 'motion' for
that two-body system (the dynamical equation in quantum field theory
*is* Schrödinger-like one).

But the equation of 'motion' is formed by single particle terms
inspired in Dirac equation (which is *wrong* in relativistic regimes,
because in RQFT the Dirac equation is an indentity for field operators
and has not wavefunction interpretation) more an interacting term.

Taking interacting term zero, one obtains two Dirac equations, which
all of us know are not correct, and this is the reason that were
abandoned in R-QFT by their field theoretic counterparts.

The interacting term is obtained from R-QFT. The term is mathematically
divergent and physically only explains scattering of particles in the
infinite past and future. What is the interaction term when particles
are not separated infinitely?

All that RQFT can do is to obtain a kind of *effective* interactions,
NEWER the real interaction for the particles in bound states. Do those
*effective* potentials derived from RQFT work? of course! but ONLY in
stationary states, where the detailed mechanism of interaction can be
ignored.

Of course, even ignoring that Coulomb-Breit U_CB or Coulomb-Gaunt U_CB
'effective' operators obtained from RQFT at second order in c^2 and e^2
are not well-founded even at stationary states. As it is well-known in
quantum chemistry, one may correct both via positive energy Casimir
type projection operators for the leptons

V_CB = Lambda_{++} U_CB Lambda_{++}

V_CG = Lambda_{++} U_CG Lambda_{++}

> > Previously, i cited Weinberg comments on bounded states but i cited
> > incorrectly the page, p560:
> >
> > "It must be said that the theory of relativistic effects and radiative
> > corrections in bound states is not yet in entirely satisfactory shape".
>
> On the same page Weinberg talks about the difficulty in finding the
> right approximation to use to reduce the full QED theory to an
> approximation in which bound state calculations are possible. The same
> situation is seen in all of theoretical physics whenever a simple
> explanation is sought for complex phenomena, which range from weather
> prediction, to high temperature superconductors, the two-body problem
> in general relativity.

There is no such one thing as full QED in bound states, because only
free particle states can be defined in QED. Moreover Weinberg said
'theory of' not 'application of'.

> > Moreover, in curved spacetimes, the particles theories can be done
> > rigorous using Synge parallel propagators. Wald 'and company' are wrong
> > about particle theories cannot 'work' in curved spacetimes.
> > Hoyle/Narkilar theory in curved spacetime is a beatiful example...
>
> Reference please.

For instance,

(1963) Proc R Soc London A 277, 1.

(1964) Proc R Soc London A 282, 184.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Igor Khavkine wrote:
> The fact that one can derive fields starting with multiparticle states
> is the main theorem of second quantization and has been known since
> Dirac, Pauli, and others. I don't understand the point you are trying
> to make about states that factor and those that don't. Multiparticle
> states that factor form a basis for all other ones. Whether you are
> dealing with only the basis vectors of the whole Hilbert space is
> irrelevant with respect to the construction of field operators.

One can derive fields begining from 'impure' multiparticle states that
can be factorized into one-body states. I was all time talking of pure
multiparticle states that does not factorize. The field description can
be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

You claim that you do not understand the point about states that
factorize and those that do not. Have you studied relativistic QFT? Why
may one study only scattering states in RQFT?

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

You are wrong. The (single particle) wave functions do not satisfy the
KG and Dirac field equations, ***in RIGOR***, and this was the reason
that both are abandoned in relativistic quantum field theory. In fact,
the evolution equation in R-QFT is a Schrödinger equation for
functionals. In RQFT, both KG and Dirac equations are field operator
identities, *newer* wavefunction equations.

> Multiparticle wave
> functions are also possible. However, the equations of motion that they
> satisfy are not closed (unlike the single particle case) unless all the
> E&M field modes are coupled in as well. This is a consequence of the no
> go theorem for the existence of a relativistically invariant
> two-particle potential. However, in approximations, the infinite
> dimensional system of equations including the E&M field can be
> truncated to a finite one, like for example the two-particle Dirac
> equation with an effective interaction given to order 1/c^2.

In rigor, multiparticle wave functions are NOT possible in relativistic
QFT, because there is not dynamical variables for those vawefunctions.
This is the reason that R-QFT can only deal with scattering states,
where N-particle states factorize and via conservation of momentum *FOR
FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
the ONE-body wavefunctions.

The so-called 'two-body wavefunctions' are not rigorous, and only in
stationary states could be partially interpreted if one is not very
precise and moreover one is only interested in some observables like
energy.

You claim that the equations of motion that they satisfy are not
closed, which is misleading. The equations of motion are *not* defined,
and all i know are mixed -totally inconsistent- approaches where, at
one hand, one defines /ad hoc/ two-body wavefunctions (for example with
16 components that are not wavefunctions of relativistic field theory)
and, at the other hand, one defines /ad hoc/ equations of 'motion' for
that two-body system (the dynamical equation in quantum field theory
*is* Schrödinger-like one).

But the equation of 'motion' is formed by single particle terms
inspired in Dirac equation (which is *wrong* in relativistic regimes,
because in RQFT the Dirac equation is an indentity for field operators
and has not wavefunction interpretation) more an interacting term.

Taking interacting term zero, one obtains two Dirac equations, which
all of us know are not correct, and this is the reason that were
abandoned in R-QFT by their field theoretic counterparts.

The interacting term is obtained from R-QFT. The term is mathematically
divergent and physically only explains scattering of particles in the
infinite past and future. What is the interaction term when particles
are not separated infinitely?

All that RQFT can do is to obtain a kind of *effective* interactions,
NEWER the real interaction for the particles in bound states. Do those
*effective* potentials derived from RQFT work? of course! but ONLY in
stationary states, where the detailed mechanism of interaction can be
ignored.

Of course, even ignoring that Coulomb-Breit U_CB or Coulomb-Gaunt U_CB
'effective' operators obtained from RQFT at second order in c^2 and e^2
are not well-founded even at stationary states. As it is well-known in
quantum chemistry, one may correct both via positive energy Casimir
type projection operators for the leptons

V_CB = Lambda_{++} U_CB Lambda_{++}

V_CG = Lambda_{++} U_CG Lambda_{++}

> > Previously, i cited Weinberg comments on bounded states but i cited
> > incorrectly the page, p560:
> >
> > "It must be said that the theory of relativistic effects and radiative
> > corrections in bound states is not yet in entirely satisfactory shape".
>
> On the same page Weinberg talks about the difficulty in finding the
> right approximation to use to reduce the full QED theory to an
> approximation in which bound state calculations are possible. The same
> situation is seen in all of theoretical physics whenever a simple
> explanation is sought for complex phenomena, which range from weather
> prediction, to high temperature superconductors, the two-body problem
> in general relativity.

There is no such one thing as full QED in bound states, because only
free particle states can be defined in QED. Moreover Weinberg said
'theory of' not 'application of'.

> > Moreover, in curved spacetimes, the particles theories can be done
> > rigorous using Synge parallel propagators. Wald 'and company' are wrong
> > about particle theories cannot 'work' in curved spacetimes.
> > Hoyle/Narkilar theory in curved spacetime is a beatiful example...
>
> Reference please.

For instance,

(1963) Proc R Soc London A 277, 1.

(1964) Proc R Soc London A 282, 184.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Igor Khavkine wrote:
> The fact that one can derive fields starting with multiparticle states
> is the main theorem of second quantization and has been known since
> Dirac, Pauli, and others. I don't understand the point you are trying
> to make about states that factor and those that don't. Multiparticle
> states that factor form a basis for all other ones. Whether you are
> dealing with only the basis vectors of the whole Hilbert space is
> irrelevant with respect to the construction of field operators.

One can derive fields begining from 'impure' multiparticle states that
can be factorized into one-body states. I was all time talking of pure
multiparticle states that does not factorize. The field description can
be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

You claim that you do not understand the point about states that
factorize and those that do not. Have you studied relativistic QFT? Why
may one study only scattering states in RQFT?

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

You are wrong. The (single particle) wave functions do not satisfy the
KG and Dirac field equations, ***in RIGOR***, and this was the reason
that both are abandoned in relativistic quantum field theory. In fact,
the evolution equation in R-QFT is a Schrödinger equation for
functionals. In RQFT, both KG and Dirac equations are field operator
identities, *newer* wavefunction equations.

> Multiparticle wave
> functions are also possible. However, the equations of motion that they
> satisfy are not closed (unlike the single particle case) unless all the
> E&M field modes are coupled in as well. This is a consequence of the no
> go theorem for the existence of a relativistically invariant
> two-particle potential. However, in approximations, the infinite
> dimensional system of equations including the E&M field can be
> truncated to a finite one, like for example the two-particle Dirac
> equation with an effective interaction given to order 1/c^2.

In rigor, multiparticle wave functions are NOT possible in relativistic
QFT, because there is not dynamical variables for those vawefunctions.
This is the reason that R-QFT can only deal with scattering states,
where N-particle states factorize and via conservation of momentum *FOR
FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
the ONE-body wavefunctions.

The so-called 'two-body wavefunctions' are not rigorous, and only in
stationary states could be partially interpreted if one is not very
precise and moreover one is only interested in some observables like
energy.

You claim that the equations of motion that they satisfy are not
closed, which is misleading. The equations of motion are *not* defined,
and all i know are mixed -totally inconsistent- approaches where, at
one hand, one defines /ad hoc/ two-body wavefunctions (for example with
16 components that are not wavefunctions of relativistic field theory)
and, at the other hand, one defines /ad hoc/ equations of 'motion' for
that two-body system (the dynamical equation in quantum field theory
*is* Schrödinger-like one).

But the equation of 'motion' is formed by single particle terms
inspired in Dirac equation (which is *wrong* in relativistic regimes,
because in RQFT the Dirac equation is an indentity for field operators
and has not wavefunction interpretation) more an interacting term.

Taking interacting term zero, one obtains two Dirac equations, which
all of us know are not correct, and this is the reason that were
abandoned in R-QFT by their field theoretic counterparts.

The interacting term is obtained from R-QFT. The term is mathematically
divergent and physically only explains scattering of particles in the
infinite past and future. What is the interaction term when particles
are not separated infinitely?

All that RQFT can do is to obtain a kind of *effective* interactions,
NEWER the real interaction for the particles in bound states. Do those
*effective* potentials derived from RQFT work? of course! but ONLY in
stationary states, where the detailed mechanism of interaction can be
ignored.

Of course, even ignoring that Coulomb-Breit U_CB or Coulomb-Gaunt U_CB
'effective' operators obtained from RQFT at second order in c^2 and e^2
are not well-founded even at stationary states. As it is well-known in
quantum chemistry, one may correct both via positive energy Casimir
type projection operators for the leptons

V_CB = Lambda_{++} U_CB Lambda_{++}

V_CG = Lambda_{++} U_CG Lambda_{++}

> > Previously, i cited Weinberg comments on bounded states but i cited
> > incorrectly the page, p560:
> >
> > "It must be said that the theory of relativistic effects and radiative
> > corrections in bound states is not yet in entirely satisfactory shape".
>
> On the same page Weinberg talks about the difficulty in finding the
> right approximation to use to reduce the full QED theory to an
> approximation in which bound state calculations are possible. The same
> situation is seen in all of theoretical physics whenever a simple
> explanation is sought for complex phenomena, which range from weather
> prediction, to high temperature superconductors, the two-body problem
> in general relativity.

There is no such one thing as full QED in bound states, because only
free particle states can be defined in QED. Moreover Weinberg said
'theory of' not 'application of'.

> > Moreover, in curved spacetimes, the particles theories can be done
> > rigorous using Synge parallel propagators. Wald 'and company' are wrong
> > about particle theories cannot 'work' in curved spacetimes.
> > Hoyle/Narkilar theory in curved spacetime is a beatiful example...
>
> Reference please.

For instance,

(1963) Proc R Soc London A 277, 1.

(1964) Proc R Soc London A 282, 184.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Igor Khavkine wrote:
> The fact that one can derive fields starting with multiparticle states
> is the main theorem of second quantization and has been known since
> Dirac, Pauli, and others. I don't understand the point you are trying
> to make about states that factor and those that don't. Multiparticle
> states that factor form a basis for all other ones. Whether you are
> dealing with only the basis vectors of the whole Hilbert space is
> irrelevant with respect to the construction of field operators.

One can derive fields begining from 'impure' multiparticle states that
can be factorized into one-body states. I was all time talking of pure
multiparticle states that does not factorize. The field description can
be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

You claim that you do not understand the point about states that
factorize and those that do not. Have you studied relativistic QFT? Why
may one study only scattering states in RQFT?

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

You are wrong. The (single particle) wave functions do not satisfy the
KG and Dirac field equations, ***in RIGOR***, and this was the reason
that both are abandoned in relativistic quantum field theory. In fact,
the evolution equation in R-QFT is a Schrödinger equation for
functionals. In RQFT, both KG and Dirac equations are field operator
identities, *newer* wavefunction equations.

> Multiparticle wave
> functions are also possible. However, the equations of motion that they
> satisfy are not closed (unlike the single particle case) unless all the
> E&M field modes are coupled in as well. This is a consequence of the no
> go theorem for the existence of a relativistically invariant
> two-particle potential. However, in approximations, the infinite
> dimensional system of equations including the E&M field can be
> truncated to a finite one, like for example the two-particle Dirac
> equation with an effective interaction given to order 1/c^2.

In rigor, multiparticle wave functions are NOT possible in relativistic
QFT, because there is not dynamical variables for those vawefunctions.
This is the reason that R-QFT can only deal with scattering states,
where N-particle states factorize and via conservation of momentum *FOR
FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
the ONE-body wavefunctions.

The so-called 'two-body wavefunctions' are not rigorous, and only in
stationary states could be partially interpreted if one is not very
precise and moreover one is only interested in some observables like
energy.

You claim that the equations of motion that they satisfy are not
closed, which is misleading. The equations of motion are *not* defined,
and all i know are mixed -totally inconsistent- approaches where, at
one hand, one defines /ad hoc/ two-body wavefunctions (for example with
16 components that are not wavefunctions of relativistic field theory)
and, at the other hand, one defines /ad hoc/ equations of 'motion' for
that two-body system (the dynamical equation in quantum field theory
*is* Schrödinger-like one).

But the equation of 'motion' is formed by single particle terms
inspired in Dirac equation (which is *wrong* in relativistic regimes,
because in RQFT the Dirac equation is an indentity for field operators
and has not wavefunction interpretation) more an interacting term.

Taking interacting term zero, one obtains two Dirac equations, which
all of us know are not correct, and this is the reason that were
abandoned in R-QFT by their field theoretic counterparts.

The interacting term is obtained from R-QFT. The term is mathematically
divergent and physically only explains scattering of particles in the
infinite past and future. What is the interaction term when particles
are not separated infinitely?

All that RQFT can do is to obtain a kind of *effective* interactions,
NEWER the real interaction for the particles in bound states. Do those
*effective* potentials derived from RQFT work? of course! but ONLY in
stationary states, where the detailed mechanism of interaction can be
ignored.

Of course, even ignoring that Coulomb-Breit U_CB or Coulomb-Gaunt U_CB
'effective' operators obtained from RQFT at second order in c^2 and e^2
are not well-founded even at stationary states. As it is well-known in
quantum chemistry, one may correct both via positive energy Casimir
type projection operators for the leptons

V_CB = Lambda_{++} U_CB Lambda_{++}

V_CG = Lambda_{++} U_CG Lambda_{++}

> > Previously, i cited Weinberg comments on bounded states but i cited
> > incorrectly the page, p560:
> >
> > "It must be said that the theory of relativistic effects and radiative
> > corrections in bound states is not yet in entirely satisfactory shape".
>
> On the same page Weinberg talks about the difficulty in finding the
> right approximation to use to reduce the full QED theory to an
> approximation in which bound state calculations are possible. The same
> situation is seen in all of theoretical physics whenever a simple
> explanation is sought for complex phenomena, which range from weather
> prediction, to high temperature superconductors, the two-body problem
> in general relativity.

There is no such one thing as full QED in bound states, because only
free particle states can be defined in QED. Moreover Weinberg said
'theory of' not 'application of'.

> > Moreover, in curved spacetimes, the particles theories can be done
> > rigorous using Synge parallel propagators. Wald 'and company' are wrong
> > about particle theories cannot 'work' in curved spacetimes.
> > Hoyle/Narkilar theory in curved spacetime is a beatiful example...
>
> Reference please.

For instance,

(1963) Proc R Soc London A 277, 1.

(1964) Proc R Soc London A 282, 184.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Igor Khavkine wrote:
> The fact that one can derive fields starting with multiparticle states
> is the main theorem of second quantization and has been known since
> Dirac, Pauli, and others. I don't understand the point you are trying
> to make about states that factor and those that don't. Multiparticle
> states that factor form a basis for all other ones. Whether you are
> dealing with only the basis vectors of the whole Hilbert space is
> irrelevant with respect to the construction of field operators.

One can derive fields begining from 'impure' multiparticle states that
can be factorized into one-body states. I was all time talking of pure
multiparticle states that does not factorize. The field description can
be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

You claim that you do not understand the point about states that
factorize and those that do not. Have you studied relativistic QFT? Why
may one study only scattering states in RQFT?

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

You are wrong. The (single particle) wave functions do not satisfy the
KG and Dirac field equations, ***in RIGOR***, and this was the reason
that both are abandoned in relativistic quantum field theory. In fact,
the evolution equation in R-QFT is a Schrödinger equation for
functionals. In RQFT, both KG and Dirac equations are field operator
identities, *newer* wavefunction equations.

> Multiparticle wave
> functions are also possible. However, the equations of motion that they
> satisfy are not closed (unlike the single particle case) unless all the
> E&M field modes are coupled in as well. This is a consequence of the no
> go theorem for the existence of a relativistically invariant
> two-particle potential. However, in approximations, the infinite
> dimensional system of equations including the E&M field can be
> truncated to a finite one, like for example the two-particle Dirac
> equation with an effective interaction given to order 1/c^2.

In rigor, multiparticle wave functions are NOT possible in relativistic
QFT, because there is not dynamical variables for those vawefunctions.
This is the reason that R-QFT can only deal with scattering states,
where N-particle states factorize and via conservation of momentum *FOR
FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
the ONE-body wavefunctions.

The so-called 'two-body wavefunctions' are not rigorous, and only in
stationary states could be partially interpreted if one is not very
precise and moreover one is only interested in some observables like
energy.

You claim that the equations of motion that they satisfy are not
closed, which is misleading. The equations of motion are *not* defined,
and all i know are mixed -totally inconsistent- approaches where, at
one hand, one defines /ad hoc/ two-body wavefunctions (for example with
16 components that are not wavefunctions of relativistic field theory)
and, at the other hand, one defines /ad hoc/ equations of 'motion' for
that two-body system (the dynamical equation in quantum field theory
*is* Schrödinger-like one).

But the equation of 'motion' is formed by single particle terms
inspired in Dirac equation (which is *wrong* in relativistic regimes,
because in RQFT the Dirac equation is an indentity for field operators
and has not wavefunction interpretation) more an interacting term.

Taking interacting term zero, one obtains two Dirac equations, which
all of us know are not correct, and this is the reason that were
abandoned in R-QFT by their field theoretic counterparts.

The interacting term is obtained from R-QFT. The term is mathematically
divergent and physically only explains scattering of particles in the
infinite past and future. What is the interaction term when particles
are not separated infinitely?

All that RQFT can do is to obtain a kind of *effective* interactions,
NEWER the real interaction for the particles in bound states. Do those
*effective* potentials derived from RQFT work? of course! but ONLY in
stationary states, where the detailed mechanism of interaction can be
ignored.

Of course, even ignoring that Coulomb-Breit U_CB or Coulomb-Gaunt U_CB
'effective' operators obtained from RQFT at second order in c^2 and e^2
are not well-founded even at stationary states. As it is well-known in
quantum chemistry, one may correct both via positive energy Casimir
type projection operators for the leptons

V_CB = Lambda_{++} U_CB Lambda_{++}

V_CG = Lambda_{++} U_CG Lambda_{++}

> > Previously, i cited Weinberg comments on bounded states but i cited
> > incorrectly the page, p560:
> >
> > "It must be said that the theory of relativistic effects and radiative
> > corrections in bound states is not yet in entirely satisfactory shape".
>
> On the same page Weinberg talks about the difficulty in finding the
> right approximation to use to reduce the full QED theory to an
> approximation in which bound state calculations are possible. The same
> situation is seen in all of theoretical physics whenever a simple
> explanation is sought for complex phenomena, which range from weather
> prediction, to high temperature superconductors, the two-body problem
> in general relativity.

There is no such one thing as full QED in bound states, because only
free particle states can be defined in QED. Moreover Weinberg said
'theory of' not 'application of'.

> > Moreover, in curved spacetimes, the particles theories can be done
> > rigorous using Synge parallel propagators. Wald 'and company' are wrong
> > about particle theories cannot 'work' in curved spacetimes.
> > Hoyle/Narkilar theory in curved spacetime is a beatiful example...
>
> Reference please.

For instance,

(1963) Proc R Soc London A 277, 1.

(1964) Proc R Soc London A 282, 184.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Igor Khavkine wrote:
> The fact that one can derive fields starting with multiparticle states
> is the main theorem of second quantization and has been known since
> Dirac, Pauli, and others. I don't understand the point you are trying
> to make about states that factor and those that don't. Multiparticle
> states that factor form a basis for all other ones. Whether you are
> dealing with only the basis vectors of the whole Hilbert space is
> irrelevant with respect to the construction of field operators.

One can derive fields begining from 'impure' multiparticle states that
can be factorized into one-body states. I was all time talking of pure
multiparticle states that does not factorize. The field description can
be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

You claim that you do not understand the point about states that
factorize and those that do not. Have you studied relativistic QFT? Why
may one study only scattering states in RQFT?

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

You are wrong. The (single particle) wave functions do not satisfy the
KG and Dirac field equations, ***in RIGOR***, and this was the reason
that both are abandoned in relativistic quantum field theory. In fact,
the evolution equation in R-QFT is a Schrödinger equation for
functionals. In RQFT, both KG and Dirac equations are field operator
identities, *newer* wavefunction equations.

> Multiparticle wave
> functions are also possible. However, the equations of motion that they
> satisfy are not closed (unlike the single particle case) unless all the
> E&M field modes are coupled in as well. This is a consequence of the no
> go theorem for the existence of a relativistically invariant
> two-particle potential. However, in approximations, the infinite
> dimensional system of equations including the E&M field can be
> truncated to a finite one, like for example the two-particle Dirac
> equation with an effective interaction given to order 1/c^2.

In rigor, multiparticle wave functions are NOT possible in relativistic
QFT, because there is not dynamical variables for those vawefunctions.
This is the reason that R-QFT can only deal with scattering states,
where N-particle states factorize and via conservation of momentum *FOR
FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
the ONE-body wavefunctions.

The so-called 'two-body wavefunctions' are not rigorous, and only in
stationary states could be partially interpreted if one is not very
precise and moreover one is only interested in some observables like
energy.

You claim that the equations of motion that they satisfy are not
closed, which is misleading. The equations of motion are *not* defined,
and all i know are mixed -totally inconsistent- approaches where, at
one hand, one defines /ad hoc/ two-body wavefunctions (for example with
16 components that are not wavefunctions of relativistic field theory)
and, at the other hand, one defines /ad hoc/ equations of 'motion' for
that two-body system (the dynamical equation in quantum field theory
*is* Schrödinger-like one).

But the equation of 'motion' is formed by single particle terms
inspired in Dirac equation (which is *wrong* in relativistic regimes,
because in RQFT the Dirac equation is an indentity for field operators
and has not wavefunction interpretation) more an interacting term.

Taking interacting term zero, one obtains two Dirac equations, which
all of us know are not correct, and this is the reason that were
abandoned in R-QFT by their field theoretic counterparts.

The interacting term is obtained from R-QFT. The term is mathematically
divergent and physically only explains scattering of particles in the
infinite past and future. What is the interaction term when particles
are not separated infinitely?

All that RQFT can do is to obtain a kind of *effective* interactions,
NEWER the real interaction for the particles in bound states. Do those
*effective* potentials derived from RQFT work? of course! but ONLY in
stationary states, where the detailed mechanism of interaction can be
ignored.

Of course, even ignoring that Coulomb-Breit U_CB or Coulomb-Gaunt U_CB
'effective' operators obtained from RQFT at second order in c^2 and e^2
are not well-founded even at stationary states. As it is well-known in
quantum chemistry, one may correct both via positive energy Casimir
type projection operators for the leptons

V_CB = Lambda_{++} U_CB Lambda_{++}

V_CG = Lambda_{++} U_CG Lambda_{++}

> > Previously, i cited Weinberg comments on bounded states but i cited
> > incorrectly the page, p560:
> >
> > "It must be said that the theory of relativistic effects and radiative
> > corrections in bound states is not yet in entirely satisfactory shape".
>
> On the same page Weinberg talks about the difficulty in finding the
> right approximation to use to reduce the full QED theory to an
> approximation in which bound state calculations are possible. The same
> situation is seen in all of theoretical physics whenever a simple
> explanation is sought for complex phenomena, which range from weather
> prediction, to high temperature superconductors, the two-body problem
> in general relativity.

There is no such one thing as full QED in bound states, because only
free particle states can be defined in QED. Moreover Weinberg said
'theory of' not 'application of'.

> > Moreover, in curved spacetimes, the particles theories can be done
> > rigorous using Synge parallel propagators. Wald 'and company' are wrong
> > about particle theories cannot 'work' in curved spacetimes.
> > Hoyle/Narkilar theory in curved spacetime is a beatiful example...
>
> Reference please.

For instance,

(1963) Proc R Soc London A 277, 1.

(1964) Proc R Soc London A 282, 184.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Igor Khavkine wrote:
> The fact that one can derive fields starting with multiparticle states
> is the main theorem of second quantization and has been known since
> Dirac, Pauli, and others. I don't understand the point you are trying
> to make about states that factor and those that don't. Multiparticle
> states that factor form a basis for all other ones. Whether you are
> dealing with only the basis vectors of the whole Hilbert space is
> irrelevant with respect to the construction of field operators.

One can derive fields begining from 'impure' multiparticle states that
can be factorized into one-body states. I was all time talking of pure
multiparticle states that does not factorize. The field description can
be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

You claim that you do not understand the point about states that
factorize and those that do not. Have you studied relativistic QFT? Why
may one study only scattering states in RQFT?

> You also seem to be confused about the KG and Dirac field equations.
> There are several distinct objects that satisfy them. Classical fields
> (which should be Grassmann valued for fermions), the quantum field
> operators, and the (single particle) wave functions.

You are wrong. The (single particle) wave functions do not satisfy the
KG and Dirac field equations, ***in RIGOR***, and this was the reason
that both are abandoned in relativistic quantum field theory. In fact,
the evolution equation in R-QFT is a Schrödinger equation for
functionals. In RQFT, both KG and Dirac equations are field operator
identities, *newer* wavefunction equations.

> Multiparticle wave
> functions are also possible. However, the equations of motion that they
> satisfy are not closed (unlike the single particle case) unless all the
> E&M field modes are coupled in as well. This is a consequence of the no
> go theorem for the existence of a relativistically invariant
> two-particle potential. However, in approximations, the infinite
> dimensional system of equations including the E&M field can be
> truncated to a finite one, like for example the two-particle Dirac
> equation with an effective interaction given to order 1/c^2.

In rigor, multiparticle wave functions are NOT possible in relativistic
QFT, because there is not dynamical variables for those vawefunctions.
This is the reason that R-QFT can only deal with scattering states,
where N-particle states factorize and via conservation of momentum *FOR
FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
the ONE-body wavefunctions.

The so-called 'two-body wavefunctions' are not rigorous, and only in
stationary states could be partially interpreted if one is not very
precise and moreover one is only interested in some observables like
energy.

You claim that the equations of motion that they satisfy are not
closed, which is misleading. The equations of motion are *not* defined,
and all i know are mixed -totally inconsistent- approaches where, at
one hand, one defines /ad hoc/ two-body wavefunctions (for example with
16 components that are not wavefunctions of relativistic field theory)
and, at the other hand, one defines /ad hoc/ equations of 'motion' for
that two-body system (the dynamical equation in quantum field theory
*is* Schrödinger-like one).

But the equation of 'motion' is formed by single particle terms
inspired in Dirac equation (which is *wrong* in relativistic regimes,
because in RQFT the Dirac equation is an indentity for field operators
and has not wavefunction interpretation) more an interacting term.

Taking interacting term zero, one obtains two Dirac equations, which
all of us know are not correct, and this is the reason that were
abandoned in R-QFT by their field theoretic counterparts.

The interacting term is obtained from R-QFT. The term is mathematically
divergent and physically only explains scattering of particles in the
infinite past and future. What is the interaction term when particles
are not separated infinitely?

All that RQFT can do is to obtain a kind of *effective* interactions,
NEWER the real interaction for the particles in bound states. Do those
*effective* potentials derived from RQFT work? of course! but ONLY in
stationary states, where the detailed mechanism of interaction can be
ignored.

Of course, even ignoring that Coulomb-Breit U_CB or Coulomb-Gaunt U_CB
'effective' operators obtained from RQFT at second order in c^2 and e^2
are not well-founded even at stationary states. As it is well-known in
quantum chemistry, one may correct both via positive energy Casimir
type projection operators for the leptons

V_CB = Lambda_{++} U_CB Lambda_{++}

V_CG = Lambda_{++} U_CG Lambda_{++}

> > Previously, i cited Weinberg comments on bounded states but i cited
> > incorrectly the page, p560:
> >
> > "It must be said that the theory of relativistic effects and radiative
> > corrections in bound states is not yet in entirely satisfactory shape".
>
> On the same page Weinberg talks about the difficulty in finding the
> right approximation to use to reduce the full QED theory to an
> approximation in which bound state calculations are possible. The same
> situation is seen in all of theoretical physics whenever a simple
> explanation is sought for complex phenomena, which range from weather
> prediction, to high temperature superconductors, the two-body problem
> in general relativity.

There is no such one thing as full QED in bound states, because only
free particle states can be defined in QED. Moreover Weinberg said
'theory of' not 'application of'.

> > Moreover, in curved spacetimes, the particles theories can be done
> > rigorous using Synge parallel propagators. Wald 'and company' are wrong
> > about particle theories cannot 'work' in curved spacetimes.
> > Hoyle/Narkilar theory in curved spacetime is a beatiful example...
>
> Reference please.

For instance,

(1963) Proc R Soc London A 277, 1.

(1964) Proc R Soc London A 282, 184.

Juan R.

Center for CANONICAL |SCIENCE)

[Eugene Stefanovich]

Juan R. wrote:

> One can derive fields begining from 'impure' multiparticle states that
> can be factorized into one-body states. I was all time talking of pure
> multiparticle states that does not factorize. The field description can
> be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

I am not sure if I understood you correctly. In my view, every
N-particle sector of the Fock space is build as a tensor product of
1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
form a basis there. Any other state is a linear combination of the
tensor product states. This is true independent on whether particles are
free or interacting. Free quantum fields can be always defined
as linear combinations of particle creation and annihilation operators.
"Interacting" (Heisenberg) quantum fields can also be defined
by switching from the
free to the interacting Hamiltonian in the formula for the time
evolution of the field. My only "problem" is that I don't know what's
the physical meaning of these "interacting fields". In my opinion, they
are not needed for a theoretical description of the system. All we want
to know about the system can be obtained without using interacting
quantum fields.

> In rigor, multiparticle wave functions are NOT possible in relativistic
> QFT, because there is not dynamical variables for those vawefunctions.

I don't see any problem with multiparticle wave functions in QFT.
In each N-particle sector of the Fock space you can define
particle observables. For example, you can define operators of momentum
P_i and spin S_i of all N particles. So, you can define momentum-spin
wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
in this sector. This wave function description works equally well for
systems with and without interaction.

> This is the reason that R-QFT can only deal with scattering states,
> where N-particle states factorize and via conservation of momentum *FOR
> FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
> the ONE-body wavefunctions.

I see a different reason why R-QFT only deals with scattering states.
The reason is that after renormalization (that is required to get
accurate S-matrix), the Hamiltonian of R-QFT
written in terms of unphysical bare particles usually contains
infinite counterterms. This Hamiltonian cannot be used directly for time
evolution calculations or for obtaining the bound states by
diagonalization. The solution of this problem is simple: change to the
physical particle representation by the "dressing transformation".

> But the equation of 'motion' is formed by single particle terms
> inspired in Dirac equation (which is *wrong* in relativistic regimes,
> because in RQFT the Dirac equation is an indentity for field operators
> and has not wavefunction interpretation) more an interacting term.

I agree with you that Dirac's field has no relationship to particle wave
functions and that Dirac's equation is not a "relativistic analog of the
Schroedinger equation". Dirac field has 4
components while 1-electron wave function has only 2 components.
The 4x4 matrix that transforms Dirac field's components is
independent on momentum. The 2x2 matrix that transforms spin components
of the 1-electron wave function depends on momentum (this is called
Wigner's rotation).

> The interacting term is obtained from R-QFT. The term is mathematically
> divergent and physically only explains scattering of particles in the
> infinite past and future. What is the interaction term when particles
> are not separated infinitely?

If you switch to the physical (or dressed) particle representation as
described above, you'll obtain a Hamiltonian with interaction that
works perfectly well for both asymptotic and intermediate regimes,
for stationary and non-stationary states.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

> One can derive fields begining from 'impure' multiparticle states that
> can be factorized into one-body states. I was all time talking of pure
> multiparticle states that does not factorize. The field description can
> be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

I am not sure if I understood you correctly. In my view, every
N-particle sector of the Fock space is build as a tensor product of
1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
form a basis there. Any other state is a linear combination of the
tensor product states. This is true independent on whether particles are
free or interacting. Free quantum fields can be always defined
as linear combinations of particle creation and annihilation operators.
"Interacting" (Heisenberg) quantum fields can also be defined
by switching from the
free to the interacting Hamiltonian in the formula for the time
evolution of the field. My only "problem" is that I don't know what's
the physical meaning of these "interacting fields". In my opinion, they
are not needed for a theoretical description of the system. All we want
to know about the system can be obtained without using interacting
quantum fields.

> In rigor, multiparticle wave functions are NOT possible in relativistic
> QFT, because there is not dynamical variables for those vawefunctions.

I don't see any problem with multiparticle wave functions in QFT.
In each N-particle sector of the Fock space you can define
particle observables. For example, you can define operators of momentum
P_i and spin S_i of all N particles. So, you can define momentum-spin
wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
in this sector. This wave function description works equally well for
systems with and without interaction.

> This is the reason that R-QFT can only deal with scattering states,
> where N-particle states factorize and via conservation of momentum *FOR
> FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
> the ONE-body wavefunctions.

I see a different reason why R-QFT only deals with scattering states.
The reason is that after renormalization (that is required to get
accurate S-matrix), the Hamiltonian of R-QFT
written in terms of unphysical bare particles usually contains
infinite counterterms. This Hamiltonian cannot be used directly for time
evolution calculations or for obtaining the bound states by
diagonalization. The solution of this problem is simple: change to the
physical particle representation by the "dressing transformation".

> But the equation of 'motion' is formed by single particle terms
> inspired in Dirac equation (which is *wrong* in relativistic regimes,
> because in RQFT the Dirac equation is an indentity for field operators
> and has not wavefunction interpretation) more an interacting term.

I agree with you that Dirac's field has no relationship to particle wave
functions and that Dirac's equation is not a "relativistic analog of the
Schroedinger equation". Dirac field has 4
components while 1-electron wave function has only 2 components.
The 4x4 matrix that transforms Dirac field's components is
independent on momentum. The 2x2 matrix that transforms spin components
of the 1-electron wave function depends on momentum (this is called
Wigner's rotation).

> The interacting term is obtained from R-QFT. The term is mathematically
> divergent and physically only explains scattering of particles in the
> infinite past and future. What is the interaction term when particles
> are not separated infinitely?

If you switch to the physical (or dressed) particle representation as
described above, you'll obtain a Hamiltonian with interaction that
works perfectly well for both asymptotic and intermediate regimes,
for stationary and non-stationary states.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

> One can derive fields begining from 'impure' multiparticle states that
> can be factorized into one-body states. I was all time talking of pure
> multiparticle states that does not factorize. The field description can
> be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

I am not sure if I understood you correctly. In my view, every
N-particle sector of the Fock space is build as a tensor product of
1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
form a basis there. Any other state is a linear combination of the
tensor product states. This is true independent on whether particles are
free or interacting. Free quantum fields can be always defined
as linear combinations of particle creation and annihilation operators.
"Interacting" (Heisenberg) quantum fields can also be defined
by switching from the
free to the interacting Hamiltonian in the formula for the time
evolution of the field. My only "problem" is that I don't know what's
the physical meaning of these "interacting fields". In my opinion, they
are not needed for a theoretical description of the system. All we want
to know about the system can be obtained without using interacting
quantum fields.

> In rigor, multiparticle wave functions are NOT possible in relativistic
> QFT, because there is not dynamical variables for those vawefunctions.

I don't see any problem with multiparticle wave functions in QFT.
In each N-particle sector of the Fock space you can define
particle observables. For example, you can define operators of momentum
P_i and spin S_i of all N particles. So, you can define momentum-spin
wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
in this sector. This wave function description works equally well for
systems with and without interaction.

> This is the reason that R-QFT can only deal with scattering states,
> where N-particle states factorize and via conservation of momentum *FOR
> FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
> the ONE-body wavefunctions.

I see a different reason why R-QFT only deals with scattering states.
The reason is that after renormalization (that is required to get
accurate S-matrix), the Hamiltonian of R-QFT
written in terms of unphysical bare particles usually contains
infinite counterterms. This Hamiltonian cannot be used directly for time
evolution calculations or for obtaining the bound states by
diagonalization. The solution of this problem is simple: change to the
physical particle representation by the "dressing transformation".

> But the equation of 'motion' is formed by single particle terms
> inspired in Dirac equation (which is *wrong* in relativistic regimes,
> because in RQFT the Dirac equation is an indentity for field operators
> and has not wavefunction interpretation) more an interacting term.

I agree with you that Dirac's field has no relationship to particle wave
functions and that Dirac's equation is not a "relativistic analog of the
Schroedinger equation". Dirac field has 4
components while 1-electron wave function has only 2 components.
The 4x4 matrix that transforms Dirac field's components is
independent on momentum. The 2x2 matrix that transforms spin components
of the 1-electron wave function depends on momentum (this is called
Wigner's rotation).

> The interacting term is obtained from R-QFT. The term is mathematically
> divergent and physically only explains scattering of particles in the
> infinite past and future. What is the interaction term when particles
> are not separated infinitely?

If you switch to the physical (or dressed) particle representation as
described above, you'll obtain a Hamiltonian with interaction that
works perfectly well for both asymptotic and intermediate regimes,
for stationary and non-stationary states.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

> One can derive fields begining from 'impure' multiparticle states that
> can be factorized into one-body states. I was all time talking of pure
> multiparticle states that does not factorize. The field description can
> be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

I am not sure if I understood you correctly. In my view, every
N-particle sector of the Fock space is build as a tensor product of
1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
form a basis there. Any other state is a linear combination of the
tensor product states. This is true independent on whether particles are
free or interacting. Free quantum fields can be always defined
as linear combinations of particle creation and annihilation operators.
"Interacting" (Heisenberg) quantum fields can also be defined
by switching from the
free to the interacting Hamiltonian in the formula for the time
evolution of the field. My only "problem" is that I don't know what's
the physical meaning of these "interacting fields". In my opinion, they
are not needed for a theoretical description of the system. All we want
to know about the system can be obtained without using interacting
quantum fields.

> In rigor, multiparticle wave functions are NOT possible in relativistic
> QFT, because there is not dynamical variables for those vawefunctions.

I don't see any problem with multiparticle wave functions in QFT.
In each N-particle sector of the Fock space you can define
particle observables. For example, you can define operators of momentum
P_i and spin S_i of all N particles. So, you can define momentum-spin
wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
in this sector. This wave function description works equally well for
systems with and without interaction.

> This is the reason that R-QFT can only deal with scattering states,
> where N-particle states factorize and via conservation of momentum *FOR
> FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
> the ONE-body wavefunctions.

I see a different reason why R-QFT only deals with scattering states.
The reason is that after renormalization (that is required to get
accurate S-matrix), the Hamiltonian of R-QFT
written in terms of unphysical bare particles usually contains
infinite counterterms. This Hamiltonian cannot be used directly for time
evolution calculations or for obtaining the bound states by
diagonalization. The solution of this problem is simple: change to the
physical particle representation by the "dressing transformation".

> But the equation of 'motion' is formed by single particle terms
> inspired in Dirac equation (which is *wrong* in relativistic regimes,
> because in RQFT the Dirac equation is an indentity for field operators
> and has not wavefunction interpretation) more an interacting term.

I agree with you that Dirac's field has no relationship to particle wave
functions and that Dirac's equation is not a "relativistic analog of the
Schroedinger equation". Dirac field has 4
components while 1-electron wave function has only 2 components.
The 4x4 matrix that transforms Dirac field's components is
independent on momentum. The 2x2 matrix that transforms spin components
of the 1-electron wave function depends on momentum (this is called
Wigner's rotation).

> The interacting term is obtained from R-QFT. The term is mathematically
> divergent and physically only explains scattering of particles in the
> infinite past and future. What is the interaction term when particles
> are not separated infinitely?

If you switch to the physical (or dressed) particle representation as
described above, you'll obtain a Hamiltonian with interaction that
works perfectly well for both asymptotic and intermediate regimes,
for stationary and non-stationary states.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

> One can derive fields begining from 'impure' multiparticle states that
> can be factorized into one-body states. I was all time talking of pure
> multiparticle states that does not factorize. The field description can
> be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

I am not sure if I understood you correctly. In my view, every
N-particle sector of the Fock space is build as a tensor product of
1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
form a basis there. Any other state is a linear combination of the
tensor product states. This is true independent on whether particles are
free or interacting. Free quantum fields can be always defined
as linear combinations of particle creation and annihilation operators.
"Interacting" (Heisenberg) quantum fields can also be defined
by switching from the
free to the interacting Hamiltonian in the formula for the time
evolution of the field. My only "problem" is that I don't know what's
the physical meaning of these "interacting fields". In my opinion, they
are not needed for a theoretical description of the system. All we want
to know about the system can be obtained without using interacting
quantum fields.

> In rigor, multiparticle wave functions are NOT possible in relativistic
> QFT, because there is not dynamical variables for those vawefunctions.

I don't see any problem with multiparticle wave functions in QFT.
In each N-particle sector of the Fock space you can define
particle observables. For example, you can define operators of momentum
P_i and spin S_i of all N particles. So, you can define momentum-spin
wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
in this sector. This wave function description works equally well for
systems with and without interaction.

> This is the reason that R-QFT can only deal with scattering states,
> where N-particle states factorize and via conservation of momentum *FOR
> FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
> the ONE-body wavefunctions.

I see a different reason why R-QFT only deals with scattering states.
The reason is that after renormalization (that is required to get
accurate S-matrix), the Hamiltonian of R-QFT
written in terms of unphysical bare particles usually contains
infinite counterterms. This Hamiltonian cannot be used directly for time
evolution calculations or for obtaining the bound states by
diagonalization. The solution of this problem is simple: change to the
physical particle representation by the "dressing transformation".

> But the equation of 'motion' is formed by single particle terms
> inspired in Dirac equation (which is *wrong* in relativistic regimes,
> because in RQFT the Dirac equation is an indentity for field operators
> and has not wavefunction interpretation) more an interacting term.

I agree with you that Dirac's field has no relationship to particle wave
functions and that Dirac's equation is not a "relativistic analog of the
Schroedinger equation". Dirac field has 4
components while 1-electron wave function has only 2 components.
The 4x4 matrix that transforms Dirac field's components is
independent on momentum. The 2x2 matrix that transforms spin components
of the 1-electron wave function depends on momentum (this is called
Wigner's rotation).

> The interacting term is obtained from R-QFT. The term is mathematically
> divergent and physically only explains scattering of particles in the
> infinite past and future. What is the interaction term when particles
> are not separated infinitely?

If you switch to the physical (or dressed) particle representation as
described above, you'll obtain a Hamiltonian with interaction that
works perfectly well for both asymptotic and intermediate regimes,
for stationary and non-stationary states.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

> One can derive fields begining from 'impure' multiparticle states that
> can be factorized into one-body states. I was all time talking of pure
> multiparticle states that does not factorize. The field description can
> be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

I am not sure if I understood you correctly. In my view, every
N-particle sector of the Fock space is build as a tensor product of
1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
form a basis there. Any other state is a linear combination of the
tensor product states. This is true independent on whether particles are
free or interacting. Free quantum fields can be always defined
as linear combinations of particle creation and annihilation operators.
"Interacting" (Heisenberg) quantum fields can also be defined
by switching from the
free to the interacting Hamiltonian in the formula for the time
evolution of the field. My only "problem" is that I don't know what's
the physical meaning of these "interacting fields". In my opinion, they
are not needed for a theoretical description of the system. All we want
to know about the system can be obtained without using interacting
quantum fields.

> In rigor, multiparticle wave functions are NOT possible in relativistic
> QFT, because there is not dynamical variables for those vawefunctions.

I don't see any problem with multiparticle wave functions in QFT.
In each N-particle sector of the Fock space you can define
particle observables. For example, you can define operators of momentum
P_i and spin S_i of all N particles. So, you can define momentum-spin
wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
in this sector. This wave function description works equally well for
systems with and without interaction.

> This is the reason that R-QFT can only deal with scattering states,
> where N-particle states factorize and via conservation of momentum *FOR
> FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
> the ONE-body wavefunctions.

I see a different reason why R-QFT only deals with scattering states.
The reason is that after renormalization (that is required to get
accurate S-matrix), the Hamiltonian of R-QFT
written in terms of unphysical bare particles usually contains
infinite counterterms. This Hamiltonian cannot be used directly for time
evolution calculations or for obtaining the bound states by
diagonalization. The solution of this problem is simple: change to the
physical particle representation by the "dressing transformation".

> But the equation of 'motion' is formed by single particle terms
> inspired in Dirac equation (which is *wrong* in relativistic regimes,
> because in RQFT the Dirac equation is an indentity for field operators
> and has not wavefunction interpretation) more an interacting term.

I agree with you that Dirac's field has no relationship to particle wave
functions and that Dirac's equation is not a "relativistic analog of the
Schroedinger equation". Dirac field has 4
components while 1-electron wave function has only 2 components.
The 4x4 matrix that transforms Dirac field's components is
independent on momentum. The 2x2 matrix that transforms spin components
of the 1-electron wave function depends on momentum (this is called
Wigner's rotation).

> The interacting term is obtained from R-QFT. The term is mathematically
> divergent and physically only explains scattering of particles in the
> infinite past and future. What is the interaction term when particles
> are not separated infinitely?

If you switch to the physical (or dressed) particle representation as
described above, you'll obtain a Hamiltonian with interaction that
works perfectly well for both asymptotic and intermediate regimes,
for stationary and non-stationary states.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

> One can derive fields begining from 'impure' multiparticle states that
> can be factorized into one-body states. I was all time talking of pure
> multiparticle states that does not factorize. The field description can
> be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

I am not sure if I understood you correctly. In my view, every
N-particle sector of the Fock space is build as a tensor product of
1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
form a basis there. Any other state is a linear combination of the
tensor product states. This is true independent on whether particles are
free or interacting. Free quantum fields can be always defined
as linear combinations of particle creation and annihilation operators.
"Interacting" (Heisenberg) quantum fields can also be defined
by switching from the
free to the interacting Hamiltonian in the formula for the time
evolution of the field. My only "problem" is that I don't know what's
the physical meaning of these "interacting fields". In my opinion, they
are not needed for a theoretical description of the system. All we want
to know about the system can be obtained without using interacting
quantum fields.

> In rigor, multiparticle wave functions are NOT possible in relativistic
> QFT, because there is not dynamical variables for those vawefunctions.

I don't see any problem with multiparticle wave functions in QFT.
In each N-particle sector of the Fock space you can define
particle observables. For example, you can define operators of momentum
P_i and spin S_i of all N particles. So, you can define momentum-spin
wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
in this sector. This wave function description works equally well for
systems with and without interaction.

> This is the reason that R-QFT can only deal with scattering states,
> where N-particle states factorize and via conservation of momentum *FOR
> FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
> the ONE-body wavefunctions.

I see a different reason why R-QFT only deals with scattering states.
The reason is that after renormalization (that is required to get
accurate S-matrix), the Hamiltonian of R-QFT
written in terms of unphysical bare particles usually contains
infinite counterterms. This Hamiltonian cannot be used directly for time
evolution calculations or for obtaining the bound states by
diagonalization. The solution of this problem is simple: change to the
physical particle representation by the "dressing transformation".

> But the equation of 'motion' is formed by single particle terms
> inspired in Dirac equation (which is *wrong* in relativistic regimes,
> because in RQFT the Dirac equation is an indentity for field operators
> and has not wavefunction interpretation) more an interacting term.

I agree with you that Dirac's field has no relationship to particle wave
functions and that Dirac's equation is not a "relativistic analog of the
Schroedinger equation". Dirac field has 4
components while 1-electron wave function has only 2 components.
The 4x4 matrix that transforms Dirac field's components is
independent on momentum. The 2x2 matrix that transforms spin components
of the 1-electron wave function depends on momentum (this is called
Wigner's rotation).

> The interacting term is obtained from R-QFT. The term is mathematically
> divergent and physically only explains scattering of particles in the
> infinite past and future. What is the interaction term when particles
> are not separated infinitely?

If you switch to the physical (or dressed) particle representation as
described above, you'll obtain a Hamiltonian with interaction that
works perfectly well for both asymptotic and intermediate regimes,
for stationary and non-stationary states.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

> One can derive fields begining from 'impure' multiparticle states that
> can be factorized into one-body states. I was all time talking of pure
> multiparticle states that does not factorize. The field description can
> be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

I am not sure if I understood you correctly. In my view, every
N-particle sector of the Fock space is build as a tensor product of
1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
form a basis there. Any other state is a linear combination of the
tensor product states. This is true independent on whether particles are
free or interacting. Free quantum fields can be always defined
as linear combinations of particle creation and annihilation operators.
"Interacting" (Heisenberg) quantum fields can also be defined
by switching from the
free to the interacting Hamiltonian in the formula for the time
evolution of the field. My only "problem" is that I don't know what's
the physical meaning of these "interacting fields". In my opinion, they
are not needed for a theoretical description of the system. All we want
to know about the system can be obtained without using interacting
quantum fields.

> In rigor, multiparticle wave functions are NOT possible in relativistic
> QFT, because there is not dynamical variables for those vawefunctions.

I don't see any problem with multiparticle wave functions in QFT.
In each N-particle sector of the Fock space you can define
particle observables. For example, you can define operators of momentum
P_i and spin S_i of all N particles. So, you can define momentum-spin
wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
in this sector. This wave function description works equally well for
systems with and without interaction.

> This is the reason that R-QFT can only deal with scattering states,
> where N-particle states factorize and via conservation of momentum *FOR
> FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
> the ONE-body wavefunctions.

I see a different reason why R-QFT only deals with scattering states.
The reason is that after renormalization (that is required to get
accurate S-matrix), the Hamiltonian of R-QFT
written in terms of unphysical bare particles usually contains
infinite counterterms. This Hamiltonian cannot be used directly for time
evolution calculations or for obtaining the bound states by
diagonalization. The solution of this problem is simple: change to the
physical particle representation by the "dressing transformation".

> But the equation of 'motion' is formed by single particle terms
> inspired in Dirac equation (which is *wrong* in relativistic regimes,
> because in RQFT the Dirac equation is an indentity for field operators
> and has not wavefunction interpretation) more an interacting term.

I agree with you that Dirac's field has no relationship to particle wave
functions and that Dirac's equation is not a "relativistic analog of the
Schroedinger equation". Dirac field has 4
components while 1-electron wave function has only 2 components.
The 4x4 matrix that transforms Dirac field's components is
independent on momentum. The 2x2 matrix that transforms spin components
of the 1-electron wave function depends on momentum (this is called
Wigner's rotation).

> The interacting term is obtained from R-QFT. The term is mathematically
> divergent and physically only explains scattering of particles in the
> infinite past and future. What is the interaction term when particles
> are not separated infinitely?

If you switch to the physical (or dressed) particle representation as
described above, you'll obtain a Hamiltonian with interaction that
works perfectly well for both asymptotic and intermediate regimes,
for stationary and non-stationary states.

Eugene.

[Eugene Stefanovich]

Juan R. wrote:

> One can derive fields begining from 'impure' multiparticle states that
> can be factorized into one-body states. I was all time talking of pure
> multiparticle states that does not factorize. The field description can
> be derived like a approximation when |Phi_N> = |1>|2>|3>...|N>.

I am not sure if I understood you correctly. In my view, every
N-particle sector of the Fock space is build as a tensor product of
1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
form a basis there. Any other state is a linear combination of the
tensor product states. This is true independent on whether particles are
free or interacting. Free quantum fields can be always defined
as linear combinations of particle creation and annihilation operators.
"Interacting" (Heisenberg) quantum fields can also be defined
by switching from the
free to the interacting Hamiltonian in the formula for the time
evolution of the field. My only "problem" is that I don't know what's
the physical meaning of these "interacting fields". In my opinion, they
are not needed for a theoretical description of the system. All we want
to know about the system can be obtained without using interacting
quantum fields.

> In rigor, multiparticle wave functions are NOT possible in relativistic
> QFT, because there is not dynamical variables for those vawefunctions.

I don't see any problem with multiparticle wave functions in QFT.
In each N-particle sector of the Fock space you can define
particle observables. For example, you can define operators of momentum
P_i and spin S_i of all N particles. So, you can define momentum-spin
wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
in this sector. This wave function description works equally well for
systems with and without interaction.

> This is the reason that R-QFT can only deal with scattering states,
> where N-particle states factorize and via conservation of momentum *FOR
> FREE PARTICLES* one can use p1 p2 p3... pN like dynamical variables for
> the ONE-body wavefunctions.

I see a different reason why R-QFT only deals with scattering states.
The reason is that after renormalization (that is required to get
accurate S-matrix), the Hamiltonian of R-QFT
written in terms of unphysical bare particles usually contains
infinite counterterms. This Hamiltonian cannot be used directly for time
evolution calculations or for obtaining the bound states by
diagonalization. The solution of this problem is simple: change to the
physical particle representation by the "dressing transformation".

> But the equation of 'motion' is formed by single particle terms
> inspired in Dirac equation (which is *wrong* in relativistic regimes,
> because in RQFT the Dirac equation is an indentity for field operators
> and has not wavefunction interpretation) more an interacting term.

I agree with you that Dirac's field has no relationship to particle wave
functions and that Dirac's equation is not a "relativistic analog of the
Schroedinger equation". Dirac field has 4
components while 1-electron wave function has only 2 components.
The 4x4 matrix that transforms Dirac field's components is
independent on momentum. The 2x2 matrix that transforms spin components
of the 1-electron wave function depends on momentum (this is called
Wigner's rotation).

> The interacting term is obtained from R-QFT. The term is mathematically
> divergent and physically only explains scattering of particles in the
> infinite past and future. What is the interaction term when particles
> are not separated infinitely?

If you switch to the physical (or dressed) particle representation as
described above, you'll obtain a Hamiltonian with interaction that
works perfectly well for both asymptotic and intermediate regimes,
for stationary and non-stationary states.

Eugene.

[Juan R.]

Eugene Stefanovich wrote:
> I am not sure if I understood you correctly. In my view, every
> N-particle sector of the Fock space is build as a tensor product of
> 1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
> form a basis there. Any other state is a linear combination of the
> tensor product states. This is true independent on whether particles are
> free or interacting. Free quantum fields can be always defined
> as linear combinations of particle creation and annihilation operators.
> "Interacting" (Heisenberg) quantum fields can also be defined
> by switching from the
> free to the interacting Hamiltonian in the formula for the time
> evolution of the field. My only "problem" is that I don't know what's
> the physical meaning of these "interacting fields". In my opinion, they
> are not needed for a theoretical description of the system. All we want
> to know about the system can be obtained without using interacting
> quantum fields.

The use of a Hilbert-Fock space is dependent of class of problems that
you was addresing and the level of rigor you need. There are well-know
situations in theoretical chemistry where a Hilbert-Fock space is not
suficient. Literature is extensive and i will not cite here. I only
call your atention to PRA 1996 53(6) 4075, where you can find some
remarks on why certain IMPORTANT effects cannot be studied in a Hilbert
space. Authors of the PRA propose an extension of scatering theory
*beyond* the Hilbert-Fock space:

"Persistent interactions require singular distribution functions which
lie outside the Hilbert space"

> > In rigor, multiparticle wave functions are NOT possible in relativistic
> > QFT, because there is not dynamical variables for those vawefunctions.
>
> I don't see any problem with multiparticle wave functions in QFT.
> In each N-particle sector of the Fock space you can define
> particle observables. For example, you can define operators of momentum
> P_i and spin S_i of all N particles. So, you can define momentum-spin
> wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
> in this sector. This wave function description works equally well for
> systems with and without interaction.

It is not so simple! The uncertainty relationships of NRQM may be
generalized to the relativistic domain. This was done by Landau in 1930
and proved recently with more mathematical rigor -at level of
Schwarzild inequality- by some authors. The physical insight of
Landau's work is that both x and p are not observable magnitudes in
relativistic regime. Whereas in classical physics there is posibility
for a measuring of x AND p, and in NRQM this is reduced to measuring of
x OR p, the trouble is that in RQM one cannot measure, in general, x
NOR p.

This is the reason that x in NRQM is an observable with asociated
operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.

The same criticism remains for the momentum representation of QM. In
general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
general, a dinamical variable in RQM.

The solution was given by Heisemberg in 1938. One may abandon partially
NRQM. In the approximation of free particles, the principle of
conservation of momentum still HOLD, and then the impulse of free
particles p_j is well defined. But particles are free only in
scattering states, due to application of the principle of decomposition
of clusters: when R --> infinite interaction is zero and particles are
free. This is reason that Weinberg emphasizes the cluster decomposition
princle as one of basic postulates of RQFT.

Then, in scatering states and *only in*, one can do |Phi(p_1, p_2,...
p_N)> = |p_1>|p_2>...|p_N>. The wavefunctions for free particles 1,
2,... N are defined for free ps, which are dinamical variables in the
*asymptotic regime*. In the full interaction regime, p are not
dinamical variables. This is one of reasons that RQFT is ONLY defined
for scattering states and cannot describe bound states.

Of course in the relativistic domain

delta p =of order= hbar / (c delta t)

and delta p zero implies delta t infinite. Scattering states in RQFT
are defined just for an infinite temporal interval [- infinite,
infinite] and RQFT cannot study the details of the dynamics, only the
scattering which is obviously trivial. As explained in PRA 1996 53(6)
4075 in many-body situations the important interval is the dynamical
one -for example TST in chemical dynamics- nor the initial and final
states and whereas computations in RQFT are trivial -see conclusion
section of above paper- the asociated computations in chemistry cannot
be done because mathematical formalism of RQFT breaks. This is the
reason of the new theory proposed by authors of PRA 1996 53(6) 4075.
Note that is not a computational problem (as Igor Khavkine appears to
think) is a *fundamental* problem still unsolved in theoretical
physics. As *emphasized* by authors of above paper, both the S-matrix
theory and basic QM structure of standard field theory break down in
the chemistry of many-body systems.

> > The interacting term is obtained from R-QFT. The term is mathematically
> > divergent and physically only explains scattering of particles in the
> > infinite past and future. What is the interaction term when particles
> > are not separated infinitely?
>
> If you switch to the physical (or dressed) particle representation as
> described above, you'll obtain a Hamiltonian with interaction that
> works perfectly well for both asymptotic and intermediate regimes,
> for stationary and non-stationary states.

In complement to my above discussion of interacting regimes in full
relativistic quantum mechanics i would add some others difficulties.
There are many of them and it would be extensive to discuss all of them
here in detail. However, i will focus on an important point: there is
NOT Hamiltonian in the full relativistic regime.

The proof of this is a bit thecnical and needs of a rigorous study of
Hamiltonian mechanics, unfortunately all of textbooks in RQFT and many
papers simply ignore this.

The discussion in the classical case is more easy. Most authors propose
-without further discussion- that the Hamiltonian with full EM
interactions is

H = (p - eA)/2m + eV

ok?

Well, this is wrong. ONLY for a single particle, in an external field,
above definition is valid, because then A = A(x, t) and the Hamiltonian
is a real Hamiltonian with functional dependence H = H(x, p).

If you adds a second electron ' a new term A(v') arises in the
Lagrangian and the nonlinearity of the Legendre transformation impides
the obtaining of a real Hamiltonian, at the best one obtains a function
h = h(x, p, v). This is the famous 'h function' named by Goldstein in
his classical textbook on mechanics, but is NOT a Hamiltonian.

But the Hamiltonian is the generator of time translations in QM.
Therefore, this is another proof of why RQFT can only deal with
one-particle states (multiparticles if are factorized) and study the
interaction via perturbation (divergent) theory. This is other of
reaosn of why there exists no one things like RQFT of a full
two-electron system and this is the reaosn that two-body relativistic
theories are based in inconsistent 'hibrids' like Bethe-Salpeter and
similar (i already explain what are some of difficulties with those
equations even if one is non rigorous and assumes that CB or CG
effective potentials derived from RQFT are correct, there is extensive
literature in why are inadequate and may be corrected -as pointed in a
previous post- via positive energy Casimir type projection operators
for the leptons, for instance.

The use of Lagrangian formalism does not change things:

1) The generator of time translations in QM is the Hamiltonian, as
well-explained by Weinberg. Feynmann path integral follow from
Hamiltonian mechanics, when interaction is small and for short times
propagator WITH THE FULL HAMILTONIAN factorizes like (free term *
interaction term). Then one can prove that this is equivalent to the
use of an action with the Lagrangian (free term minus potential).

2) The N-body full relativistic Lagrangian is unphysical (QFT theory
deals only with approximated Lagrangians). For example, the potential
in the Lagrangian U contains unphysical terms associated to velocity
terms. The only real potential is eV as proved by the Hamiltonian
formalism (as explained by Goldstein in his celebrated textbook on
mechanics any potential with velocity dependendent terms is
unphysical). In fact, the P = (partial L / partial v) derived from
aditional velocity-dependent terms in the Lagrangian is also
'unphysical'. This is the reason that the canonical momentum contains
'correction' terms as eA that may be eliminated in the Hamiltonian
formalism. The minimal coupling rule of usual literature p -> p - eA is
misleading because 'p' at the left is mv but 'p' at the right is NOT mv
is P: the canonical momentum which is NOT mv.

The minimal coupling rule looks like p -> P - eA or mv --> mv.

3) Path integral methods relies in the asumption of quasi-free motion
(one expands the propagator H around the free motion propagator T) and
focuses only in the interaction representation V_int. The true
generator of the dynamics in QM is, as stated by Weinberg the
Hamiltonian -see Weinberg discussion on his volume 1- Moreover, in some
cases explicitely cited by Weinberg, the path integral method offer
*incorrect* replies for measured observables whereas the canonical
method based in a Hamiltonian offers the *right* replies.

Etc.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> I am not sure if I understood you correctly. In my view, every
> N-particle sector of the Fock space is build as a tensor product of
> 1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
> form a basis there. Any other state is a linear combination of the
> tensor product states. This is true independent on whether particles are
> free or interacting. Free quantum fields can be always defined
> as linear combinations of particle creation and annihilation operators.
> "Interacting" (Heisenberg) quantum fields can also be defined
> by switching from the
> free to the interacting Hamiltonian in the formula for the time
> evolution of the field. My only "problem" is that I don't know what's
> the physical meaning of these "interacting fields". In my opinion, they
> are not needed for a theoretical description of the system. All we want
> to know about the system can be obtained without using interacting
> quantum fields.

The use of a Hilbert-Fock space is dependent of class of problems that
you was addresing and the level of rigor you need. There are well-know
situations in theoretical chemistry where a Hilbert-Fock space is not
suficient. Literature is extensive and i will not cite here. I only
call your atention to PRA 1996 53(6) 4075, where you can find some
remarks on why certain IMPORTANT effects cannot be studied in a Hilbert
space. Authors of the PRA propose an extension of scatering theory
*beyond* the Hilbert-Fock space:

"Persistent interactions require singular distribution functions which
lie outside the Hilbert space"

> > In rigor, multiparticle wave functions are NOT possible in relativistic
> > QFT, because there is not dynamical variables for those vawefunctions.
>
> I don't see any problem with multiparticle wave functions in QFT.
> In each N-particle sector of the Fock space you can define
> particle observables. For example, you can define operators of momentum
> P_i and spin S_i of all N particles. So, you can define momentum-spin
> wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
> in this sector. This wave function description works equally well for
> systems with and without interaction.

It is not so simple! The uncertainty relationships of NRQM may be
generalized to the relativistic domain. This was done by Landau in 1930
and proved recently with more mathematical rigor -at level of
Schwarzild inequality- by some authors. The physical insight of
Landau's work is that both x and p are not observable magnitudes in
relativistic regime. Whereas in classical physics there is posibility
for a measuring of x AND p, and in NRQM this is reduced to measuring of
x OR p, the trouble is that in RQM one cannot measure, in general, x
NOR p.

This is the reason that x in NRQM is an observable with asociated
operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.

The same criticism remains for the momentum representation of QM. In
general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
general, a dinamical variable in RQM.

The solution was given by Heisemberg in 1938. One may abandon partially
NRQM. In the approximation of free particles, the principle of
conservation of momentum still HOLD, and then the impulse of free
particles p_j is well defined. But particles are free only in
scattering states, due to application of the principle of decomposition
of clusters: when R --> infinite interaction is zero and particles are
free. This is reason that Weinberg emphasizes the cluster decomposition
princle as one of basic postulates of RQFT.

Then, in scatering states and *only in*, one can do |Phi(p_1, p_2,...
p_N)> = |p_1>|p_2>...|p_N>. The wavefunctions for free particles 1,
2,... N are defined for free ps, which are dinamical variables in the
*asymptotic regime*. In the full interaction regime, p are not
dinamical variables. This is one of reasons that RQFT is ONLY defined
for scattering states and cannot describe bound states.

Of course in the relativistic domain

delta p =of order= hbar / (c delta t)

and delta p zero implies delta t infinite. Scattering states in RQFT
are defined just for an infinite temporal interval [- infinite,
infinite] and RQFT cannot study the details of the dynamics, only the
scattering which is obviously trivial. As explained in PRA 1996 53(6)
4075 in many-body situations the important interval is the dynamical
one -for example TST in chemical dynamics- nor the initial and final
states and whereas computations in RQFT are trivial -see conclusion
section of above paper- the asociated computations in chemistry cannot
be done because mathematical formalism of RQFT breaks. This is the
reason of the new theory proposed by authors of PRA 1996 53(6) 4075.
Note that is not a computational problem (as Igor Khavkine appears to
think) is a *fundamental* problem still unsolved in theoretical
physics. As *emphasized* by authors of above paper, both the S-matrix
theory and basic QM structure of standard field theory break down in
the chemistry of many-body systems.

> > The interacting term is obtained from R-QFT. The term is mathematically
> > divergent and physically only explains scattering of particles in the
> > infinite past and future. What is the interaction term when particles
> > are not separated infinitely?
>
> If you switch to the physical (or dressed) particle representation as
> described above, you'll obtain a Hamiltonian with interaction that
> works perfectly well for both asymptotic and intermediate regimes,
> for stationary and non-stationary states.

In complement to my above discussion of interacting regimes in full
relativistic quantum mechanics i would add some others difficulties.
There are many of them and it would be extensive to discuss all of them
here in detail. However, i will focus on an important point: there is
NOT Hamiltonian in the full relativistic regime.

The proof of this is a bit thecnical and needs of a rigorous study of
Hamiltonian mechanics, unfortunately all of textbooks in RQFT and many
papers simply ignore this.

The discussion in the classical case is more easy. Most authors propose
-without further discussion- that the Hamiltonian with full EM
interactions is

H = (p - eA)/2m + eV

ok?

Well, this is wrong. ONLY for a single particle, in an external field,
above definition is valid, because then A = A(x, t) and the Hamiltonian
is a real Hamiltonian with functional dependence H = H(x, p).

If you adds a second electron ' a new term A(v') arises in the
Lagrangian and the nonlinearity of the Legendre transformation impides
the obtaining of a real Hamiltonian, at the best one obtains a function
h = h(x, p, v). This is the famous 'h function' named by Goldstein in
his classical textbook on mechanics, but is NOT a Hamiltonian.

But the Hamiltonian is the generator of time translations in QM.
Therefore, this is another proof of why RQFT can only deal with
one-particle states (multiparticles if are factorized) and study the
interaction via perturbation (divergent) theory. This is other of
reaosn of why there exists no one things like RQFT of a full
two-electron system and this is the reaosn that two-body relativistic
theories are based in inconsistent 'hibrids' like Bethe-Salpeter and
similar (i already explain what are some of difficulties with those
equations even if one is non rigorous and assumes that CB or CG
effective potentials derived from RQFT are correct, there is extensive
literature in why are inadequate and may be corrected -as pointed in a
previous post- via positive energy Casimir type projection operators
for the leptons, for instance.

The use of Lagrangian formalism does not change things:

1) The generator of time translations in QM is the Hamiltonian, as
well-explained by Weinberg. Feynmann path integral follow from
Hamiltonian mechanics, when interaction is small and for short times
propagator WITH THE FULL HAMILTONIAN factorizes like (free term *
interaction term). Then one can prove that this is equivalent to the
use of an action with the Lagrangian (free term minus potential).

2) The N-body full relativistic Lagrangian is unphysical (QFT theory
deals only with approximated Lagrangians). For example, the potential
in the Lagrangian U contains unphysical terms associated to velocity
terms. The only real potential is eV as proved by the Hamiltonian
formalism (as explained by Goldstein in his celebrated textbook on
mechanics any potential with velocity dependendent terms is
unphysical). In fact, the P = (partial L / partial v) derived from
aditional velocity-dependent terms in the Lagrangian is also
'unphysical'. This is the reason that the canonical momentum contains
'correction' terms as eA that may be eliminated in the Hamiltonian
formalism. The minimal coupling rule of usual literature p -> p - eA is
misleading because 'p' at the left is mv but 'p' at the right is NOT mv
is P: the canonical momentum which is NOT mv.

The minimal coupling rule looks like p -> P - eA or mv --> mv.

3) Path integral methods relies in the asumption of quasi-free motion
(one expands the propagator H around the free motion propagator T) and
focuses only in the interaction representation V_int. The true
generator of the dynamics in QM is, as stated by Weinberg the
Hamiltonian -see Weinberg discussion on his volume 1- Moreover, in some
cases explicitely cited by Weinberg, the path integral method offer
*incorrect* replies for measured observables whereas the canonical
method based in a Hamiltonian offers the *right* replies.

Etc.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> I am not sure if I understood you correctly. In my view, every
> N-particle sector of the Fock space is build as a tensor product of
> 1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
> form a basis there. Any other state is a linear combination of the
> tensor product states. This is true independent on whether particles are
> free or interacting. Free quantum fields can be always defined
> as linear combinations of particle creation and annihilation operators.
> "Interacting" (Heisenberg) quantum fields can also be defined
> by switching from the
> free to the interacting Hamiltonian in the formula for the time
> evolution of the field. My only "problem" is that I don't know what's
> the physical meaning of these "interacting fields". In my opinion, they
> are not needed for a theoretical description of the system. All we want
> to know about the system can be obtained without using interacting
> quantum fields.

The use of a Hilbert-Fock space is dependent of class of problems that
you was addresing and the level of rigor you need. There are well-know
situations in theoretical chemistry where a Hilbert-Fock space is not
suficient. Literature is extensive and i will not cite here. I only
call your atention to PRA 1996 53(6) 4075, where you can find some
remarks on why certain IMPORTANT effects cannot be studied in a Hilbert
space. Authors of the PRA propose an extension of scatering theory
*beyond* the Hilbert-Fock space:

"Persistent interactions require singular distribution functions which
lie outside the Hilbert space"

> > In rigor, multiparticle wave functions are NOT possible in relativistic
> > QFT, because there is not dynamical variables for those vawefunctions.
>
> I don't see any problem with multiparticle wave functions in QFT.
> In each N-particle sector of the Fock space you can define
> particle observables. For example, you can define operators of momentum
> P_i and spin S_i of all N particles. So, you can define momentum-spin
> wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
> in this sector. This wave function description works equally well for
> systems with and without interaction.

It is not so simple! The uncertainty relationships of NRQM may be
generalized to the relativistic domain. This was done by Landau in 1930
and proved recently with more mathematical rigor -at level of
Schwarzild inequality- by some authors. The physical insight of
Landau's work is that both x and p are not observable magnitudes in
relativistic regime. Whereas in classical physics there is posibility
for a measuring of x AND p, and in NRQM this is reduced to measuring of
x OR p, the trouble is that in RQM one cannot measure, in general, x
NOR p.

This is the reason that x in NRQM is an observable with asociated
operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.

The same criticism remains for the momentum representation of QM. In
general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
general, a dinamical variable in RQM.

The solution was given by Heisemberg in 1938. One may abandon partially
NRQM. In the approximation of free particles, the principle of
conservation of momentum still HOLD, and then the impulse of free
particles p_j is well defined. But particles are free only in
scattering states, due to application of the principle of decomposition
of clusters: when R --> infinite interaction is zero and particles are
free. This is reason that Weinberg emphasizes the cluster decomposition
princle as one of basic postulates of RQFT.

Then, in scatering states and *only in*, one can do |Phi(p_1, p_2,...
p_N)> = |p_1>|p_2>...|p_N>. The wavefunctions for free particles 1,
2,... N are defined for free ps, which are dinamical variables in the
*asymptotic regime*. In the full interaction regime, p are not
dinamical variables. This is one of reasons that RQFT is ONLY defined
for scattering states and cannot describe bound states.

Of course in the relativistic domain

delta p =of order= hbar / (c delta t)

and delta p zero implies delta t infinite. Scattering states in RQFT
are defined just for an infinite temporal interval [- infinite,
infinite] and RQFT cannot study the details of the dynamics, only the
scattering which is obviously trivial. As explained in PRA 1996 53(6)
4075 in many-body situations the important interval is the dynamical
one -for example TST in chemical dynamics- nor the initial and final
states and whereas computations in RQFT are trivial -see conclusion
section of above paper- the asociated computations in chemistry cannot
be done because mathematical formalism of RQFT breaks. This is the
reason of the new theory proposed by authors of PRA 1996 53(6) 4075.
Note that is not a computational problem (as Igor Khavkine appears to
think) is a *fundamental* problem still unsolved in theoretical
physics. As *emphasized* by authors of above paper, both the S-matrix
theory and basic QM structure of standard field theory break down in
the chemistry of many-body systems.

> > The interacting term is obtained from R-QFT. The term is mathematically
> > divergent and physically only explains scattering of particles in the
> > infinite past and future. What is the interaction term when particles
> > are not separated infinitely?
>
> If you switch to the physical (or dressed) particle representation as
> described above, you'll obtain a Hamiltonian with interaction that
> works perfectly well for both asymptotic and intermediate regimes,
> for stationary and non-stationary states.

In complement to my above discussion of interacting regimes in full
relativistic quantum mechanics i would add some others difficulties.
There are many of them and it would be extensive to discuss all of them
here in detail. However, i will focus on an important point: there is
NOT Hamiltonian in the full relativistic regime.

The proof of this is a bit thecnical and needs of a rigorous study of
Hamiltonian mechanics, unfortunately all of textbooks in RQFT and many
papers simply ignore this.

The discussion in the classical case is more easy. Most authors propose
-without further discussion- that the Hamiltonian with full EM
interactions is

H = (p - eA)/2m + eV

ok?

Well, this is wrong. ONLY for a single particle, in an external field,
above definition is valid, because then A = A(x, t) and the Hamiltonian
is a real Hamiltonian with functional dependence H = H(x, p).

If you adds a second electron ' a new term A(v') arises in the
Lagrangian and the nonlinearity of the Legendre transformation impides
the obtaining of a real Hamiltonian, at the best one obtains a function
h = h(x, p, v). This is the famous 'h function' named by Goldstein in
his classical textbook on mechanics, but is NOT a Hamiltonian.

But the Hamiltonian is the generator of time translations in QM.
Therefore, this is another proof of why RQFT can only deal with
one-particle states (multiparticles if are factorized) and study the
interaction via perturbation (divergent) theory. This is other of
reaosn of why there exists no one things like RQFT of a full
two-electron system and this is the reaosn that two-body relativistic
theories are based in inconsistent 'hibrids' like Bethe-Salpeter and
similar (i already explain what are some of difficulties with those
equations even if one is non rigorous and assumes that CB or CG
effective potentials derived from RQFT are correct, there is extensive
literature in why are inadequate and may be corrected -as pointed in a
previous post- via positive energy Casimir type projection operators
for the leptons, for instance.

The use of Lagrangian formalism does not change things:

1) The generator of time translations in QM is the Hamiltonian, as
well-explained by Weinberg. Feynmann path integral follow from
Hamiltonian mechanics, when interaction is small and for short times
propagator WITH THE FULL HAMILTONIAN factorizes like (free term *
interaction term). Then one can prove that this is equivalent to the
use of an action with the Lagrangian (free term minus potential).

2) The N-body full relativistic Lagrangian is unphysical (QFT theory
deals only with approximated Lagrangians). For example, the potential
in the Lagrangian U contains unphysical terms associated to velocity
terms. The only real potential is eV as proved by the Hamiltonian
formalism (as explained by Goldstein in his celebrated textbook on
mechanics any potential with velocity dependendent terms is
unphysical). In fact, the P = (partial L / partial v) derived from
aditional velocity-dependent terms in the Lagrangian is also
'unphysical'. This is the reason that the canonical momentum contains
'correction' terms as eA that may be eliminated in the Hamiltonian
formalism. The minimal coupling rule of usual literature p -> p - eA is
misleading because 'p' at the left is mv but 'p' at the right is NOT mv
is P: the canonical momentum which is NOT mv.

The minimal coupling rule looks like p -> P - eA or mv --> mv.

3) Path integral methods relies in the asumption of quasi-free motion
(one expands the propagator H around the free motion propagator T) and
focuses only in the interaction representation V_int. The true
generator of the dynamics in QM is, as stated by Weinberg the
Hamiltonian -see Weinberg discussion on his volume 1- Moreover, in some
cases explicitely cited by Weinberg, the path integral method offer
*incorrect* replies for measured observables whereas the canonical
method based in a Hamiltonian offers the *right* replies.

Etc.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> I am not sure if I understood you correctly. In my view, every
> N-particle sector of the Fock space is build as a tensor product of
> 1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
> form a basis there. Any other state is a linear combination of the
> tensor product states. This is true independent on whether particles are
> free or interacting. Free quantum fields can be always defined
> as linear combinations of particle creation and annihilation operators.
> "Interacting" (Heisenberg) quantum fields can also be defined
> by switching from the
> free to the interacting Hamiltonian in the formula for the time
> evolution of the field. My only "problem" is that I don't know what's
> the physical meaning of these "interacting fields". In my opinion, they
> are not needed for a theoretical description of the system. All we want
> to know about the system can be obtained without using interacting
> quantum fields.

The use of a Hilbert-Fock space is dependent of class of problems that
you was addresing and the level of rigor you need. There are well-know
situations in theoretical chemistry where a Hilbert-Fock space is not
suficient. Literature is extensive and i will not cite here. I only
call your atention to PRA 1996 53(6) 4075, where you can find some
remarks on why certain IMPORTANT effects cannot be studied in a Hilbert
space. Authors of the PRA propose an extension of scatering theory
*beyond* the Hilbert-Fock space:

"Persistent interactions require singular distribution functions which
lie outside the Hilbert space"

> > In rigor, multiparticle wave functions are NOT possible in relativistic
> > QFT, because there is not dynamical variables for those vawefunctions.
>
> I don't see any problem with multiparticle wave functions in QFT.
> In each N-particle sector of the Fock space you can define
> particle observables. For example, you can define operators of momentum
> P_i and spin S_i of all N particles. So, you can define momentum-spin
> wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
> in this sector. This wave function description works equally well for
> systems with and without interaction.

It is not so simple! The uncertainty relationships of NRQM may be
generalized to the relativistic domain. This was done by Landau in 1930
and proved recently with more mathematical rigor -at level of
Schwarzild inequality- by some authors. The physical insight of
Landau's work is that both x and p are not observable magnitudes in
relativistic regime. Whereas in classical physics there is posibility
for a measuring of x AND p, and in NRQM this is reduced to measuring of
x OR p, the trouble is that in RQM one cannot measure, in general, x
NOR p.

This is the reason that x in NRQM is an observable with asociated
operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.

The same criticism remains for the momentum representation of QM. In
general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
general, a dinamical variable in RQM.

The solution was given by Heisemberg in 1938. One may abandon partially
NRQM. In the approximation of free particles, the principle of
conservation of momentum still HOLD, and then the impulse of free
particles p_j is well defined. But particles are free only in
scattering states, due to application of the principle of decomposition
of clusters: when R --> infinite interaction is zero and particles are
free. This is reason that Weinberg emphasizes the cluster decomposition
princle as one of basic postulates of RQFT.

Then, in scatering states and *only in*, one can do |Phi(p_1, p_2,...
p_N)> = |p_1>|p_2>...|p_N>. The wavefunctions for free particles 1,
2,... N are defined for free ps, which are dinamical variables in the
*asymptotic regime*. In the full interaction regime, p are not
dinamical variables. This is one of reasons that RQFT is ONLY defined
for scattering states and cannot describe bound states.

Of course in the relativistic domain

delta p =of order= hbar / (c delta t)

and delta p zero implies delta t infinite. Scattering states in RQFT
are defined just for an infinite temporal interval [- infinite,
infinite] and RQFT cannot study the details of the dynamics, only the
scattering which is obviously trivial. As explained in PRA 1996 53(6)
4075 in many-body situations the important interval is the dynamical
one -for example TST in chemical dynamics- nor the initial and final
states and whereas computations in RQFT are trivial -see conclusion
section of above paper- the asociated computations in chemistry cannot
be done because mathematical formalism of RQFT breaks. This is the
reason of the new theory proposed by authors of PRA 1996 53(6) 4075.
Note that is not a computational problem (as Igor Khavkine appears to
think) is a *fundamental* problem still unsolved in theoretical
physics. As *emphasized* by authors of above paper, both the S-matrix
theory and basic QM structure of standard field theory break down in
the chemistry of many-body systems.

> > The interacting term is obtained from R-QFT. The term is mathematically
> > divergent and physically only explains scattering of particles in the
> > infinite past and future. What is the interaction term when particles
> > are not separated infinitely?
>
> If you switch to the physical (or dressed) particle representation as
> described above, you'll obtain a Hamiltonian with interaction that
> works perfectly well for both asymptotic and intermediate regimes,
> for stationary and non-stationary states.

In complement to my above discussion of interacting regimes in full
relativistic quantum mechanics i would add some others difficulties.
There are many of them and it would be extensive to discuss all of them
here in detail. However, i will focus on an important point: there is
NOT Hamiltonian in the full relativistic regime.

The proof of this is a bit thecnical and needs of a rigorous study of
Hamiltonian mechanics, unfortunately all of textbooks in RQFT and many
papers simply ignore this.

The discussion in the classical case is more easy. Most authors propose
-without further discussion- that the Hamiltonian with full EM
interactions is

H = (p - eA)/2m + eV

ok?

Well, this is wrong. ONLY for a single particle, in an external field,
above definition is valid, because then A = A(x, t) and the Hamiltonian
is a real Hamiltonian with functional dependence H = H(x, p).

If you adds a second electron ' a new term A(v') arises in the
Lagrangian and the nonlinearity of the Legendre transformation impides
the obtaining of a real Hamiltonian, at the best one obtains a function
h = h(x, p, v). This is the famous 'h function' named by Goldstein in
his classical textbook on mechanics, but is NOT a Hamiltonian.

But the Hamiltonian is the generator of time translations in QM.
Therefore, this is another proof of why RQFT can only deal with
one-particle states (multiparticles if are factorized) and study the
interaction via perturbation (divergent) theory. This is other of
reaosn of why there exists no one things like RQFT of a full
two-electron system and this is the reaosn that two-body relativistic
theories are based in inconsistent 'hibrids' like Bethe-Salpeter and
similar (i already explain what are some of difficulties with those
equations even if one is non rigorous and assumes that CB or CG
effective potentials derived from RQFT are correct, there is extensive
literature in why are inadequate and may be corrected -as pointed in a
previous post- via positive energy Casimir type projection operators
for the leptons, for instance.

The use of Lagrangian formalism does not change things:

1) The generator of time translations in QM is the Hamiltonian, as
well-explained by Weinberg. Feynmann path integral follow from
Hamiltonian mechanics, when interaction is small and for short times
propagator WITH THE FULL HAMILTONIAN factorizes like (free term *
interaction term). Then one can prove that this is equivalent to the
use of an action with the Lagrangian (free term minus potential).

2) The N-body full relativistic Lagrangian is unphysical (QFT theory
deals only with approximated Lagrangians). For example, the potential
in the Lagrangian U contains unphysical terms associated to velocity
terms. The only real potential is eV as proved by the Hamiltonian
formalism (as explained by Goldstein in his celebrated textbook on
mechanics any potential with velocity dependendent terms is
unphysical). In fact, the P = (partial L / partial v) derived from
aditional velocity-dependent terms in the Lagrangian is also
'unphysical'. This is the reason that the canonical momentum contains
'correction' terms as eA that may be eliminated in the Hamiltonian
formalism. The minimal coupling rule of usual literature p -> p - eA is
misleading because 'p' at the left is mv but 'p' at the right is NOT mv
is P: the canonical momentum which is NOT mv.

The minimal coupling rule looks like p -> P - eA or mv --> mv.

3) Path integral methods relies in the asumption of quasi-free motion
(one expands the propagator H around the free motion propagator T) and
focuses only in the interaction representation V_int. The true
generator of the dynamics in QM is, as stated by Weinberg the
Hamiltonian -see Weinberg discussion on his volume 1- Moreover, in some
cases explicitely cited by Weinberg, the path integral method offer
*incorrect* replies for measured observables whereas the canonical
method based in a Hamiltonian offers the *right* replies.

Etc.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> I am not sure if I understood you correctly. In my view, every
> N-particle sector of the Fock space is build as a tensor product of
> 1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
> form a basis there. Any other state is a linear combination of the
> tensor product states. This is true independent on whether particles are
> free or interacting. Free quantum fields can be always defined
> as linear combinations of particle creation and annihilation operators.
> "Interacting" (Heisenberg) quantum fields can also be defined
> by switching from the
> free to the interacting Hamiltonian in the formula for the time
> evolution of the field. My only "problem" is that I don't know what's
> the physical meaning of these "interacting fields". In my opinion, they
> are not needed for a theoretical description of the system. All we want
> to know about the system can be obtained without using interacting
> quantum fields.

The use of a Hilbert-Fock space is dependent of class of problems that
you was addresing and the level of rigor you need. There are well-know
situations in theoretical chemistry where a Hilbert-Fock space is not
suficient. Literature is extensive and i will not cite here. I only
call your atention to PRA 1996 53(6) 4075, where you can find some
remarks on why certain IMPORTANT effects cannot be studied in a Hilbert
space. Authors of the PRA propose an extension of scatering theory
*beyond* the Hilbert-Fock space:

"Persistent interactions require singular distribution functions which
lie outside the Hilbert space"

> > In rigor, multiparticle wave functions are NOT possible in relativistic
> > QFT, because there is not dynamical variables for those vawefunctions.
>
> I don't see any problem with multiparticle wave functions in QFT.
> In each N-particle sector of the Fock space you can define
> particle observables. For example, you can define operators of momentum
> P_i and spin S_i of all N particles. So, you can define momentum-spin
> wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
> in this sector. This wave function description works equally well for
> systems with and without interaction.

It is not so simple! The uncertainty relationships of NRQM may be
generalized to the relativistic domain. This was done by Landau in 1930
and proved recently with more mathematical rigor -at level of
Schwarzild inequality- by some authors. The physical insight of
Landau's work is that both x and p are not observable magnitudes in
relativistic regime. Whereas in classical physics there is posibility
for a measuring of x AND p, and in NRQM this is reduced to measuring of
x OR p, the trouble is that in RQM one cannot measure, in general, x
NOR p.

This is the reason that x in NRQM is an observable with asociated
operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.

The same criticism remains for the momentum representation of QM. In
general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
general, a dinamical variable in RQM.

The solution was given by Heisemberg in 1938. One may abandon partially
NRQM. In the approximation of free particles, the principle of
conservation of momentum still HOLD, and then the impulse of free
particles p_j is well defined. But particles are free only in
scattering states, due to application of the principle of decomposition
of clusters: when R --> infinite interaction is zero and particles are
free. This is reason that Weinberg emphasizes the cluster decomposition
princle as one of basic postulates of RQFT.

Then, in scatering states and *only in*, one can do |Phi(p_1, p_2,...
p_N)> = |p_1>|p_2>...|p_N>. The wavefunctions for free particles 1,
2,... N are defined for free ps, which are dinamical variables in the
*asymptotic regime*. In the full interaction regime, p are not
dinamical variables. This is one of reasons that RQFT is ONLY defined
for scattering states and cannot describe bound states.

Of course in the relativistic domain

delta p =of order= hbar / (c delta t)

and delta p zero implies delta t infinite. Scattering states in RQFT
are defined just for an infinite temporal interval [- infinite,
infinite] and RQFT cannot study the details of the dynamics, only the
scattering which is obviously trivial. As explained in PRA 1996 53(6)
4075 in many-body situations the important interval is the dynamical
one -for example TST in chemical dynamics- nor the initial and final
states and whereas computations in RQFT are trivial -see conclusion
section of above paper- the asociated computations in chemistry cannot
be done because mathematical formalism of RQFT breaks. This is the
reason of the new theory proposed by authors of PRA 1996 53(6) 4075.
Note that is not a computational problem (as Igor Khavkine appears to
think) is a *fundamental* problem still unsolved in theoretical
physics. As *emphasized* by authors of above paper, both the S-matrix
theory and basic QM structure of standard field theory break down in
the chemistry of many-body systems.

> > The interacting term is obtained from R-QFT. The term is mathematically
> > divergent and physically only explains scattering of particles in the
> > infinite past and future. What is the interaction term when particles
> > are not separated infinitely?
>
> If you switch to the physical (or dressed) particle representation as
> described above, you'll obtain a Hamiltonian with interaction that
> works perfectly well for both asymptotic and intermediate regimes,
> for stationary and non-stationary states.

In complement to my above discussion of interacting regimes in full
relativistic quantum mechanics i would add some others difficulties.
There are many of them and it would be extensive to discuss all of them
here in detail. However, i will focus on an important point: there is
NOT Hamiltonian in the full relativistic regime.

The proof of this is a bit thecnical and needs of a rigorous study of
Hamiltonian mechanics, unfortunately all of textbooks in RQFT and many
papers simply ignore this.

The discussion in the classical case is more easy. Most authors propose
-without further discussion- that the Hamiltonian with full EM
interactions is

H = (p - eA)/2m + eV

ok?

Well, this is wrong. ONLY for a single particle, in an external field,
above definition is valid, because then A = A(x, t) and the Hamiltonian
is a real Hamiltonian with functional dependence H = H(x, p).

If you adds a second electron ' a new term A(v') arises in the
Lagrangian and the nonlinearity of the Legendre transformation impides
the obtaining of a real Hamiltonian, at the best one obtains a function
h = h(x, p, v). This is the famous 'h function' named by Goldstein in
his classical textbook on mechanics, but is NOT a Hamiltonian.

But the Hamiltonian is the generator of time translations in QM.
Therefore, this is another proof of why RQFT can only deal with
one-particle states (multiparticles if are factorized) and study the
interaction via perturbation (divergent) theory. This is other of
reaosn of why there exists no one things like RQFT of a full
two-electron system and this is the reaosn that two-body relativistic
theories are based in inconsistent 'hibrids' like Bethe-Salpeter and
similar (i already explain what are some of difficulties with those
equations even if one is non rigorous and assumes that CB or CG
effective potentials derived from RQFT are correct, there is extensive
literature in why are inadequate and may be corrected -as pointed in a
previous post- via positive energy Casimir type projection operators
for the leptons, for instance.

The use of Lagrangian formalism does not change things:

1) The generator of time translations in QM is the Hamiltonian, as
well-explained by Weinberg. Feynmann path integral follow from
Hamiltonian mechanics, when interaction is small and for short times
propagator WITH THE FULL HAMILTONIAN factorizes like (free term *
interaction term). Then one can prove that this is equivalent to the
use of an action with the Lagrangian (free term minus potential).

2) The N-body full relativistic Lagrangian is unphysical (QFT theory
deals only with approximated Lagrangians). For example, the potential
in the Lagrangian U contains unphysical terms associated to velocity
terms. The only real potential is eV as proved by the Hamiltonian
formalism (as explained by Goldstein in his celebrated textbook on
mechanics any potential with velocity dependendent terms is
unphysical). In fact, the P = (partial L / partial v) derived from
aditional velocity-dependent terms in the Lagrangian is also
'unphysical'. This is the reason that the canonical momentum contains
'correction' terms as eA that may be eliminated in the Hamiltonian
formalism. The minimal coupling rule of usual literature p -> p - eA is
misleading because 'p' at the left is mv but 'p' at the right is NOT mv
is P: the canonical momentum which is NOT mv.

The minimal coupling rule looks like p -> P - eA or mv --> mv.

3) Path integral methods relies in the asumption of quasi-free motion
(one expands the propagator H around the free motion propagator T) and
focuses only in the interaction representation V_int. The true
generator of the dynamics in QM is, as stated by Weinberg the
Hamiltonian -see Weinberg discussion on his volume 1- Moreover, in some
cases explicitely cited by Weinberg, the path integral method offer
*incorrect* replies for measured observables whereas the canonical
method based in a Hamiltonian offers the *right* replies.

Etc.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> I am not sure if I understood you correctly. In my view, every
> N-particle sector of the Fock space is build as a tensor product of
> 1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
> form a basis there. Any other state is a linear combination of the
> tensor product states. This is true independent on whether particles are
> free or interacting. Free quantum fields can be always defined
> as linear combinations of particle creation and annihilation operators.
> "Interacting" (Heisenberg) quantum fields can also be defined
> by switching from the
> free to the interacting Hamiltonian in the formula for the time
> evolution of the field. My only "problem" is that I don't know what's
> the physical meaning of these "interacting fields". In my opinion, they
> are not needed for a theoretical description of the system. All we want
> to know about the system can be obtained without using interacting
> quantum fields.

The use of a Hilbert-Fock space is dependent of class of problems that
you was addresing and the level of rigor you need. There are well-know
situations in theoretical chemistry where a Hilbert-Fock space is not
suficient. Literature is extensive and i will not cite here. I only
call your atention to PRA 1996 53(6) 4075, where you can find some
remarks on why certain IMPORTANT effects cannot be studied in a Hilbert
space. Authors of the PRA propose an extension of scatering theory
*beyond* the Hilbert-Fock space:

"Persistent interactions require singular distribution functions which
lie outside the Hilbert space"

> > In rigor, multiparticle wave functions are NOT possible in relativistic
> > QFT, because there is not dynamical variables for those vawefunctions.
>
> I don't see any problem with multiparticle wave functions in QFT.
> In each N-particle sector of the Fock space you can define
> particle observables. For example, you can define operators of momentum
> P_i and spin S_i of all N particles. So, you can define momentum-spin
> wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
> in this sector. This wave function description works equally well for
> systems with and without interaction.

It is not so simple! The uncertainty relationships of NRQM may be
generalized to the relativistic domain. This was done by Landau in 1930
and proved recently with more mathematical rigor -at level of
Schwarzild inequality- by some authors. The physical insight of
Landau's work is that both x and p are not observable magnitudes in
relativistic regime. Whereas in classical physics there is posibility
for a measuring of x AND p, and in NRQM this is reduced to measuring of
x OR p, the trouble is that in RQM one cannot measure, in general, x
NOR p.

This is the reason that x in NRQM is an observable with asociated
operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.

The same criticism remains for the momentum representation of QM. In
general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
general, a dinamical variable in RQM.

The solution was given by Heisemberg in 1938. One may abandon partially
NRQM. In the approximation of free particles, the principle of
conservation of momentum still HOLD, and then the impulse of free
particles p_j is well defined. But particles are free only in
scattering states, due to application of the principle of decomposition
of clusters: when R --> infinite interaction is zero and particles are
free. This is reason that Weinberg emphasizes the cluster decomposition
princle as one of basic postulates of RQFT.

Then, in scatering states and *only in*, one can do |Phi(p_1, p_2,...
p_N)> = |p_1>|p_2>...|p_N>. The wavefunctions for free particles 1,
2,... N are defined for free ps, which are dinamical variables in the
*asymptotic regime*. In the full interaction regime, p are not
dinamical variables. This is one of reasons that RQFT is ONLY defined
for scattering states and cannot describe bound states.

Of course in the relativistic domain

delta p =of order= hbar / (c delta t)

and delta p zero implies delta t infinite. Scattering states in RQFT
are defined just for an infinite temporal interval [- infinite,
infinite] and RQFT cannot study the details of the dynamics, only the
scattering which is obviously trivial. As explained in PRA 1996 53(6)
4075 in many-body situations the important interval is the dynamical
one -for example TST in chemical dynamics- nor the initial and final
states and whereas computations in RQFT are trivial -see conclusion
section of above paper- the asociated computations in chemistry cannot
be done because mathematical formalism of RQFT breaks. This is the
reason of the new theory proposed by authors of PRA 1996 53(6) 4075.
Note that is not a computational problem (as Igor Khavkine appears to
think) is a *fundamental* problem still unsolved in theoretical
physics. As *emphasized* by authors of above paper, both the S-matrix
theory and basic QM structure of standard field theory break down in
the chemistry of many-body systems.

> > The interacting term is obtained from R-QFT. The term is mathematically
> > divergent and physically only explains scattering of particles in the
> > infinite past and future. What is the interaction term when particles
> > are not separated infinitely?
>
> If you switch to the physical (or dressed) particle representation as
> described above, you'll obtain a Hamiltonian with interaction that
> works perfectly well for both asymptotic and intermediate regimes,
> for stationary and non-stationary states.

In complement to my above discussion of interacting regimes in full
relativistic quantum mechanics i would add some others difficulties.
There are many of them and it would be extensive to discuss all of them
here in detail. However, i will focus on an important point: there is
NOT Hamiltonian in the full relativistic regime.

The proof of this is a bit thecnical and needs of a rigorous study of
Hamiltonian mechanics, unfortunately all of textbooks in RQFT and many
papers simply ignore this.

The discussion in the classical case is more easy. Most authors propose
-without further discussion- that the Hamiltonian with full EM
interactions is

H = (p - eA)/2m + eV

ok?

Well, this is wrong. ONLY for a single particle, in an external field,
above definition is valid, because then A = A(x, t) and the Hamiltonian
is a real Hamiltonian with functional dependence H = H(x, p).

If you adds a second electron ' a new term A(v') arises in the
Lagrangian and the nonlinearity of the Legendre transformation impides
the obtaining of a real Hamiltonian, at the best one obtains a function
h = h(x, p, v). This is the famous 'h function' named by Goldstein in
his classical textbook on mechanics, but is NOT a Hamiltonian.

But the Hamiltonian is the generator of time translations in QM.
Therefore, this is another proof of why RQFT can only deal with
one-particle states (multiparticles if are factorized) and study the
interaction via perturbation (divergent) theory. This is other of
reaosn of why there exists no one things like RQFT of a full
two-electron system and this is the reaosn that two-body relativistic
theories are based in inconsistent 'hibrids' like Bethe-Salpeter and
similar (i already explain what are some of difficulties with those
equations even if one is non rigorous and assumes that CB or CG
effective potentials derived from RQFT are correct, there is extensive
literature in why are inadequate and may be corrected -as pointed in a
previous post- via positive energy Casimir type projection operators
for the leptons, for instance.

The use of Lagrangian formalism does not change things:

1) The generator of time translations in QM is the Hamiltonian, as
well-explained by Weinberg. Feynmann path integral follow from
Hamiltonian mechanics, when interaction is small and for short times
propagator WITH THE FULL HAMILTONIAN factorizes like (free term *
interaction term). Then one can prove that this is equivalent to the
use of an action with the Lagrangian (free term minus potential).

2) The N-body full relativistic Lagrangian is unphysical (QFT theory
deals only with approximated Lagrangians). For example, the potential
in the Lagrangian U contains unphysical terms associated to velocity
terms. The only real potential is eV as proved by the Hamiltonian
formalism (as explained by Goldstein in his celebrated textbook on
mechanics any potential with velocity dependendent terms is
unphysical). In fact, the P = (partial L / partial v) derived from
aditional velocity-dependent terms in the Lagrangian is also
'unphysical'. This is the reason that the canonical momentum contains
'correction' terms as eA that may be eliminated in the Hamiltonian
formalism. The minimal coupling rule of usual literature p -> p - eA is
misleading because 'p' at the left is mv but 'p' at the right is NOT mv
is P: the canonical momentum which is NOT mv.

The minimal coupling rule looks like p -> P - eA or mv --> mv.

3) Path integral methods relies in the asumption of quasi-free motion
(one expands the propagator H around the free motion propagator T) and
focuses only in the interaction representation V_int. The true
generator of the dynamics in QM is, as stated by Weinberg the
Hamiltonian -see Weinberg discussion on his volume 1- Moreover, in some
cases explicitely cited by Weinberg, the path integral method offer
*incorrect* replies for measured observables whereas the canonical
method based in a Hamiltonian offers the *right* replies.

Etc.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> I am not sure if I understood you correctly. In my view, every
> N-particle sector of the Fock space is build as a tensor product of
> 1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
> form a basis there. Any other state is a linear combination of the
> tensor product states. This is true independent on whether particles are
> free or interacting. Free quantum fields can be always defined
> as linear combinations of particle creation and annihilation operators.
> "Interacting" (Heisenberg) quantum fields can also be defined
> by switching from the
> free to the interacting Hamiltonian in the formula for the time
> evolution of the field. My only "problem" is that I don't know what's
> the physical meaning of these "interacting fields". In my opinion, they
> are not needed for a theoretical description of the system. All we want
> to know about the system can be obtained without using interacting
> quantum fields.

The use of a Hilbert-Fock space is dependent of class of problems that
you was addresing and the level of rigor you need. There are well-know
situations in theoretical chemistry where a Hilbert-Fock space is not
suficient. Literature is extensive and i will not cite here. I only
call your atention to PRA 1996 53(6) 4075, where you can find some
remarks on why certain IMPORTANT effects cannot be studied in a Hilbert
space. Authors of the PRA propose an extension of scatering theory
*beyond* the Hilbert-Fock space:

"Persistent interactions require singular distribution functions which
lie outside the Hilbert space"

> > In rigor, multiparticle wave functions are NOT possible in relativistic
> > QFT, because there is not dynamical variables for those vawefunctions.
>
> I don't see any problem with multiparticle wave functions in QFT.
> In each N-particle sector of the Fock space you can define
> particle observables. For example, you can define operators of momentum
> P_i and spin S_i of all N particles. So, you can define momentum-spin
> wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
> in this sector. This wave function description works equally well for
> systems with and without interaction.

It is not so simple! The uncertainty relationships of NRQM may be
generalized to the relativistic domain. This was done by Landau in 1930
and proved recently with more mathematical rigor -at level of
Schwarzild inequality- by some authors. The physical insight of
Landau's work is that both x and p are not observable magnitudes in
relativistic regime. Whereas in classical physics there is posibility
for a measuring of x AND p, and in NRQM this is reduced to measuring of
x OR p, the trouble is that in RQM one cannot measure, in general, x
NOR p.

This is the reason that x in NRQM is an observable with asociated
operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.

The same criticism remains for the momentum representation of QM. In
general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
general, a dinamical variable in RQM.

The solution was given by Heisemberg in 1938. One may abandon partially
NRQM. In the approximation of free particles, the principle of
conservation of momentum still HOLD, and then the impulse of free
particles p_j is well defined. But particles are free only in
scattering states, due to application of the principle of decomposition
of clusters: when R --> infinite interaction is zero and particles are
free. This is reason that Weinberg emphasizes the cluster decomposition
princle as one of basic postulates of RQFT.

Then, in scatering states and *only in*, one can do |Phi(p_1, p_2,...
p_N)> = |p_1>|p_2>...|p_N>. The wavefunctions for free particles 1,
2,... N are defined for free ps, which are dinamical variables in the
*asymptotic regime*. In the full interaction regime, p are not
dinamical variables. This is one of reasons that RQFT is ONLY defined
for scattering states and cannot describe bound states.

Of course in the relativistic domain

delta p =of order= hbar / (c delta t)

and delta p zero implies delta t infinite. Scattering states in RQFT
are defined just for an infinite temporal interval [- infinite,
infinite] and RQFT cannot study the details of the dynamics, only the
scattering which is obviously trivial. As explained in PRA 1996 53(6)
4075 in many-body situations the important interval is the dynamical
one -for example TST in chemical dynamics- nor the initial and final
states and whereas computations in RQFT are trivial -see conclusion
section of above paper- the asociated computations in chemistry cannot
be done because mathematical formalism of RQFT breaks. This is the
reason of the new theory proposed by authors of PRA 1996 53(6) 4075.
Note that is not a computational problem (as Igor Khavkine appears to
think) is a *fundamental* problem still unsolved in theoretical
physics. As *emphasized* by authors of above paper, both the S-matrix
theory and basic QM structure of standard field theory break down in
the chemistry of many-body systems.

> > The interacting term is obtained from R-QFT. The term is mathematically
> > divergent and physically only explains scattering of particles in the
> > infinite past and future. What is the interaction term when particles
> > are not separated infinitely?
>
> If you switch to the physical (or dressed) particle representation as
> described above, you'll obtain a Hamiltonian with interaction that
> works perfectly well for both asymptotic and intermediate regimes,
> for stationary and non-stationary states.

In complement to my above discussion of interacting regimes in full
relativistic quantum mechanics i would add some others difficulties.
There are many of them and it would be extensive to discuss all of them
here in detail. However, i will focus on an important point: there is
NOT Hamiltonian in the full relativistic regime.

The proof of this is a bit thecnical and needs of a rigorous study of
Hamiltonian mechanics, unfortunately all of textbooks in RQFT and many
papers simply ignore this.

The discussion in the classical case is more easy. Most authors propose
-without further discussion- that the Hamiltonian with full EM
interactions is

H = (p - eA)/2m + eV

ok?

Well, this is wrong. ONLY for a single particle, in an external field,
above definition is valid, because then A = A(x, t) and the Hamiltonian
is a real Hamiltonian with functional dependence H = H(x, p).

If you adds a second electron ' a new term A(v') arises in the
Lagrangian and the nonlinearity of the Legendre transformation impides
the obtaining of a real Hamiltonian, at the best one obtains a function
h = h(x, p, v). This is the famous 'h function' named by Goldstein in
his classical textbook on mechanics, but is NOT a Hamiltonian.

But the Hamiltonian is the generator of time translations in QM.
Therefore, this is another proof of why RQFT can only deal with
one-particle states (multiparticles if are factorized) and study the
interaction via perturbation (divergent) theory. This is other of
reaosn of why there exists no one things like RQFT of a full
two-electron system and this is the reaosn that two-body relativistic
theories are based in inconsistent 'hibrids' like Bethe-Salpeter and
similar (i already explain what are some of difficulties with those
equations even if one is non rigorous and assumes that CB or CG
effective potentials derived from RQFT are correct, there is extensive
literature in why are inadequate and may be corrected -as pointed in a
previous post- via positive energy Casimir type projection operators
for the leptons, for instance.

The use of Lagrangian formalism does not change things:

1) The generator of time translations in QM is the Hamiltonian, as
well-explained by Weinberg. Feynmann path integral follow from
Hamiltonian mechanics, when interaction is small and for short times
propagator WITH THE FULL HAMILTONIAN factorizes like (free term *
interaction term). Then one can prove that this is equivalent to the
use of an action with the Lagrangian (free term minus potential).

2) The N-body full relativistic Lagrangian is unphysical (QFT theory
deals only with approximated Lagrangians). For example, the potential
in the Lagrangian U contains unphysical terms associated to velocity
terms. The only real potential is eV as proved by the Hamiltonian
formalism (as explained by Goldstein in his celebrated textbook on
mechanics any potential with velocity dependendent terms is
unphysical). In fact, the P = (partial L / partial v) derived from
aditional velocity-dependent terms in the Lagrangian is also
'unphysical'. This is the reason that the canonical momentum contains
'correction' terms as eA that may be eliminated in the Hamiltonian
formalism. The minimal coupling rule of usual literature p -> p - eA is
misleading because 'p' at the left is mv but 'p' at the right is NOT mv
is P: the canonical momentum which is NOT mv.

The minimal coupling rule looks like p -> P - eA or mv --> mv.

3) Path integral methods relies in the asumption of quasi-free motion
(one expands the propagator H around the free motion propagator T) and
focuses only in the interaction representation V_int. The true
generator of the dynamics in QM is, as stated by Weinberg the
Hamiltonian -see Weinberg discussion on his volume 1- Moreover, in some
cases explicitely cited by Weinberg, the path integral method offer
*incorrect* replies for measured observables whereas the canonical
method based in a Hamiltonian offers the *right* replies.

Etc.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> I am not sure if I understood you correctly. In my view, every
> N-particle sector of the Fock space is build as a tensor product of
> 1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
> form a basis there. Any other state is a linear combination of the
> tensor product states. This is true independent on whether particles are
> free or interacting. Free quantum fields can be always defined
> as linear combinations of particle creation and annihilation operators.
> "Interacting" (Heisenberg) quantum fields can also be defined
> by switching from the
> free to the interacting Hamiltonian in the formula for the time
> evolution of the field. My only "problem" is that I don't know what's
> the physical meaning of these "interacting fields". In my opinion, they
> are not needed for a theoretical description of the system. All we want
> to know about the system can be obtained without using interacting
> quantum fields.

The use of a Hilbert-Fock space is dependent of class of problems that
you was addresing and the level of rigor you need. There are well-know
situations in theoretical chemistry where a Hilbert-Fock space is not
suficient. Literature is extensive and i will not cite here. I only
call your atention to PRA 1996 53(6) 4075, where you can find some
remarks on why certain IMPORTANT effects cannot be studied in a Hilbert
space. Authors of the PRA propose an extension of scatering theory
*beyond* the Hilbert-Fock space:

"Persistent interactions require singular distribution functions which
lie outside the Hilbert space"

> > In rigor, multiparticle wave functions are NOT possible in relativistic
> > QFT, because there is not dynamical variables for those vawefunctions.
>
> I don't see any problem with multiparticle wave functions in QFT.
> In each N-particle sector of the Fock space you can define
> particle observables. For example, you can define operators of momentum
> P_i and spin S_i of all N particles. So, you can define momentum-spin
> wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
> in this sector. This wave function description works equally well for
> systems with and without interaction.

It is not so simple! The uncertainty relationships of NRQM may be
generalized to the relativistic domain. This was done by Landau in 1930
and proved recently with more mathematical rigor -at level of
Schwarzild inequality- by some authors. The physical insight of
Landau's work is that both x and p are not observable magnitudes in
relativistic regime. Whereas in classical physics there is posibility
for a measuring of x AND p, and in NRQM this is reduced to measuring of
x OR p, the trouble is that in RQM one cannot measure, in general, x
NOR p.

This is the reason that x in NRQM is an observable with asociated
operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.

The same criticism remains for the momentum representation of QM. In
general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
general, a dinamical variable in RQM.

The solution was given by Heisemberg in 1938. One may abandon partially
NRQM. In the approximation of free particles, the principle of
conservation of momentum still HOLD, and then the impulse of free
particles p_j is well defined. But particles are free only in
scattering states, due to application of the principle of decomposition
of clusters: when R --> infinite interaction is zero and particles are
free. This is reason that Weinberg emphasizes the cluster decomposition
princle as one of basic postulates of RQFT.

Then, in scatering states and *only in*, one can do |Phi(p_1, p_2,...
p_N)> = |p_1>|p_2>...|p_N>. The wavefunctions for free particles 1,
2,... N are defined for free ps, which are dinamical variables in the
*asymptotic regime*. In the full interaction regime, p are not
dinamical variables. This is one of reasons that RQFT is ONLY defined
for scattering states and cannot describe bound states.

Of course in the relativistic domain

delta p =of order= hbar / (c delta t)

and delta p zero implies delta t infinite. Scattering states in RQFT
are defined just for an infinite temporal interval [- infinite,
infinite] and RQFT cannot study the details of the dynamics, only the
scattering which is obviously trivial. As explained in PRA 1996 53(6)
4075 in many-body situations the important interval is the dynamical
one -for example TST in chemical dynamics- nor the initial and final
states and whereas computations in RQFT are trivial -see conclusion
section of above paper- the asociated computations in chemistry cannot
be done because mathematical formalism of RQFT breaks. This is the
reason of the new theory proposed by authors of PRA 1996 53(6) 4075.
Note that is not a computational problem (as Igor Khavkine appears to
think) is a *fundamental* problem still unsolved in theoretical
physics. As *emphasized* by authors of above paper, both the S-matrix
theory and basic QM structure of standard field theory break down in
the chemistry of many-body systems.

> > The interacting term is obtained from R-QFT. The term is mathematically
> > divergent and physically only explains scattering of particles in the
> > infinite past and future. What is the interaction term when particles
> > are not separated infinitely?
>
> If you switch to the physical (or dressed) particle representation as
> described above, you'll obtain a Hamiltonian with interaction that
> works perfectly well for both asymptotic and intermediate regimes,
> for stationary and non-stationary states.

In complement to my above discussion of interacting regimes in full
relativistic quantum mechanics i would add some others difficulties.
There are many of them and it would be extensive to discuss all of them
here in detail. However, i will focus on an important point: there is
NOT Hamiltonian in the full relativistic regime.

The proof of this is a bit thecnical and needs of a rigorous study of
Hamiltonian mechanics, unfortunately all of textbooks in RQFT and many
papers simply ignore this.

The discussion in the classical case is more easy. Most authors propose
-without further discussion- that the Hamiltonian with full EM
interactions is

H = (p - eA)/2m + eV

ok?

Well, this is wrong. ONLY for a single particle, in an external field,
above definition is valid, because then A = A(x, t) and the Hamiltonian
is a real Hamiltonian with functional dependence H = H(x, p).

If you adds a second electron ' a new term A(v') arises in the
Lagrangian and the nonlinearity of the Legendre transformation impides
the obtaining of a real Hamiltonian, at the best one obtains a function
h = h(x, p, v). This is the famous 'h function' named by Goldstein in
his classical textbook on mechanics, but is NOT a Hamiltonian.

But the Hamiltonian is the generator of time translations in QM.
Therefore, this is another proof of why RQFT can only deal with
one-particle states (multiparticles if are factorized) and study the
interaction via perturbation (divergent) theory. This is other of
reaosn of why there exists no one things like RQFT of a full
two-electron system and this is the reaosn that two-body relativistic
theories are based in inconsistent 'hibrids' like Bethe-Salpeter and
similar (i already explain what are some of difficulties with those
equations even if one is non rigorous and assumes that CB or CG
effective potentials derived from RQFT are correct, there is extensive
literature in why are inadequate and may be corrected -as pointed in a
previous post- via positive energy Casimir type projection operators
for the leptons, for instance.

The use of Lagrangian formalism does not change things:

1) The generator of time translations in QM is the Hamiltonian, as
well-explained by Weinberg. Feynmann path integral follow from
Hamiltonian mechanics, when interaction is small and for short times
propagator WITH THE FULL HAMILTONIAN factorizes like (free term *
interaction term). Then one can prove that this is equivalent to the
use of an action with the Lagrangian (free term minus potential).

2) The N-body full relativistic Lagrangian is unphysical (QFT theory
deals only with approximated Lagrangians). For example, the potential
in the Lagrangian U contains unphysical terms associated to velocity
terms. The only real potential is eV as proved by the Hamiltonian
formalism (as explained by Goldstein in his celebrated textbook on
mechanics any potential with velocity dependendent terms is
unphysical). In fact, the P = (partial L / partial v) derived from
aditional velocity-dependent terms in the Lagrangian is also
'unphysical'. This is the reason that the canonical momentum contains
'correction' terms as eA that may be eliminated in the Hamiltonian
formalism. The minimal coupling rule of usual literature p -> p - eA is
misleading because 'p' at the left is mv but 'p' at the right is NOT mv
is P: the canonical momentum which is NOT mv.

The minimal coupling rule looks like p -> P - eA or mv --> mv.

3) Path integral methods relies in the asumption of quasi-free motion
(one expands the propagator H around the free motion propagator T) and
focuses only in the interaction representation V_int. The true
generator of the dynamics in QM is, as stated by Weinberg the
Hamiltonian -see Weinberg discussion on his volume 1- Moreover, in some
cases explicitely cited by Weinberg, the path integral method offer
*incorrect* replies for measured observables whereas the canonical
method based in a Hamiltonian offers the *right* replies.

Etc.

Juan R.

Center for CANONICAL |SCIENCE)

[Juan R.]

Eugene Stefanovich wrote:
> I am not sure if I understood you correctly. In my view, every
> N-particle sector of the Fock space is build as a tensor product of
> 1-particle spaces. Therefore, tensor product states |1>|2>|3>...|N>
> form a basis there. Any other state is a linear combination of the
> tensor product states. This is true independent on whether particles are
> free or interacting. Free quantum fields can be always defined
> as linear combinations of particle creation and annihilation operators.
> "Interacting" (Heisenberg) quantum fields can also be defined
> by switching from the
> free to the interacting Hamiltonian in the formula for the time
> evolution of the field. My only "problem" is that I don't know what's
> the physical meaning of these "interacting fields". In my opinion, they
> are not needed for a theoretical description of the system. All we want
> to know about the system can be obtained without using interacting
> quantum fields.

The use of a Hilbert-Fock space is dependent of class of problems that
you was addresing and the level of rigor you need. There are well-know
situations in theoretical chemistry where a Hilbert-Fock space is not
suficient. Literature is extensive and i will not cite here. I only
call your atention to PRA 1996 53(6) 4075, where you can find some
remarks on why certain IMPORTANT effects cannot be studied in a Hilbert
space. Authors of the PRA propose an extension of scatering theory
*beyond* the Hilbert-Fock space:

"Persistent interactions require singular distribution functions which
lie outside the Hilbert space"

> > In rigor, multiparticle wave functions are NOT possible in relativistic
> > QFT, because there is not dynamical variables for those vawefunctions.
>
> I don't see any problem with multiparticle wave functions in QFT.
> In each N-particle sector of the Fock space you can define
> particle observables. For example, you can define operators of momentum
> P_i and spin S_i of all N particles. So, you can define momentum-spin
> wavefunction psi(p_1, s_1; p_2, s_2; ..., p_N, s_N) for any state vector
> in this sector. This wave function description works equally well for
> systems with and without interaction.

It is not so simple! The uncertainty relationships of NRQM may be
generalized to the relativistic domain. This was done by Landau in 1930
and proved recently with more mathematical rigor -at level of
Schwarzild inequality- by some authors. The physical insight of
Landau's work is that both x and p are not observable magnitudes in
relativistic regime. Whereas in classical physics there is posibility
for a measuring of x AND p, and in NRQM this is reduced to measuring of
x OR p, the trouble is that in RQM one cannot measure, in general, x
NOR p.

This is the reason that x in NRQM is an observable with asociated
operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.

The same criticism remains for the momentum representation of QM. In
general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
general, a dinamical variable in RQM.

The solution was given by Heisemberg in 1938. One may abandon partially
NRQM. In the approximation of free particles, the principle of
conservation of momentum still HOLD, and then the impulse of free
particles p_j is well defined. But particles are free only in
scattering states, due to application of the principle of decomposition
of clusters: when R --> infinite interaction is zero and particles are
free. This is reason that Weinberg emphasizes the cluster decomposition
princle as one of basic postulates of RQFT.

Then, in scatering states and *only in*, one can do |Phi(p_1, p_2,...
p_N)> = |p_1>|p_2>...|p_N>. The wavefunctions for free particles 1,
2,... N are defined for free ps, which are dinamical variables in the
*asymptotic regime*. In the full interaction regime, p are not
dinamical variables. This is one of reasons that RQFT is ONLY defined
for scattering states and cannot describe bound states.

Of course in the relativistic domain

delta p =of order= hbar / (c delta t)

and delta p zero implies delta t infinite. Scattering states in RQFT
are defined just for an infinite temporal interval [- infinite,
infinite] and RQFT cannot study the details of the dynamics, only the
scattering which is obviously trivial. As explained in PRA 1996 53(6)
4075 in many-body situations the important interval is the dynamical
one -for example TST in chemical dynamics- nor the initial and final
states and whereas computations in RQFT are trivial -see conclusion
section of above paper- the asociated computations in chemistry cannot
be done because mathematical formalism of RQFT breaks. This is the
reason of the new theory proposed by authors of PRA 1996 53(6) 4075.
Note that is not a computational problem (as Igor Khavkine appears to
think) is a *fundamental* problem still unsolved in theoretical
physics. As *emphasized* by authors of above paper, both the S-matrix
theory and basic QM structure of standard field theory break down in
the chemistry of many-body systems.

> > The interacting term is obtained from R-QFT. The term is mathematically
> > divergent and physically only explains scattering of particles in the
> > infinite past and future. What is the interaction term when particles
> > are not separated infinitely?
>
> If you switch to the physical (or dressed) particle representation as
> described above, you'll obtain a Hamiltonian with interaction that
> works perfectly well for both asymptotic and intermediate regimes,
> for stationary and non-stationary states.

In complement to my above discussion of interacting regimes in full
relativistic quantum mechanics i would add some others difficulties.
There are many of them and it would be extensive to discuss all of them
here in detail. However, i will focus on an important point: there is
NOT Hamiltonian in the full relativistic regime.

The proof of this is a bit thecnical and needs of a rigorous study of
Hamiltonian mechanics, unfortunately all of textbooks in RQFT and many
papers simply ignore this.

The discussion in the classical case is more easy. Most authors propose
-without further discussion- that the Hamiltonian with full EM
interactions is

H = (p - eA)/2m + eV

ok?

Well, this is wrong. ONLY for a single particle, in an external field,
above definition is valid, because then A = A(x, t) and the Hamiltonian
is a real Hamiltonian with functional dependence H = H(x, p).

If you adds a second electron ' a new term A(v') arises in the
Lagrangian and the nonlinearity of the Legendre transformation impides
the obtaining of a real Hamiltonian, at the best one obtains a function
h = h(x, p, v). This is the famous 'h function' named by Goldstein in
his classical textbook on mechanics, but is NOT a Hamiltonian.

But the Hamiltonian is the generator of time translations in QM.
Therefore, this is another proof of why RQFT can only deal with
one-particle states (multiparticles if are factorized) and study the
interaction via perturbation (divergent) theory. This is other of
reaosn of why there exists no one things like RQFT of a full
two-electron system and this is the reaosn that two-body relativistic
theories are based in inconsistent 'hibrids' like Bethe-Salpeter and
similar (i already explain what are some of difficulties with those
equations even if one is non rigorous and assumes that CB or CG
effective potentials derived from RQFT are correct, there is extensive
literature in why are inadequate and may be corrected -as pointed in a
previous post- via positive energy Casimir type projection operators
for the leptons, for instance.

The use of Lagrangian formalism does not change things:

1) The generator of time translations in QM is the Hamiltonian, as
well-explained by Weinberg. Feynmann path integral follow from
Hamiltonian mechanics, when interaction is small and for short times
propagator WITH THE FULL HAMILTONIAN factorizes like (free term *
interaction term). Then one can prove that this is equivalent to the
use of an action with the Lagrangian (free term minus potential).

2) The N-body full relativistic Lagrangian is unphysical (QFT theory
deals only with approximated Lagrangians). For example, the potential
in the Lagrangian U contains unphysical terms associated to velocity
terms. The only real potential is eV as proved by the Hamiltonian
formalism (as explained by Goldstein in his celebrated textbook on
mechanics any potential with velocity dependendent terms is
unphysical). In fact, the P = (partial L / partial v) derived from
aditional velocity-dependent terms in the Lagrangian is also
'unphysical'. This is the reason that the canonical momentum contains
'correction' terms as eA that may be eliminated in the Hamiltonian
formalism. The minimal coupling rule of usual literature p -> p - eA is
misleading because 'p' at the left is mv but 'p' at the right is NOT mv
is P: the canonical momentum which is NOT mv.

The minimal coupling rule looks like p -> P - eA or mv --> mv.

3) Path integral methods relies in the asumption of quasi-free motion
(one expands the propagator H around the free motion propagator T) and
focuses only in the interaction representation V_int. The true
generator of the dynamics in QM is, as stated by Weinberg the
Hamiltonian -see Weinberg discussion on his volume 1- Moreover, in some
cases explicitely cited by Weinberg, the path integral method offer
*incorrect* replies for measured observables whereas the canonical
method based in a Hamiltonian offers the *right* replies.

Etc.

Juan R.

Center for CANONICAL |SCIENCE)

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1129294765.036332.72810@g49g2000cwa.googlegro ups.com...
> It is not so simple! The uncertainty relationships of NRQM may be
> generalized to the relativistic domain. This was done by Landau in 1930
> and proved recently with more mathematical rigor -at level of
> Schwarzild inequality- by some authors. The physical insight of
> Landau's work is that both x and p are not observable magnitudes in
> relativistic regime. Whereas in classical physics there is posibility
> for a measuring of x AND p, and in NRQM this is reduced to measuring of
> x OR p, the trouble is that in RQM one cannot measure, in general, x
> NOR p.
>
> This is the reason that x in NRQM is an observable with asociated
> operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.
>
> The same criticism remains for the momentum representation of QM. In
> general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
> general, a dinamical variable in RQM.
>
> The solution was given by Heisemberg in 1938. One may abandon partially
> NRQM. In the approximation of free particles, the principle of
> conservation of momentum still HOLD, and then the impulse of free
> particles p_j is well defined. But particles are free only in
> scattering states, due to application of the principle of decomposition
> of clusters: when R --> infinite interaction is zero and particles are
> free. This is reason that Weinberg emphasizes the cluster decomposition
> princle as one of basic postulates of RQFT.

That's not how I understand quantum mechanics (non-relativistic or
relativistic, it doesn't matter much). The very basic principle of QM
requires that if we have observable (position, momentum, or whatever)
then there always exist states (called eigenstates) in which this
observable can be measured with absolute certainty. This fact is
completely independent on what are the interactions in the system. If
you deny this basic principle, then you are not doing quantum mechanics.
It surely looks like modern relativistic QFT abandons some basic
postulates of QM. People don't like to talk about position-space wave
functions, Hamiltonian, time evolution, etc. However, it only seems that
way. In fact, RQFT is nothing more than application of good old QM to
systems invariant with respect to the Poincare group in which the number
of particles is not conserved.

The "only" major problem of traditional RQFT is that it permits weird
interaction terms (i.e., the famous tri-linear electron-photon
coupling). These bad interactions lead to "non-trivial" vacuum, real
particles dressed by virtual particles, renormalization, and bunch of
other things that make RQFT almost incomprehensible. If you forbid the
presence of these unphysical interactions in RQFT, then you'll get a
nice logical theory that can explain everything using the language of
good old QM.

> As explained in PRA 1996 53(6)
> 4075 in many-body situations the important interval is the dynamical
> one -for example TST in chemical dynamics- nor the initial and final
> states and whereas computations in RQFT are trivial -see conclusion
> section of above paper- the asociated computations in chemistry cannot
> be done because mathematical formalism of RQFT breaks.

Does TST mean "transition state"? I've done some quantum chemistry
calculations. There are some technical difficulties associated with
transition state calculations, but fundamentally it's all crystal clear.

>
> The discussion in the classical case is more easy. Most authors propose
> -without further discussion- that the Hamiltonian with full EM
> interactions is
>
> H = (p - eA)/2m + eV
>
> ok?
>
> Well, this is wrong. ONLY for a single particle, in an external field,
> above definition is valid, because then A = A(x, t) and the Hamiltonian
> is a real Hamiltonian with functional dependence H = H(x, p).
>
> If you adds a second electron ' a new term A(v') arises in the
> Lagrangian and the nonlinearity of the Legendre transformation impides
> the obtaining of a real Hamiltonian, at the best one obtains a function
> h = h(x, p, v). This is the famous 'h function' named by Goldstein in
> his classical textbook on mechanics, but is NOT a Hamiltonian.

Your H is not a good Hamiltonian. At the risk of annoying our
moderators, I'll again direct you to chapter 12 of my book where an
appropriate electromagnetic Hamiltonian for "dressed particle" QED is
constructed.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1129294765.036332.72810@g49g2000cwa.googlegro ups.com...
> It is not so simple! The uncertainty relationships of NRQM may be
> generalized to the relativistic domain. This was done by Landau in 1930
> and proved recently with more mathematical rigor -at level of
> Schwarzild inequality- by some authors. The physical insight of
> Landau's work is that both x and p are not observable magnitudes in
> relativistic regime. Whereas in classical physics there is posibility
> for a measuring of x AND p, and in NRQM this is reduced to measuring of
> x OR p, the trouble is that in RQM one cannot measure, in general, x
> NOR p.
>
> This is the reason that x in NRQM is an observable with asociated
> operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.
>
> The same criticism remains for the momentum representation of QM. In
> general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
> general, a dinamical variable in RQM.
>
> The solution was given by Heisemberg in 1938. One may abandon partially
> NRQM. In the approximation of free particles, the principle of
> conservation of momentum still HOLD, and then the impulse of free
> particles p_j is well defined. But particles are free only in
> scattering states, due to application of the principle of decomposition
> of clusters: when R --> infinite interaction is zero and particles are
> free. This is reason that Weinberg emphasizes the cluster decomposition
> princle as one of basic postulates of RQFT.

That's not how I understand quantum mechanics (non-relativistic or
relativistic, it doesn't matter much). The very basic principle of QM
requires that if we have observable (position, momentum, or whatever)
then there always exist states (called eigenstates) in which this
observable can be measured with absolute certainty. This fact is
completely independent on what are the interactions in the system. If
you deny this basic principle, then you are not doing quantum mechanics.
It surely looks like modern relativistic QFT abandons some basic
postulates of QM. People don't like to talk about position-space wave
functions, Hamiltonian, time evolution, etc. However, it only seems that
way. In fact, RQFT is nothing more than application of good old QM to
systems invariant with respect to the Poincare group in which the number
of particles is not conserved.

The "only" major problem of traditional RQFT is that it permits weird
interaction terms (i.e., the famous tri-linear electron-photon
coupling). These bad interactions lead to "non-trivial" vacuum, real
particles dressed by virtual particles, renormalization, and bunch of
other things that make RQFT almost incomprehensible. If you forbid the
presence of these unphysical interactions in RQFT, then you'll get a
nice logical theory that can explain everything using the language of
good old QM.

> As explained in PRA 1996 53(6)
> 4075 in many-body situations the important interval is the dynamical
> one -for example TST in chemical dynamics- nor the initial and final
> states and whereas computations in RQFT are trivial -see conclusion
> section of above paper- the asociated computations in chemistry cannot
> be done because mathematical formalism of RQFT breaks.

Does TST mean "transition state"? I've done some quantum chemistry
calculations. There are some technical difficulties associated with
transition state calculations, but fundamentally it's all crystal clear.

>
> The discussion in the classical case is more easy. Most authors propose
> -without further discussion- that the Hamiltonian with full EM
> interactions is
>
> H = (p - eA)/2m + eV
>
> ok?
>
> Well, this is wrong. ONLY for a single particle, in an external field,
> above definition is valid, because then A = A(x, t) and the Hamiltonian
> is a real Hamiltonian with functional dependence H = H(x, p).
>
> If you adds a second electron ' a new term A(v') arises in the
> Lagrangian and the nonlinearity of the Legendre transformation impides
> the obtaining of a real Hamiltonian, at the best one obtains a function
> h = h(x, p, v). This is the famous 'h function' named by Goldstein in
> his classical textbook on mechanics, but is NOT a Hamiltonian.

Your H is not a good Hamiltonian. At the risk of annoying our
moderators, I'll again direct you to chapter 12 of my book where an
appropriate electromagnetic Hamiltonian for "dressed particle" QED is
constructed.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1129294765.036332.72810@g49g2000cwa.googlegro ups.com...
> It is not so simple! The uncertainty relationships of NRQM may be
> generalized to the relativistic domain. This was done by Landau in 1930
> and proved recently with more mathematical rigor -at level of
> Schwarzild inequality- by some authors. The physical insight of
> Landau's work is that both x and p are not observable magnitudes in
> relativistic regime. Whereas in classical physics there is posibility
> for a measuring of x AND p, and in NRQM this is reduced to measuring of
> x OR p, the trouble is that in RQM one cannot measure, in general, x
> NOR p.
>
> This is the reason that x in NRQM is an observable with asociated
> operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.
>
> The same criticism remains for the momentum representation of QM. In
> general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
> general, a dinamical variable in RQM.
>
> The solution was given by Heisemberg in 1938. One may abandon partially
> NRQM. In the approximation of free particles, the principle of
> conservation of momentum still HOLD, and then the impulse of free
> particles p_j is well defined. But particles are free only in
> scattering states, due to application of the principle of decomposition
> of clusters: when R --> infinite interaction is zero and particles are
> free. This is reason that Weinberg emphasizes the cluster decomposition
> princle as one of basic postulates of RQFT.

That's not how I understand quantum mechanics (non-relativistic or
relativistic, it doesn't matter much). The very basic principle of QM
requires that if we have observable (position, momentum, or whatever)
then there always exist states (called eigenstates) in which this
observable can be measured with absolute certainty. This fact is
completely independent on what are the interactions in the system. If
you deny this basic principle, then you are not doing quantum mechanics.
It surely looks like modern relativistic QFT abandons some basic
postulates of QM. People don't like to talk about position-space wave
functions, Hamiltonian, time evolution, etc. However, it only seems that
way. In fact, RQFT is nothing more than application of good old QM to
systems invariant with respect to the Poincare group in which the number
of particles is not conserved.

The "only" major problem of traditional RQFT is that it permits weird
interaction terms (i.e., the famous tri-linear electron-photon
coupling). These bad interactions lead to "non-trivial" vacuum, real
particles dressed by virtual particles, renormalization, and bunch of
other things that make RQFT almost incomprehensible. If you forbid the
presence of these unphysical interactions in RQFT, then you'll get a
nice logical theory that can explain everything using the language of
good old QM.

> As explained in PRA 1996 53(6)
> 4075 in many-body situations the important interval is the dynamical
> one -for example TST in chemical dynamics- nor the initial and final
> states and whereas computations in RQFT are trivial -see conclusion
> section of above paper- the asociated computations in chemistry cannot
> be done because mathematical formalism of RQFT breaks.

Does TST mean "transition state"? I've done some quantum chemistry
calculations. There are some technical difficulties associated with
transition state calculations, but fundamentally it's all crystal clear.

>
> The discussion in the classical case is more easy. Most authors propose
> -without further discussion- that the Hamiltonian with full EM
> interactions is
>
> H = (p - eA)/2m + eV
>
> ok?
>
> Well, this is wrong. ONLY for a single particle, in an external field,
> above definition is valid, because then A = A(x, t) and the Hamiltonian
> is a real Hamiltonian with functional dependence H = H(x, p).
>
> If you adds a second electron ' a new term A(v') arises in the
> Lagrangian and the nonlinearity of the Legendre transformation impides
> the obtaining of a real Hamiltonian, at the best one obtains a function
> h = h(x, p, v). This is the famous 'h function' named by Goldstein in
> his classical textbook on mechanics, but is NOT a Hamiltonian.

Your H is not a good Hamiltonian. At the risk of annoying our
moderators, I'll again direct you to chapter 12 of my book where an
appropriate electromagnetic Hamiltonian for "dressed particle" QED is
constructed.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1129294765.036332.72810@g49g2000cwa.googlegro ups.com...
> It is not so simple! The uncertainty relationships of NRQM may be
> generalized to the relativistic domain. This was done by Landau in 1930
> and proved recently with more mathematical rigor -at level of
> Schwarzild inequality- by some authors. The physical insight of
> Landau's work is that both x and p are not observable magnitudes in
> relativistic regime. Whereas in classical physics there is posibility
> for a measuring of x AND p, and in NRQM this is reduced to measuring of
> x OR p, the trouble is that in RQM one cannot measure, in general, x
> NOR p.
>
> This is the reason that x in NRQM is an observable with asociated
> operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.
>
> The same criticism remains for the momentum representation of QM. In
> general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
> general, a dinamical variable in RQM.
>
> The solution was given by Heisemberg in 1938. One may abandon partially
> NRQM. In the approximation of free particles, the principle of
> conservation of momentum still HOLD, and then the impulse of free
> particles p_j is well defined. But particles are free only in
> scattering states, due to application of the principle of decomposition
> of clusters: when R --> infinite interaction is zero and particles are
> free. This is reason that Weinberg emphasizes the cluster decomposition
> princle as one of basic postulates of RQFT.

That's not how I understand quantum mechanics (non-relativistic or
relativistic, it doesn't matter much). The very basic principle of QM
requires that if we have observable (position, momentum, or whatever)
then there always exist states (called eigenstates) in which this
observable can be measured with absolute certainty. This fact is
completely independent on what are the interactions in the system. If
you deny this basic principle, then you are not doing quantum mechanics.
It surely looks like modern relativistic QFT abandons some basic
postulates of QM. People don't like to talk about position-space wave
functions, Hamiltonian, time evolution, etc. However, it only seems that
way. In fact, RQFT is nothing more than application of good old QM to
systems invariant with respect to the Poincare group in which the number
of particles is not conserved.

The "only" major problem of traditional RQFT is that it permits weird
interaction terms (i.e., the famous tri-linear electron-photon
coupling). These bad interactions lead to "non-trivial" vacuum, real
particles dressed by virtual particles, renormalization, and bunch of
other things that make RQFT almost incomprehensible. If you forbid the
presence of these unphysical interactions in RQFT, then you'll get a
nice logical theory that can explain everything using the language of
good old QM.

> As explained in PRA 1996 53(6)
> 4075 in many-body situations the important interval is the dynamical
> one -for example TST in chemical dynamics- nor the initial and final
> states and whereas computations in RQFT are trivial -see conclusion
> section of above paper- the asociated computations in chemistry cannot
> be done because mathematical formalism of RQFT breaks.

Does TST mean "transition state"? I've done some quantum chemistry
calculations. There are some technical difficulties associated with
transition state calculations, but fundamentally it's all crystal clear.

>
> The discussion in the classical case is more easy. Most authors propose
> -without further discussion- that the Hamiltonian with full EM
> interactions is
>
> H = (p - eA)/2m + eV
>
> ok?
>
> Well, this is wrong. ONLY for a single particle, in an external field,
> above definition is valid, because then A = A(x, t) and the Hamiltonian
> is a real Hamiltonian with functional dependence H = H(x, p).
>
> If you adds a second electron ' a new term A(v') arises in the
> Lagrangian and the nonlinearity of the Legendre transformation impides
> the obtaining of a real Hamiltonian, at the best one obtains a function
> h = h(x, p, v). This is the famous 'h function' named by Goldstein in
> his classical textbook on mechanics, but is NOT a Hamiltonian.

Your H is not a good Hamiltonian. At the risk of annoying our
moderators, I'll again direct you to chapter 12 of my book where an
appropriate electromagnetic Hamiltonian for "dressed particle" QED is
constructed.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1129294765.036332.72810@g49g2000cwa.googlegro ups.com...
> It is not so simple! The uncertainty relationships of NRQM may be
> generalized to the relativistic domain. This was done by Landau in 1930
> and proved recently with more mathematical rigor -at level of
> Schwarzild inequality- by some authors. The physical insight of
> Landau's work is that both x and p are not observable magnitudes in
> relativistic regime. Whereas in classical physics there is posibility
> for a measuring of x AND p, and in NRQM this is reduced to measuring of
> x OR p, the trouble is that in RQM one cannot measure, in general, x
> NOR p.
>
> This is the reason that x in NRQM is an observable with asociated
> operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.
>
> The same criticism remains for the momentum representation of QM. In
> general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
> general, a dinamical variable in RQM.
>
> The solution was given by Heisemberg in 1938. One may abandon partially
> NRQM. In the approximation of free particles, the principle of
> conservation of momentum still HOLD, and then the impulse of free
> particles p_j is well defined. But particles are free only in
> scattering states, due to application of the principle of decomposition
> of clusters: when R --> infinite interaction is zero and particles are
> free. This is reason that Weinberg emphasizes the cluster decomposition
> princle as one of basic postulates of RQFT.

That's not how I understand quantum mechanics (non-relativistic or
relativistic, it doesn't matter much). The very basic principle of QM
requires that if we have observable (position, momentum, or whatever)
then there always exist states (called eigenstates) in which this
observable can be measured with absolute certainty. This fact is
completely independent on what are the interactions in the system. If
you deny this basic principle, then you are not doing quantum mechanics.
It surely looks like modern relativistic QFT abandons some basic
postulates of QM. People don't like to talk about position-space wave
functions, Hamiltonian, time evolution, etc. However, it only seems that
way. In fact, RQFT is nothing more than application of good old QM to
systems invariant with respect to the Poincare group in which the number
of particles is not conserved.

The "only" major problem of traditional RQFT is that it permits weird
interaction terms (i.e., the famous tri-linear electron-photon
coupling). These bad interactions lead to "non-trivial" vacuum, real
particles dressed by virtual particles, renormalization, and bunch of
other things that make RQFT almost incomprehensible. If you forbid the
presence of these unphysical interactions in RQFT, then you'll get a
nice logical theory that can explain everything using the language of
good old QM.

> As explained in PRA 1996 53(6)
> 4075 in many-body situations the important interval is the dynamical
> one -for example TST in chemical dynamics- nor the initial and final
> states and whereas computations in RQFT are trivial -see conclusion
> section of above paper- the asociated computations in chemistry cannot
> be done because mathematical formalism of RQFT breaks.

Does TST mean "transition state"? I've done some quantum chemistry
calculations. There are some technical difficulties associated with
transition state calculations, but fundamentally it's all crystal clear.

>
> The discussion in the classical case is more easy. Most authors propose
> -without further discussion- that the Hamiltonian with full EM
> interactions is
>
> H = (p - eA)/2m + eV
>
> ok?
>
> Well, this is wrong. ONLY for a single particle, in an external field,
> above definition is valid, because then A = A(x, t) and the Hamiltonian
> is a real Hamiltonian with functional dependence H = H(x, p).
>
> If you adds a second electron ' a new term A(v') arises in the
> Lagrangian and the nonlinearity of the Legendre transformation impides
> the obtaining of a real Hamiltonian, at the best one obtains a function
> h = h(x, p, v). This is the famous 'h function' named by Goldstein in
> his classical textbook on mechanics, but is NOT a Hamiltonian.

Your H is not a good Hamiltonian. At the risk of annoying our
moderators, I'll again direct you to chapter 12 of my book where an
appropriate electromagnetic Hamiltonian for "dressed particle" QED is
constructed.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1129294765.036332.72810@g49g2000cwa.googlegro ups.com...
> It is not so simple! The uncertainty relationships of NRQM may be
> generalized to the relativistic domain. This was done by Landau in 1930
> and proved recently with more mathematical rigor -at level of
> Schwarzild inequality- by some authors. The physical insight of
> Landau's work is that both x and p are not observable magnitudes in
> relativistic regime. Whereas in classical physics there is posibility
> for a measuring of x AND p, and in NRQM this is reduced to measuring of
> x OR p, the trouble is that in RQM one cannot measure, in general, x
> NOR p.
>
> This is the reason that x in NRQM is an observable with asociated
> operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.
>
> The same criticism remains for the momentum representation of QM. In
> general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
> general, a dinamical variable in RQM.
>
> The solution was given by Heisemberg in 1938. One may abandon partially
> NRQM. In the approximation of free particles, the principle of
> conservation of momentum still HOLD, and then the impulse of free
> particles p_j is well defined. But particles are free only in
> scattering states, due to application of the principle of decomposition
> of clusters: when R --> infinite interaction is zero and particles are
> free. This is reason that Weinberg emphasizes the cluster decomposition
> princle as one of basic postulates of RQFT.

That's not how I understand quantum mechanics (non-relativistic or
relativistic, it doesn't matter much). The very basic principle of QM
requires that if we have observable (position, momentum, or whatever)
then there always exist states (called eigenstates) in which this
observable can be measured with absolute certainty. This fact is
completely independent on what are the interactions in the system. If
you deny this basic principle, then you are not doing quantum mechanics.
It surely looks like modern relativistic QFT abandons some basic
postulates of QM. People don't like to talk about position-space wave
functions, Hamiltonian, time evolution, etc. However, it only seems that
way. In fact, RQFT is nothing more than application of good old QM to
systems invariant with respect to the Poincare group in which the number
of particles is not conserved.

The "only" major problem of traditional RQFT is that it permits weird
interaction terms (i.e., the famous tri-linear electron-photon
coupling). These bad interactions lead to "non-trivial" vacuum, real
particles dressed by virtual particles, renormalization, and bunch of
other things that make RQFT almost incomprehensible. If you forbid the
presence of these unphysical interactions in RQFT, then you'll get a
nice logical theory that can explain everything using the language of
good old QM.

> As explained in PRA 1996 53(6)
> 4075 in many-body situations the important interval is the dynamical
> one -for example TST in chemical dynamics- nor the initial and final
> states and whereas computations in RQFT are trivial -see conclusion
> section of above paper- the asociated computations in chemistry cannot
> be done because mathematical formalism of RQFT breaks.

Does TST mean "transition state"? I've done some quantum chemistry
calculations. There are some technical difficulties associated with
transition state calculations, but fundamentally it's all crystal clear.

>
> The discussion in the classical case is more easy. Most authors propose
> -without further discussion- that the Hamiltonian with full EM
> interactions is
>
> H = (p - eA)/2m + eV
>
> ok?
>
> Well, this is wrong. ONLY for a single particle, in an external field,
> above definition is valid, because then A = A(x, t) and the Hamiltonian
> is a real Hamiltonian with functional dependence H = H(x, p).
>
> If you adds a second electron ' a new term A(v') arises in the
> Lagrangian and the nonlinearity of the Legendre transformation impides
> the obtaining of a real Hamiltonian, at the best one obtains a function
> h = h(x, p, v). This is the famous 'h function' named by Goldstein in
> his classical textbook on mechanics, but is NOT a Hamiltonian.

Your H is not a good Hamiltonian. At the risk of annoying our
moderators, I'll again direct you to chapter 12 of my book where an
appropriate electromagnetic Hamiltonian for "dressed particle" QED is
constructed.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1129294765.036332.72810@g49g2000cwa.googlegro ups.com...
> It is not so simple! The uncertainty relationships of NRQM may be
> generalized to the relativistic domain. This was done by Landau in 1930
> and proved recently with more mathematical rigor -at level of
> Schwarzild inequality- by some authors. The physical insight of
> Landau's work is that both x and p are not observable magnitudes in
> relativistic regime. Whereas in classical physics there is posibility
> for a measuring of x AND p, and in NRQM this is reduced to measuring of
> x OR p, the trouble is that in RQM one cannot measure, in general, x
> NOR p.
>
> This is the reason that x in NRQM is an observable with asociated
> operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.
>
> The same criticism remains for the momentum representation of QM. In
> general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
> general, a dinamical variable in RQM.
>
> The solution was given by Heisemberg in 1938. One may abandon partially
> NRQM. In the approximation of free particles, the principle of
> conservation of momentum still HOLD, and then the impulse of free
> particles p_j is well defined. But particles are free only in
> scattering states, due to application of the principle of decomposition
> of clusters: when R --> infinite interaction is zero and particles are
> free. This is reason that Weinberg emphasizes the cluster decomposition
> princle as one of basic postulates of RQFT.

That's not how I understand quantum mechanics (non-relativistic or
relativistic, it doesn't matter much). The very basic principle of QM
requires that if we have observable (position, momentum, or whatever)
then there always exist states (called eigenstates) in which this
observable can be measured with absolute certainty. This fact is
completely independent on what are the interactions in the system. If
you deny this basic principle, then you are not doing quantum mechanics.
It surely looks like modern relativistic QFT abandons some basic
postulates of QM. People don't like to talk about position-space wave
functions, Hamiltonian, time evolution, etc. However, it only seems that
way. In fact, RQFT is nothing more than application of good old QM to
systems invariant with respect to the Poincare group in which the number
of particles is not conserved.

The "only" major problem of traditional RQFT is that it permits weird
interaction terms (i.e., the famous tri-linear electron-photon
coupling). These bad interactions lead to "non-trivial" vacuum, real
particles dressed by virtual particles, renormalization, and bunch of
other things that make RQFT almost incomprehensible. If you forbid the
presence of these unphysical interactions in RQFT, then you'll get a
nice logical theory that can explain everything using the language of
good old QM.

> As explained in PRA 1996 53(6)
> 4075 in many-body situations the important interval is the dynamical
> one -for example TST in chemical dynamics- nor the initial and final
> states and whereas computations in RQFT are trivial -see conclusion
> section of above paper- the asociated computations in chemistry cannot
> be done because mathematical formalism of RQFT breaks.

Does TST mean "transition state"? I've done some quantum chemistry
calculations. There are some technical difficulties associated with
transition state calculations, but fundamentally it's all crystal clear.

>
> The discussion in the classical case is more easy. Most authors propose
> -without further discussion- that the Hamiltonian with full EM
> interactions is
>
> H = (p - eA)/2m + eV
>
> ok?
>
> Well, this is wrong. ONLY for a single particle, in an external field,
> above definition is valid, because then A = A(x, t) and the Hamiltonian
> is a real Hamiltonian with functional dependence H = H(x, p).
>
> If you adds a second electron ' a new term A(v') arises in the
> Lagrangian and the nonlinearity of the Legendre transformation impides
> the obtaining of a real Hamiltonian, at the best one obtains a function
> h = h(x, p, v). This is the famous 'h function' named by Goldstein in
> his classical textbook on mechanics, but is NOT a Hamiltonian.

Your H is not a good Hamiltonian. At the risk of annoying our
moderators, I'll again direct you to chapter 12 of my book where an
appropriate electromagnetic Hamiltonian for "dressed particle" QED is
constructed.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1129294765.036332.72810@g49g2000cwa.googlegro ups.com...
> It is not so simple! The uncertainty relationships of NRQM may be
> generalized to the relativistic domain. This was done by Landau in 1930
> and proved recently with more mathematical rigor -at level of
> Schwarzild inequality- by some authors. The physical insight of
> Landau's work is that both x and p are not observable magnitudes in
> relativistic regime. Whereas in classical physics there is posibility
> for a measuring of x AND p, and in NRQM this is reduced to measuring of
> x OR p, the trouble is that in RQM one cannot measure, in general, x
> NOR p.
>
> This is the reason that x in NRQM is an observable with asociated
> operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.
>
> The same criticism remains for the momentum representation of QM. In
> general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
> general, a dinamical variable in RQM.
>
> The solution was given by Heisemberg in 1938. One may abandon partially
> NRQM. In the approximation of free particles, the principle of
> conservation of momentum still HOLD, and then the impulse of free
> particles p_j is well defined. But particles are free only in
> scattering states, due to application of the principle of decomposition
> of clusters: when R --> infinite interaction is zero and particles are
> free. This is reason that Weinberg emphasizes the cluster decomposition
> princle as one of basic postulates of RQFT.

That's not how I understand quantum mechanics (non-relativistic or
relativistic, it doesn't matter much). The very basic principle of QM
requires that if we have observable (position, momentum, or whatever)
then there always exist states (called eigenstates) in which this
observable can be measured with absolute certainty. This fact is
completely independent on what are the interactions in the system. If
you deny this basic principle, then you are not doing quantum mechanics.
It surely looks like modern relativistic QFT abandons some basic
postulates of QM. People don't like to talk about position-space wave
functions, Hamiltonian, time evolution, etc. However, it only seems that
way. In fact, RQFT is nothing more than application of good old QM to
systems invariant with respect to the Poincare group in which the number
of particles is not conserved.

The "only" major problem of traditional RQFT is that it permits weird
interaction terms (i.e., the famous tri-linear electron-photon
coupling). These bad interactions lead to "non-trivial" vacuum, real
particles dressed by virtual particles, renormalization, and bunch of
other things that make RQFT almost incomprehensible. If you forbid the
presence of these unphysical interactions in RQFT, then you'll get a
nice logical theory that can explain everything using the language of
good old QM.

> As explained in PRA 1996 53(6)
> 4075 in many-body situations the important interval is the dynamical
> one -for example TST in chemical dynamics- nor the initial and final
> states and whereas computations in RQFT are trivial -see conclusion
> section of above paper- the asociated computations in chemistry cannot
> be done because mathematical formalism of RQFT breaks.

Does TST mean "transition state"? I've done some quantum chemistry
calculations. There are some technical difficulties associated with
transition state calculations, but fundamentally it's all crystal clear.

>
> The discussion in the classical case is more easy. Most authors propose
> -without further discussion- that the Hamiltonian with full EM
> interactions is
>
> H = (p - eA)/2m + eV
>
> ok?
>
> Well, this is wrong. ONLY for a single particle, in an external field,
> above definition is valid, because then A = A(x, t) and the Hamiltonian
> is a real Hamiltonian with functional dependence H = H(x, p).
>
> If you adds a second electron ' a new term A(v') arises in the
> Lagrangian and the nonlinearity of the Legendre transformation impides
> the obtaining of a real Hamiltonian, at the best one obtains a function
> h = h(x, p, v). This is the famous 'h function' named by Goldstein in
> his classical textbook on mechanics, but is NOT a Hamiltonian.

Your H is not a good Hamiltonian. At the risk of annoying our
moderators, I'll again direct you to chapter 12 of my book where an
appropriate electromagnetic Hamiltonian for "dressed particle" QED is
constructed.

Eugene.

[Eugene Stefanovich]

"Juan R." <juanrgonzaleza@canonicalscience.com> wrote in message
news:1129294765.036332.72810@g49g2000cwa.googlegro ups.com...
> It is not so simple! The uncertainty relationships of NRQM may be
> generalized to the relativistic domain. This was done by Landau in 1930
> and proved recently with more mathematical rigor -at level of
> Schwarzild inequality- by some authors. The physical insight of
> Landau's work is that both x and p are not observable magnitudes in
> relativistic regime. Whereas in classical physics there is posibility
> for a measuring of x AND p, and in NRQM this is reduced to measuring of
> x OR p, the trouble is that in RQM one cannot measure, in general, x
> NOR p.
>
> This is the reason that x in NRQM is an observable with asociated
> operator x_op, whereas in RQFT x is a parameter and there is NOT x_op.
>
> The same criticism remains for the momentum representation of QM. In
> general, there is not |Phi(p_1, p_2,... p_N)> because p is *not*, in
> general, a dinamical variable in RQM.
>
> The solution was given by Heisemberg in 1938. One may abandon partially
> NRQM. In the approximation of free particles, the principle of
> conservation of momentum still HOLD, and then the impulse of free
> particles p_j is well defined. But particles are free only in
> scattering states, due to application of the principle of decomposition
> of clusters: when R --> infinite interaction is zero and particles are
> free. This is reason that Weinberg emphasizes the cluster decomposition
> princle as one of basic postulates of RQFT.

That's not how I understand quantum mechanics (non-relativistic or
relativistic, it doesn't matter much). The very basic principle of QM
requires that if we have observable (position, momentum, or whatever)
then there always exist states (called eigenstates) in which this
observable can be measured with absolute certainty. This fact is
completely independent on what are the interactions in the system. If
you deny this basic principle, then you are not doing quantum mechanics.
It surely looks like modern relativistic QFT abandons some basic
postulates of QM. People don't like to talk about position-space wave
functions, Hamiltonian, time evolution, etc. However, it only seems that
way. In fact, RQFT is nothing more than application of good old QM to
systems invariant with respect to the Poincare group in which the number
of particles is not conserved.

The "only" major problem of traditional RQFT is that it permits weird
interaction terms (i.e., the famous tri-linear electron-photon
coupling). These bad interactions lead to "non-trivial" vacuum, real
particles dressed by virtual particles, renormalization, and bunch of
other things that make RQFT almost incomprehensible. If you forbid the
presence of these unphysical interactions in RQFT, then you'll get a
nice logical theory that can explain everything using the language of
good old QM.

> As explained in PRA 1996 53(6)
> 4075 in many-body situations the important interval is the dynamical
> one -for example TST in chemical dynamics- nor the initial and final
> states and whereas computations in RQFT are trivial -see conclusion
> section of above paper- the asociated computations in chemistry cannot
> be done because mathematical formalism of RQFT breaks.

Does TST mean "transition state"? I've done some quantum chemistry
calculations. There are some technical difficulties associated with
transition state calculations, but fundamentally it's all crystal clear.

>
> The discussion in the classical case is more easy. Most authors propose
> -without further discussion- that the Hamiltonian with full EM
> interactions is
>
> H = (p - eA)/2m + eV
>
> ok?
>
> Well, this is wrong. ONLY for a single particle, in an external field,
> above definition is valid, because then A = A(x, t) and the Hamiltonian
> is a real Hamiltonian with functional dependence H = H(x, p).
>
> If you adds a second electron ' a new term A(v') arises in the
> Lagrangian and the nonlinearity of the Legendre transformation impides
> the obtaining of a real Hamiltonian, at the best one obtains a function
> h = h(x, p, v). This is the famous 'h function' named by Goldstein in
> his classical textbook on mechanics, but is NOT a Hamiltonian.

Your H is not a good Hamiltonian. At the risk of annoying our
moderators, I'll again direct you to chapter 12 of my book where an
appropriate electromagnetic Hamiltonian for "dressed particle" QED is
constructed.

Eugene.