Why no tensors in quantum mechanics?

[cfgauss]

I have been studying tensor calculus in relativity a little bit before
school starts, and I have a question that I haven't been able to answer
by myself. I know that we can rewrite Maxwell's equations using the
electromagnetic field tensor,
F^(mu nu) =
(0 E^1 E^2 E^3)
(-E^1 0 B^3 -B^2)
(-E^2 -B^3 0 B^1)
(-E^3 B^2 -B^1 0)
And we can re-write Maxwell's equations as, with d meaning curly-d, and
J is the 4-current
d_nu F^(mu nu) = 4 pi J^mu
d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) = 0

With which we can do all kinds of nice special relativity-stuff, and, I
presume, general relativity stuff, too (although I haven't gotten that
far in what I've been studying).

Now, I know that in quantum mechanics, we have a complex valued state
vector Psi, satisfying some differential equation (which we'll assume
is not the Schrödinger equation, but is relativistically correct,
because I've been thinking about relativity). We can write Psi in
terms of real and imaginary parts, say Psi = A + i B. Now, presumably
if we wanted we could re-write the differential equation that Psi
satisfies as two separate differential equations, one in terms of A and
one in terms of B. Now, it seems like we could form another "quantum
mechanical field tensor" in terms of the components of A and B, just
like we did for E and B, and re-write our differential equations as
tensor equations like we did before, and do more relativity-stuff.
Now, it seems to me that since we've got something in the tensor
language of general relativity, we should be able to do general
relativity with this. But, obviously, we can't, or people would be
doing this. Why don't we do this (or do we, and no one has told me
about it)? At what point does this break and not make any sense
anymore? Can we at least do relativistic quantum mechanics like this
if we wanted to?

I'd really appreciate any comments anyone has about this!

Thanks,
Jeremy Price
cfgauss@u.washington.edu

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[Benjamin Schulz]

cfgauss wrote:

> Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore?

The Correspondence Principle from which operators are formed, does per
definition only hold in Cartesian Coordinates. For example, you could
write an Hamiltionian in spherical coordinates. If you want to put the
definitions of the momentum operator, you get the wrong hamiltonian,
even if the space is the same. Simply by definition you must start with
cartesian coordinates, putting in your operators and then making the
conversion to other coordinates. When the operators are only defined for
a special space with metric, then this is so, with the whole
schroedinger equations and its solutions. A way beyond this, are path
integrals, for example, which one can try to formulate for other
metrics.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[mikem@despammed.com]

Jeremy Price wrote:

> We can write Psi in terms of real and
> imaginary parts, say Psi = A + i B.
> [...]
> Can we at least do relativistic quantum
> mechanics like this if we wanted to?

I can't think why you'd want to. The
probability density involves Psi* Psi
which means both A^2 and B^2 contribute
to the experimentally-meaningful
probability. I doubt there's much
value in separating the two things the
way you suggest.

As for QM and relativity, look in your
nearest physics library for a book
on relativistic QM, which will tell you
about the Klein-Gordon equation and
Dirac equation. Then take a look at
full-on quantum field theory. These
theories are compatible with special
relativity. General relativity is
another matter though.

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

cfgauss wrote:
> I have been studying tensor calculus in relativity a little bit before
> school starts, and I have a question that I haven't been able to answer
> by myself. I know that we can rewrite Maxwell's equations using the
> electromagnetic field tensor,
> F^(mu nu) =3D
> ( 0 E^1 E^2 E^3)
> (-E^1 0 B^3 -B^2)
> (-E^2 -B^3 0 B^1)
> (-E^3 B^2 -B^1 0)
> And we can re-write Maxwell's equations as, with d meaning curly-d, and
> J is the 4-current
> d_nu F^(mu nu) =3D 4 pi J^mu
> d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) =3D 0
>
> With which we can do all kinds of nice special relativity-stuff, and, I
> presume, general relativity stuff, too (although I haven't gotten that
> far in what I've been studying).
>
> Now, I know that in quantum mechanics, we have a complex valued state
> vector Psi, satisfying some differential equation (which we'll assume
> is not the Schr=F6dinger equation, but is relativistically correct,
> because I've been thinking about relativity). We can write Psi in
> terms of real and imaginary parts, say Psi =3D A + i B. Now, presumabl=
y
> if we wanted we could re-write the differential equation that Psi
> satisfies as two separate differential equations, one in terms of A and
> one in terms of B. Now, it seems like we could form another "quantum
> mechanical field tensor" in terms of the components of A and B, just
> like we did for E and B, and re-write our differential equations as
> tensor equations like we did before, and do more relativity-stuff.
> Now, it seems to me that since we've got something in the tensor
> language of general relativity, we should be able to do general
> relativity with this. But, obviously, we can't, or people would be
> doing this. Why don't we do this (or do we, and no one has told me
> about it)? At what point does this break and not make any sense
> anymore? Can we at least do relativistic quantum mechanics like this
> if we wanted to?
>
> I'd really appreciate any comments anyone has about this!

That's an interesting question. And can be asked in a much more general
way. Namely, in many cases in physics, a dynamical variable (such as
the
components of a field, or even just the coordinates of a particle) can
be described either as a collection of real valued parameters or
equivalently as a grouping of these parameters into a certain algebraic
structure (a vector, a tensor, a complex number, a spinor, etc.). How
do
we know which structure to pick to do the grouping?

To answer this question, one has to examine the transformation
properties of these parameters. For example, the velocity of a particle
is described by three real numbers, its X, Y and Z components. You then
notice that these components mix between themselves under rotations,
translations, and other changes of coordinates (for example, X -> X +
theta*Y, Y -> Y - theta*X, under a rotation about the Z axis by a small
angle theta). Therefore, you find that it is advantageous to group
these
numbers into the components of a vector. The fact that this can be done
is somewhat obvious in this case, and perhaps even in the case of the
electric and magnetic field components. But in the latter case, we find
that the components of the E and B fields still mix between each other
when we apply boost transformations. In fact, their transformation
properties are exactly such that we can group them together into the
components of a (0,2)-tensor, the Faraday tensor. The components of the
Faraday tensor don't mix with any other physical quantities, so it
doesn't seem fruitful to group them with anything else. (This is not
strictly true, but I don't want to bring the weak force into this).

Now the case you were wondering about. The Schroedinger wave function
can be written as Psi =3D A + iB, where A and B are real. All of quantum
mechanics can be expressed in terms of A and B instead of Psi. So,
what's the reason for combining them into one complex number? One
reason
is that time evolution mixes them. For a solution of the
time-independent Schroedinger equation of energy E, the time-dependent
solutions will evolve with as A -> A + E*dt*B and B -> B - E*dt*A, for
a
small time interval dt. But for the complex wave function Psi, the time
evolution is written simply as exp(-iEt)*Psi.

There is another important reason to regard the electron wave function
as a complex quantity. It is related to the gauge theory formulation of
electrodynamics. Given the scalar and vector electromagnetic potentials
V and A, a gauge transformation changes them as V -> V + dC/dt, and A
->
grad C, where C is an arbitrary function. At the same time, consistency
requires that the electron wave function transforms as
Psi->exp(-iC)*Psi.
This mixes the real and imaginary parts of Psi, but acts on the complex
Psi by mere multiplication. It can even be said that the electron is
described by a complex wave function *because* it is a charged particle
and interacts with the electromagnetic field. For example, a neutral
pion would be described by a real wave function satisfying the
Klein-Gordon equation, instead of the Schroedinger one.

In the 1920 Stern and Gerlach performed experiments on the interaction
of electrons with a magnetic field. They noticed that there were two
kinds of electrons, which could be distinguished by the direction of
deflection in a magnetic field. This situation can be described by
increasing the number of components of the electron wave function from
one to two (which was done by Pauli). However, they also found that
when
the magnetic field is rotated relative to the electrons (or the othe
way
around), the two kinds of electrons lost their distinction and became
mixed. In other words, rotations mixed the components of the Pauli
2-component wave function. In the end it turns out that Pauli found
that
the two kinds of electrons mixed under rotations precisely how a spinor
would. That is why combining them into one spinor wave function is
advantageous. Note that this is a slight reinterpretation of history
that serves to illustrate my point.

There are many other examples where seemingly unrelated fields or other
physical quantities mix together under some kind of transformation.
This
is often an indication that they can be grouped together into some
larger object that will significatly simplify their treatment. There
are
powerful tools that have been developed to detect these mixings and the
possible groupings that could be exploited. It goes under the name of
group representation theory.

Hope this helps. Unfortunately, I must run now, but I may
say more about this later.

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Igor Khavkine]

On 2005-09-10, Igor Khavkine <igor.kh@gmail.com> wrote:

> In the 1920 Stern and Gerlach performed experiments on the interaction
> of electrons with a magnetic field. They noticed that there were two
> kinds of electrons, which could be distinguished by the direction of
> deflection in a magnetic field. This situation can be described by
> increasing the number of components of the electron wave function from
> one to two (which was done by Pauli). However, they also found that
> when the magnetic field is rotated relative to the electrons (or the
> othe way around), the two kinds of electrons lost their distinction
> and became mixed. In other words, rotations mixed the components of
> the Pauli 2-component wave function. In the end it turns out that
> Pauli found that the two kinds of electrons mixed under rotations
> precisely how a spinor would. That is why combining them into one
> spinor wave function is advantageous. Note that this is a slight
> reinterpretation of history that serves to illustrate my point.

Of course, in the case of the Stern-Gerlach experiment, it was pretty
clear from the start that there shouldn't be "two kinds" of electrons,
but only one kind of electron with different states describing its spin.
However, in other cases, this grouping of degrees of freedom wasn't
obvious. For example, there are three pions, pi^0, pi^+, and pi^-, with
the superscripts describing the electric charge of each species. At the
time when they were discovered, people treated them as distinct
particles. However, using the same trick that Pauli used for the
electron, we can group the wave functions of these "particles" into a
single 3-component wave function describing a single particle, the pion,
with three possible internal states (similar to the two possible
orientations of the electron's spin). These three components will mix
between each other under so-called isospin transformations. (I'm not
completely sure about some of the details, since particle physics is not
one of my strong points.) The same thing happens with the W^0 and Z^+
and Z^- bosons, which mix together under gauge transformations
associated to the weak force.

> There are many other examples where seemingly unrelated fields or
> other physical quantities mix together under some kind of
> transformation. This is often an indication that they can be grouped
> together into some larger object that will significatly simplify their
> treatment. There are powerful tools that have been developed to detect
> these mixings and the possible groupings that could be exploited. It
> goes under the name of group representation theory.

Basically, one no longer needs to grope around in the dark to try to
identify the best groupings of degrees of freedom that get mixed
together.

The first step is to identify the transformations that induce
the mixing. These usually form a group, in the mathematical sense of the
word. A good example of that is the set of all permutations of a set of
n identical objects, and another one is the set of all rotations of 3D
space about a fixed point.

Next one has to look at the kind of mixing that takes place. If relevant
degrees of freedom transform between each other with linear formulas (as
in all the examples I've given so far), then the action of the group of
transformations identified in the previous paragraph is said to be
linear. Such a linear action is also called a representation of the
group of transformations. Under fairly general circumstances, it can be
proven that, for any representation, we can choose groups (in the
colloquial sense) of linear combinations of the relevant degrees of
freedom which only transform between themselves and don't mix with other
such groups and further cannot be made any smaller. In the language of
representation theory, we split the representation we started with into
irreducible components. Even better yet, there is an algorithm to do
this that relies on something called Schur's lemma and group averaging
to construct intertwiners. In other words, if we identify all the
transformations that mix some physical degrees of freedom, such that
the mixing is done linearly, under fairly general circumstances, we can
find optimal groupings of these degrees of freedom (or rather linear
combinations thereof) with respect to thse transformations.

If the mixing is not linear, then you can get into trouble since I don't
think there is a theory of non-linear representations developed to the
same degree as the linear case, so you'll have to go back to guessing
again.

I hope I've shed some light on the reason for which field components and
other degrees of freedom sometimes get grouped in unexpected ways
(tensors, complex numbers, quaternions, Clifford algebra, etc.).

Igor

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Eugene Stefanovich]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...

> For example, a neutral
> pion would be described by a real wave function satisfying the
> Klein-Gordon equation, instead of the Schroedinger one.

I am afraid you confused the quantum field used to describe interactions of
pions in QFT with the wave function of a single pion.

The pion's quantum field is a real function on the Minkowski space-time.
It satisfies the Klein-Gordon equation.

The wave function of a single pion is a complex-valued function.
This function expresses the components of the pion's state vector in
the 1-particle complex Hilbert space wrt a (position or momentum or
any other) basis.

Eugene.

[Eugene Stefanovich]

Eugene Stefanovich wrote:

> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.

Since quantum fields are, actually, operator functions, let me add a
minor correction here:

"The pion's quantum field is a Hermitian (self-adjoint) operator
function on the Minkowski space-time. It satisfies the Klein-Gordon
equation.

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Igor Khavkine]

On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
> "Igor Khavkine" <igor.kh@gmail.com> wrote in message
> news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>
>> For example, a neutral
>> pion would be described by a real wave function satisfying the
>> Klein-Gordon equation, instead of the Schroedinger one.
>
> I am afraid you confused the quantum field used to describe interactions of
> pions in QFT with the wave function of a single pion.
>
> The pion's quantum field is a real function on the Minkowski space-time.
> It satisfies the Klein-Gordon equation.
>
> The wave function of a single pion is a complex-valued function.
> This function expresses the components of the pion's state vector in
> the 1-particle complex Hilbert space wrt a (position or momentum or
> any other) basis.

You are right, the neutral pion wave function is complex. However, it
still satisfies the Klein-Gordon equation and is not coupled to a gauge
field. Hence its real imaginar components decouple and can be treated
separately as real solutions to the Klein-Gordon equation.

Igor

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.

[Eugene Stefanovich]

Igor Khavkine wrote:
> On 2005-09-12, Eugene Stefanovich <eugene_stefanovich@usa.net> wrote:
>
>>"Igor Khavkine" <igor.kh@gmail.com> wrote in message
>>news:1126309587.003819.200000@z14g2000cwz.googlegr oups.com...
>>
>>
>>> For example, a neutral
>>>pion would be described by a real wave function satisfying the
>>>Klein-Gordon equation, instead of the Schroedinger one.
>>
>>I am afraid you confused the quantum field used to describe interactions of
>>pions in QFT with the wave function of a single pion.
>>
>>The pion's quantum field is a real function on the Minkowski space-time.
>>It satisfies the Klein-Gordon equation.
>>
>>The wave function of a single pion is a complex-valued function.
>>This function expresses the components of the pion's state vector in
>>the 1-particle complex Hilbert space wrt a (position or momentum or
>>any other) basis.
>
>
> You are right, the neutral pion wave function is complex. However, it
> still satisfies the Klein-Gordon equation and is not coupled to a gauge
> field. Hence its real imaginar components decouple and can be treated
> separately as real solutions to the Klein-Gordon equation.

I think it is important to clearly distinguish two completely different
objects in QFT: the quantum field and particle's wave function.
This distinction is not clearly ephasised in textbooks, and, in my view,
this creates a lot of misunderstandings.

Quantum field is an operator function on an abstract 4D Minkowski
space-time.
Quantum fields are used to build interaction operators
(e.g., in the Hamiltonian). Quantum field has no relationship
whatsoever to the state of the physical system. It is not some kind
of "second-quantized wave function", as often stated.
Given expression for the quantum
field we can say nothing about the state of the physical system.
We even cannot say how many particles there are in the system.
Pion's quantum field satisfies the Klein-Gordon equation.
This equation has no relationship (even distant) to the
Schroedinger equation that describes the time evolution of the wave
function (see below).

On the other hand, particle's wave function gives a full
description of the state in the 1-particle sector of the
Fock space (for 2-particle, 3-particle, etc. sectors you need
to consider wave functions of 2-, 3-, etc. arguments). Different
states have different wave functions.
Wave function is always a complex function defined on the
common spectrum of a set of mutually commuting observables,
e.g., position, momentum, or any other. The wave function
satisfies the time-dependent Schroedinger equation that simply
expresses the transformation law of the state vector
wrt time translations:

-i d |psi>/dt = H |psi>

The relativistic Hamiltonian of a free massive
particle is H = sqrt(p^2 + m^2)
Therefore, the relativistic Schroedinger equation for the pion
wave function in the momentum representation is

-i d psi(p)/dt = sqrt(p^2 + m^2) psi(p) (1)

In the position representation we should simply use the operator
-i d/dx instead of p in (1). Note that, in contrast to
the Klein-Gordon equation, the correct Schroedinger
equation (1) can only contain the 1-st time derivative of the
wave function.

Eugene.