What Are the States in Quantum Field Theory?

[cgoakley]

Eugene,

I do not understand your question. As you point out, what you have written down is just a definition.

 

[meopemuk]

Eugene,

I do not understand your question. As you point out, what you have written down is just a definition.

Why do you think this definition makes physical sense?

Don't you find it suspicious that we use *interacting* representation of the Poincare group to transform *interacting* fields and then assume that the transformation formula does not depend on interaction at all? This is either a remarkable coincidence or simply a wrong assumption (definition).

There is a paper

H. Kita, "A non-trivial example of a relativistic quantum theory of particles without divergence difficulties" Prog. Theor. Phys. 98 (1966), 934

where it is shown that the above field transformation formula does not hold in a model interacting theory.

Eugene.

 

[A. Neumaier]

Where was I trying to evade Haag's theorem?

As I said, the interacting c/a operators in Stückelberg's covariant perturbation theory cannot be unitarily transformed to the non-interacting ones, even though they live in the same Fock space.

Since the construction cannot be made rigorous, it is irrelevant for the purposes of constructive quantum theory (which is under discussion in this thread).

There are other reasons why Fock space cannot be the answer for real QFT, even in its approximate version as used by most physicists: Fock space cannot accommodate any of the nonperturbative stuff that is being discussed (e.g., solitons and instantons) .

 

[A. Neumaier]

As far as I know, the proof of Haag's theorem uses the assumption that *interacting* fields have specific transformation laws under the *interacting* representation of the Lorentz group. Namely, the transformation is assumed to be of the form

\Psi_i(x) \to \sum_j D_{ij}(\Lambda^{-1}) \Psi_j(\Lambda x)

Can somebody explain me the reason for this formula? In particular, it is interesting why there are no traces of interaction there?

This is the meaning of carrying a representation of the Lorentz group preserving the causal commutation relations.

Informally (and in 1+1D and 1+2D without bound states rigorously locally - ignoring large volume questions), the interacting Psi_i(x) is a unitary transform of the free Psi(x), hence satisfies the same transformation rules.

 

[cgoakley]

Since the construction cannot be made rigorous, it is irrelevant for the purposes of constructive quantum theory (which is under discussion in this thread).

I hope so, as this constructive QFT seems not to be able to generate cross-sections in 3+1 dimensions.

We seem not even to be able to agree about the basic rules of logic. Stückelberg's covariant P.T. uses free field theory as its framework. You agree that free theory is rigorous. Yet you say that Stückelberg's methods are not. Why?

As there seems to be little danger of you (or anyone else) actually looking at the references I gave I will once again give up. Maybe I will come back in another five years, though I am not optimistic that constructive/axiomatic/algebraic/whatever field theorists will be calculating cross-sections for real processes even then.

 

[A. Neumaier]

We seem not even to be able to agree about the basic rules of logic. Stückelberg's covariant P.T. uses free field theory as its framework. You agree that free theory is rigorous. Yet you say that Stückelberg's methods are not. Why?

Stueckelberg constructs only the leading terms in a formal power seires. But it is well-known that formal power series never define a function: There are always infinitely many functions whose Taylor expansion agrees with the formal power series.

What is missing to make his methods rigorous is a recipe for picking the right function in a way that is consistent with the action of some self-adjoint Hamiltonian acting on Fock space (or another space).

 

[meopemuk]

the interacting Psi_i(x) is a unitary transform of the free Psi(x), hence satisfies the same transformation rules.

Can you prove this statement or, at least, point me to the reference, where it is proved? It doesn't look obvious to me.

Eugene.

 

[strangerep]

As far as I know, the proof of Haag's theorem uses the assumption that
*interacting* fields have specific transformation laws under the
*interacting* representation of the Lorentz group. Namely, the
transformation is assumed to be of the form

\Psi_i(x) \to \sum_j D_{ij}(\Lambda^{-1}) \Psi_j(\Lambda x)

Can somebody explain me the reason for this formula?
In particular, it is interesting why there are no traces of
interaction there?

This is the meaning of carrying a representation of the Lorentz group
preserving the causal commutation relations.

I don't see how it's reasonable to insist that an interacting theory
(involving accelerations of particles wrt each other) must preserve
the same causal structure of spacetime as the free theory.
Consider Rindler horizons for mutually accelerated observers...
this implies a very different causal structure compared to that
perceived by inertial observers.

Informally (and in 1+1D and 1+2D without bound states rigorously
locally - ignoring large volume questions), the interacting Psi_i(x) is
a unitary transform of the free Psi(x), hence satisfies the same
transformation rules.

Was there a typo in your last sentence above? (If the free and interacting
fields are related by a unitary transform then the spectra of the two
theories are the same, aren't they?)

 

[A. Neumaier]

I don't see how it's reasonable to insist that an interacting theory
(involving accelerations of particles wrt each other) must preserve
the same causal structure of spacetime as the free theory.
Consider Rindler horizons for mutually accelerated observers...
this implies a very different causal structure compared to that
perceived by inertial observers.

Quantum gravity is beyond my expertise. I am assuming a renormalizable QFT in flat space. This has a well-defined causal structure independent of any interactions.

Informally (and in 1+1D and 1+2D without bound states rigorously locally - ignoring large volume questions), the interacting Psi_i(x) is a unitary transform of the free Psi(x), hence satisfies the same transformation rules.

Was there a typo in your last sentence above?

Not a typo but I was too sloppy. The interaction Psi_i(x) is a limit of a sequence of unitary transform of the free Psi(x), and the conclusion still holds. For all known relativistic QFTs in 1+d dimensions (d=1,2) for which the analysis could be made rigorous, the infinite volume limit changes the representation. The same is expected to hold for
the case d=3 where no rigorous analysis has been completed so far.

(If the free and interacting fields are related by a unitary transform then the spectra of the two theories are the same, aren't they?)

No, since the Hamiltonians are different. But if the free and interacting fields are related by a unitary transform then the representations of the equal-time CCRs of the two theories are the same, and both would be Fock spaces, which contradicts Haag's theorem.

 

[A. Neumaier]

Can you prove this statement or, at least, point me to the reference, where it is proved? It doesn't look obvious to me.

As just mentioned, the correct, intended statement was ''The interaction Psi_i(x) is a limit of a sequence of unitary transform of the free Psi(x), hence satisfies the same transformation rules.'' This is indeed a nontrivial statement; for the 1+1d case, see, e.g.,
the paper by Glimm and Jaffe in
Acta Mathematica 125, 1970, 203-267
http://www.springerlink.com/content/t044kq0072712664/
and the two papes that preceded their part III.

 

[cgoakley]

Stueckelberg constructs only the leading terms in a formal power series.

The power series is obtained by expanding the QED equations of motion. If one simply defines one's theory such that the interacting fields are the free fields plus these leading terms then everything will be perfectly finite and rigorous, and the simple scattering amplitudes will be correctly reproduced. The cost will just be that the interacting CCRs will not be the expected ones.

But it is well-known that formal power series never define a function: There are always infinitely many functions whose Taylor expansion agrees with the formal power series.

True, but irrelevant.

What is missing to make his methods rigorous is a recipe for picking the right function in a way that is consistent with the action of some self-adjoint Hamiltonian acting on Fock space (or another space).

This is not in my view the issue. The Hamiltonian for the free fields works just as well for the interacting fields in this approach. The thing that does not exist is a "free Hamiltonian" that time-displaces the interacting field as though it was the free field (and if it did, it would violate Haag's theorem).

 

[A. Neumaier]

The power series is obtained by expanding the QED equations of motion. If one simply defines one's theory such that the interacting fields are the free fields plus these leading terms then everything will be perfectly finite and rigorous, and the simple scattering amplitudes will be correctly reproduced. The cost will just be that the interacting CCRs will not be the expected ones.

But then the commutation relations are not causal, and what you get is not Poincare invariant. It is very easy to construct theories that are well-defined and approximate QED one way or another. It is also easy to construct Poincare invariant theories if you make compromises with causality.

But real QED at the same time
(i) is Poincare invariant,
(ii) is causal, i.e., field commutators at space-like related arguments vanish,
(iii) satisfies the cluster decomposition property,
and no amount of tinkering a la Stueckelberg so far has lead to a consistent and nontrivial 4D field theory with these properties.

 

[cgoakley]

But then the commutation relations are not causal,

Spacelike (anti)commutativity is not guaranteed, unless one introduces higher-order terms, certainly. What this has to do with causality is not so clear, though, as defining what one means by this in the quantum world is a lot harder than in the classical world.

and what you get is not Poincare invariant.

This I absolutely do not get. How and why is this not Poincare invariant?

It is very easy to construct theories that are well-defined and approximate QED one way or another. It is also easy to construct Poincare invariant theories if you make compromises with causality.

Examples?

 

[meopemuk]

As just mentioned, the correct, intended statement was ''The interaction Psi_i(x) is a limit of a sequence of unitary transform of the free Psi(x), hence satisfies the same transformation rules.'' This is indeed a nontrivial statement; for the 1+1d case, see, e.g.,
the paper by Glimm and Jaffe in
Acta Mathematica 125, 1970, 203-267
http://www.springerlink.com/content/t044kq0072712664/
and the two papes that preceded their part III.

Thanks. I'll check it out.

Since this nontrivial statement (which was assumed to be self-evident by Haag) was proven only much later, does it mean that Haag's proof is not complete? Shall we call it Haag-Glimm-Jaffe theorem now? Does it mean that this theorem is rigorously valid only in the 1+1d case? The story of Haag's theorem becomes rather confusing.

Eugene.

 

[A. Neumaier]

Since this nontrivial statement (which was assumed to be self-evident by Haag) was proven only much later, does it mean that Haag's proof is not complete? Shall we call it Haag-Glimm-Jaffe theorem now? Does it mean that this theorem is rigorously valid only in the 1+1d case? The story of Haag's theorem becomes rather confusing.

No. Haag's theorem is a theorem that holds in general when the Wightman axioms are satisfied, while the statement under discussion is a significantly stronger statement that can be proved for the specific theories that were explicitly constructed.

 

[A. Neumaier]

Spacelike (anti)commutativity is not guaranteed, unless one introduces higher-order terms, certainly. What this has to do with causality is not so clear, though, as defining what one means by this in the quantum world is a lot harder than in the classical world.

It means that one can prepare a state with given exact values of a Hermitian fields
everywhere at any particular time, in any Lorentz frame. Violation of the causal commutation rules mean that this is impossible (due to the uncertainty relation for m=noncommuting observables), so that there must be an instantaneous influence of part of the world to other parts of the world that would forbid this.

Therefore, at least for the electromagnetic field which is observable and preparable, the causal commutation rules ar a necessity for a consistent relativistic QFT.

How and why is this not Poincare invariant?

Poincare invariance of a quantum field theory is a very nontrivial statement that is not easy to get. Thus as long as no proof is available that a given construction is Poincare invariant (by giving the interacting generators with P_0 and verifying that they satisfy the Lie algebra of Poincare) it is very likely that it is not Poincare invariant. In particular, truncating the Hamiltonian in a field theory generally destroys Poincare invariance, since there is no matching truncation of the other generators that would preserve the Lie algebra. (This can be made more rigorous in terms of cohomology...)

Indeed, Weinberg argues in his book that Poincare invariance of a field theory requires causal commutation rules.

 

[meopemuk]

No. Haag's theorem is a theorem that holds in general when the Wightman axioms are satisfied, while the statement under discussion is a significantly stronger statement that can be proved for the specific theories that were explicitly constructed.

Well, now I am confused even more. Isn't it true that one of Wightman axioms (the one named W2 in the Wikipedia article http://en.wikipedia.org/wiki/Wightman_axioms ) defines exactly the Lorentz transformations of the fields. Then it appears that Glimm and Jaffe have proven one of Wightman axioms?

Eugene.

 

[A. Neumaier]

Well, now I am confused even more. Isn't it true that one of Wightman axioms (the one named W2 in the Wikipedia article http://en.wikipedia.org/wiki/Wightman_axioms ) defines exactly the Lorentz transformations of the fields. Then it appears that Glimm and Jaffe have proven one of Wightman axioms?

What Glimm and Jaffe proved in part iV, based on the results in part iii (my reference),
is that P(Phi)_2 quantum field theories satisfy the Wightman axioms. This was a major achievement at the time since before their work it wasn't known whether any interacting field theory satisfying Wightmans's axiom exist at all.

In case your confusion is about why one should prove an ''axiom'' (which is supposed to be an assumption): This is not more strange than when verifying that the integers form a group under addition - you need to prove for them the axioms of group theory.

 

[cgoakley]

How and why is this not Poincare invariant?

Poincare invariance of a quantum field theory is a very nontrivial statement that is not easy to get. Thus as long as no proof is available that a given construction is Poincare invariant (by giving the interacting generators with P_0 and verifying that they satisfy the Lie algebra of Poincare) it is very likely that it is not Poincare invariant. In particular, truncating the Hamiltonian in a field theory generally destroys Poincare invariance, since there is no matching truncation of the other generators that would preserve the Lie algebra. (This can be made more rigorous in terms of cohomology...)

Indeed, Weinberg argues in his book that Poincare invariance of a field theory requires causal commutation rules.

If you are saying this, then with respect, I do not think that you have understood the approach at all.

The steps are these:

1. Build a free field relativistic quantum field theory

We are not going to argue about this - or are we?

The Hamiltonian and other Poincare generators are perfectly well defined in terms of the free field creation and annihilation operators (Noether's theorem). The Poincare algebra is obeyed and a faithful representation of the Poincare algebra is obtained on the space of physical states.

2. Construct interacting fields as sums of products of free fields, the zeroth order in each case being the free field. As long as the coefficient functions in each multilinear product are chosen correctly (i.e. covariantly, with spacetime co-ordinates only appearing as differences), then the transformation properties of the interacting field under the Poincare group will be the same as for the free fields. The time displacement generator, a.k.a. the Hamiltonian, will be the same for both. There are no "free" and "interacting" Hamiltonians - there is just a Hamiltonian.

3. Following Stueckelberg, the matrix elements for elementary processes can then be read off directly, after applying the interacting fields to the vacuum to create particle states. The free vacuum is the same as the interacting one.

Note: no interaction picture, no time-ordered products. The approach is NOT equivalent to the one given (e.g.) in Weinberg's books.

 

[A. Neumaier]

1. Build a free field relativistic quantum field theory

The Hamiltonian and other Poincare generators are perfectly well defined in terms of the free field creation and annihilation operators (Noether's theorem). The Poincare algebra is obeyed and a faithful representation of the Poincare algebra is obtained on the space of physical states.

2. Construct interacting fields as sums of products of free fields, the zeroth order in each case being the free field. As long as the coefficient functions in each multilinear product are chosen correctly (i.e. covariantly, with spacetime co-ordinates only appearing as differences), then the transformation properties of the interacting field under the Poincare group will be the same as for the free fields. The time displacement generator, a.k.a. the Hamiltonian, will be the same for both. There are no "free" and "interacting" Hamiltonians - there is just a Hamiltonian.

If the Hamiltonian is still the Hamiltonian of the free field, then the dynamics is trivial.
You get exactly the same eigenstates and asymptotic behavior, there are no bound states, the scattering is trivial.

Renaming the field observables is not enough to create a nontrivial dynamics.
Of course one can use the revised field operators to create a mock scattering scenario, but this scenario has nothing to do anymore with Schroedinger equations.

If Stueckelberg's idea had been the breakthrough that your interpretation claims it is, it would have had far more impact.

 

[cgoakley]

If the Hamiltonian is still the Hamiltonian of the free field, then the dynamics is trivial.
You get exactly the same eigenstates and asymptotic behavior, there are no bound states, the scattering is trivial.

Renaming the field observables is not enough to create a nontrivial dynamics.
Of course one can use the revised field operators to create a mock scattering scenario,

Yes, of course if you put free fields into the scattering calculations, you will get trivial results. That is not what he does. The interacting electron field contains, in higher order, a (free) electron combined with a (free) photon, an electron combined with an electron-positron pair and so on. Similarly, the interacting photon field contains, in higher-order, an electron-positron pair, a photon combined with an electron-positron pair, and so forth. It is the non-zero matrix elements between the higher-order pieces of the interacting fields that enable one to correctly obtain all tree-level scattering amplitudes for QED.

but this scenario has nothing to do anymore with Schroedinger equations.

Why should this be a requirement? Schroedinger equations are very resistant to being made relativistic when there are interactions - this is what Haag's theorem is all about. Stueckelberg's method is relativistic right from the start.

If Stueckelberg's idea had been the breakthrough that your interpretation claims it is, it would have had far more impact.

"Already in 1934 [...] it seemed that a systematic theory could be developed in which these infinities [divergent radiative corrections] are circumvented. At that time nobody attempted to formulate such a theory [...].There was one tragic exception [...], and that was Ernst C.G. Stueckelberg. He wrote several important papers in 1934-38 putting forward a manifestly invariant formulation of field theory. This could have been a perfect basis for developing the ideas of renormalization. Later on, he actually carried out a complete renormalization procedure in papers with D. Rivier, independently of the efforts of other authors. Unfortunately, his writings and his talks were rather obscure,and it was very difficult to understand them or to make use of his methods. He came frequently to Zurich in the years 1934-6, when I was working with Pauli, but we could not follow his way of presentation. Had Pauli and I myself been capable of grasping his ideas, we might well have calculated the Lamb shift and the correction to the magnetic moment of the electron at the time."

Weisskopf's words (in 1981) - not mine.

 

[A. Neumaier]

It is the non-zero matrix elements between the higher-order pieces of the interacting fields that enable one to correctly obtain all tree-level scattering amplitudes for QED.

Tree level is considered trivial. It doesn't even explain the anomalous magnetic moment of the lectron, or the Lamb shift - the successes that made modern QED respectable.

Why should this be a requirement? Schroedinger equations are very resistant to being made relativistic when there are interactions - this is what Haag's theorem is all about. Stueckelberg's method is relativistic right from the start.

The Schroedinger equation is still the thing that makes quantum field theory consistent (at least on a formal level). As you can read in any QFT textbook, it is needed to derive the form of the S-matrix and its unitarity. Therefore everyone (except you) requires the Schroedinger equation, though it is no longer very practical to use it in computations since its use breaks manifest Lorentz covariance.

All relativistic QFTs that have been constructed rigorously have a Hamiltonian generating both the time evolution and relating to the S-matrix in the same way as in simple scattering at external potentials. Therefore, if you can't construct the Hamiltonian as part of a nontrivial representation of the Poincare group, you didn't construct a relativistic quantum field theory according to today's standards. In particular, Stueckelberg didn't construct one.

"Ernst C.G. Stueckelberg. He wrote several important papers in 1934-38 putting forward a manifestly invariant formulation of field theory. This could have been a perfect basis for developing the ideas of renormalization. Later on, he actually carried out a complete renormalization procedure in papers with D. Rivier, independently of the efforts of other authors. Unfortunately, his writings and his talks were rather obscure,and it was very difficult to understand them or to make use of his methods."

Weisskopf's words (in 1981) - not mine.

It is still very difficult to understand them or to make use of his methods. In the 30 years since this revelation, nobody found them useful enough to develop his methods further. They are not useful - they are far less powerful than the real thing, and they are approximate only, violating causality. You may find that this is irrelevant, but the experts know better.

 

[meopemuk]

What Glimm and Jaffe proved in part iV, based on the results in part iii (my reference),
is that P(Phi)_2 quantum field theories satisfy the Wightman axioms. This was a major achievement at the time since before their work it wasn't known whether any interacting field theory satisfying Wightmans's axiom exist at all.

Is it possible to verify the Wightman axiom about Lorentz transformations in more realistic theories, such as QED? Perhaps P(Phi)_2 quantum field theories (where, according to Glimm and Jaffe, the axiom is true) are some exceptional pathological cases, where the transformation law becomes simple due to some cancellations? Still, I don't see any *physical* reason to believe in this simple transformation law. I agree, it makes a nice formula, but what is the physics of it?

Eugene.

 

[A. Neumaier]

Is it possible to verify the Wightman axiom about Lorentz transformations in more realistic theories, such as QED?

QED is currently still too hard for mathematical physicists, though they can construct various reasonable approximations to QED, but not one satisfying the Wightman axioms.
The problem is still open. (On the other hand, they can prove that Phi^4 theory exists in dimensions 2 and 3, and is trivial in dimensions >4. The 4D case is borderline and therefore hardest.)

Perhaps P(Phi)_2 quantum field theories (where, according to Glimm and Jaffe, the axiom is true) are some exceptional pathological cases, where the transformation law becomes simple due to some cancellations?

There is nothing pathological for 2D theories. If you can prove the Wightman axioms, you can apply all the nice results that can be derived from these axioms, including the existence of a good scattering theory with a covariant S-matrix.

Still, I don't see any *physical* reason to believe in this simple transformation law. I agree, it makes a nice formula, but what is the physics of it?

The necessity for the Wightman axioms stems from the belief in fundamental physical principles - relativity, causality, the existence of fields and a vacuum, and a separable Hilbert space accommodating all these. These together make the Wightman axioms essentially unescapable.

A transformation law that does not satisfy the commutation rules of the Poincare algebra
(your nice formulas) has no representation of the Poincare group in which H=P_0 (the interacting Hamiltonian) generates the physical time translations.

Anyway, I don't understand why you complain about the transformation laws that you champion yourself in your book, though in perturbation theory, and in the instant form only. This restricted form of the representation has no effect on the transformation law.

 

[meopemuk]

QED is currently still too hard for mathematical physicists, though they can construct various reasonable approximations to QED, but not one satisfying the Wightman axioms.
The problem is still open.

A rigorous proof would be very difficult, I agree. But one can still try to verify the transformation formula in low perturbation orders, I think. In QED we have both the interacting Hamiltonian and the boost operator. So, in principle, we should be able to insert them in the Wightman's transformation formula, make the perturbation expansion and see directly whether this formula holds, at least in low orders.

If this formula does hold, I would be very surprised. If the formula is violated, then I wouldn't blame QED, which is our best physical theory, after all. I would rather say that Wightman's assumption is unrealistic.

A transformation law that does not satisfy the commutation rules of the Poincare algebra
(your nice formulas) has no representation of the Poincare group in which H=P_0 (the interacting Hamiltonian) generates the physical time translations.

I don't quite understand your logic. According to Weinberg, a theory is relativistically invariant if it has a unitary representation of the Poincare group. In other words, if there exist 10 Hermitian operators, satisfying the corresponding Lie algebra commutators. This says nothing about the explicit transformation law of the interacting field. If you can prove that Wightman's formula follows directly from commutation relations of Poincare generators, then I would agree with you. Can you prove that?

Anyway, I don't understand why you complain about the transformation laws that you champion yourself in your book, though in perturbation theory, and in the instant form only. This restricted form of the representation has no effect on the transformation law.

The transformation law in question is postulated for free quantum fields. However, this is not a reflection of any physical principle, like the principle of relativity. According to Weinberg, free fields are intentionally defined in such a way that this transformation law is valid. Then Weinberg proves that if one builds the interacting Hamiltonian and boost operators out of products of such free fields, then one obtains 10 non-trivial Poincare generators with required commutation relations. This is enough to obtain a satisfactory interacting quantum field theory (apart from renormalization, which we do not discuss here). The behavior of the *interacting* field is not relevant in this construction. If I remember correctly, Weinberg does not mention Lorentz transformations of the interacting field anywhere in his book. They are simply not needed for calculations of scattering amplitudes.

In my opinion, free quantum fields (and their covariant transformation laws) have no relationship to any physical object observed in experiments. They are just mathematical entities, which are useful for constructing interaction operators. *Interacting* fields don't have even this limited meaning. A full interacting theory can be constructed without mentioning interacting fields at all. Yes, one can formally build these objects and study their transformation laws. But I don't understand why one should a priori assume some specific form of these transformations?

Eugene.

 

[strangerep]

[...]
The necessity for the Wightman axioms stems from the belief in fundamental physical principles - relativity, causality, the existence of fields and a vacuum, and a separable Hilbert space accommodating all these. These together make the Wightman axioms essentially unescapable.

There's one more: the belief that multi-particle physics takes place in a common
Minkowski spacetime -- the truth/falsehood of that belief is what this discussion
centers on. In Haag's original paper, he says (in effect) that any other choice is
"unnatural", but gives no further justification. OTOH, the fact that 2-particle
non-relativistic QM requires a tensor product space (i.e., does not work properly
if the particles are assumed to occupy a common position space) gives me reason
to doubt whether Haag's "natural" choice is indeed physically correct.

 

[A. Neumaier]

There's one more: the belief that multi-particle physics takes place in a common
Minkowski spacetime -- the truth/falsehood of that belief is what this discussion
centers on. In Haag's original paper, he says (in effect) that any other choice is
"unnatural", but gives no further justification. OTOH, the fact that 2-particle
non-relativistic QM requires a tensor product space (i.e., does not work properly
if the particles are assumed to occupy a common position space) gives me reason
to doubt whether Haag's "natural" choice is indeed physically correct.

Minkowski space-time is not to be confused with configuration space. The configuration space of a field theory is i(n the absence of gauge invariance and assuming asymptotic completeness) the disjoint union of a (particle content dependent) number of copies of R^3N for N=0,1,2,... This accounts sufficiently for your tensor products.

On the other hand, Minkowski space is the space of arguments of the fields, and this is just 4D for any traditional field theory (excluding Kaluza-Klein, strings, etc.).

 

[A. Neumaier]

But one can still try to verify the transformation formula in low perturbation orders, I think. In QED we have both the interacting Hamiltonian and the boost operator. So, in principle, we should be able to insert them in the Wightman's transformation formula, make the perturbation expansion and see directly whether this formula holds, at least in low orders.

If this formula does hold, I would be very surprised.

So be surprised! Weinberg does so to all orders in Chapter 3.3 of Volume 1.
If the formula were violated, it would have been the end of QED or relativity.

According to Weinberg, a theory is relativistically invariant if it has a unitary representation of the Poincare group. In other words, if there exist 10 Hermitian operators, satisfying the corresponding Lie algebra commutators.

If this were the only condition then even a nonrelativistic field theory would be relativistically invariant. For one can define on nonrelativistic Fock space over R^3 the free relativistic fields. But they are unrelated to all the other structure of the nonrelativistic field theory, and hence meaningless.

But Weinberg is not so stupid to make meaningless definitions.

The theory is relativistic if the physical creation operators (that create physical particles from the vacuum and satisfy causal commutation relations) generate n-point vacuum expectation values that are Poincare covariant. But this is just what the Wightman axioms require.

Weinberg proves this condition in a heuristic fashion in Section 3.3 (there for the S-matrix, which, in view of the the LSZ-formula p.430 proves it for the time-ordered expectation values, which is only little weaker than the Wightman axioms. Using closed-time-path integrals, one can extend the argument to contour-ordered expectation values, which include the Wightman functions. of course, this ''proof'' is only in perturbation theory, and not a mathematical proof but only one according to the usual standards of theoretical physics.

This says nothing about the explicit transformation law of the interacting field. If you can prove that Wightman's formula follows directly from commutation relations of Poincare generators, then I would agree with you. Can you prove that?

Since (as I can see from your book) you accept the usual standards of theoretical physics as sufficient for proofs, you should now agree.

The transformation law in question is postulated for free quantum fields.

This is completely irrelevant. The free operators are only the scaffolding of the building.What counts is the transformation law for the interacting fields.

Weinberg proves that if one builds the interacting Hamiltonian and boost operators out of products of such free fields, then one obtains 10 non-trivial Poincare generators with required commutation relations. This is enough to obtain a satisfactory interacting quantum field theory (apart from renormalization, which we do not discuss here). The behavior of the *interacting* field is not relevant in this construction.

Of course it is, since the interacting field is defined using the interacting representation of the Poincare group.

If I remember correctly, Weinberg does not mention Lorentz transformations of the interacting field anywhere in his book. They are simply not needed for calculations of scattering amplitudes.

You don't remember correctly. The nontrivial ones are constructed in Section 3.3.; see formulas (3.3.18), (3.3.20), and more explicitly (3.5.17) and (7.4.20). They are not needed for the calculations, but they are essential for the proof of covariance of the S-matrix.

In my opinion, [...] *Interacting* fields don't have even this limited meaning. A full interacting theory can be constructed without mentioning interacting fields at all.

The challence is not to construct some interacting theory but to construct one that has the physically verifiable properties - giving a Lorentz invariant and cluster separable scattering matrix. To verify this you need all the stuff you despise.

 

[meopemuk]

But one can still try to verify the transformation formula in low perturbation orders, I think. In QED we have both the interacting Hamiltonian and the boost operator. So, in principle, we should be able to insert them in the Wightman's transformation formula, make the perturbation expansion and see directly whether this formula holds, at least in low orders.

If this formula does hold, I would be very surprised.

So be surprised! Weinberg does so to all orders in Chapter 3.3 of Volume 1.
If the formula were violated, it would have been the end of QED or relativity.

The theory is relativistic if the physical creation operators (that create physical particles from the vacuum and satisfy causal commutation relations) generate n-point vacuum expectation values that are Poincare covariant. But this is just what the Wightman axioms require.

Weinberg proves this condition in a heuristic fashion in Section 3.3 (there for the S-matrix, which, in view of the the LSZ-formula p.430 proves it for the time-ordered expectation values, which is only little weaker than the Wightman axioms. Using closed-time-path integrals, one can extend the argument to contour-ordered expectation values, which include the Wightman functions. of course, this ''proof'' is only in perturbation theory, and not a mathematical proof but only one according to the usual standards of theoretical physics.

Arnold, I don't see any explicit proof of the covariant transformation law for interacting fields in the places you mentioned. Section 3.3 is titled "Symmetries of the S-matrix". There is not a word there about fields and their transformations. Actually, the whole concept of a *free* quantum field is introduced much later in chapter 5. On page 430 Weinberg discusses the pole structure of the S-matrix. Again, not a word about field transformations. It seems that you are reading something between Weinberg's lines. I would like to see a more explicit proof.

If I remember correctly, Weinberg does not mention Lorentz transformations of the interacting field anywhere in his book. They are simply not needed for calculations of scattering amplitudes.

You don't remember correctly. The nontrivial ones are constructed in Section 3.3.; see formulas (3.3.18), (3.3.20), and more explicitly (3.5.17) and (7.4.20). They are not needed for the calculations, but they are essential for the proof of covariance of the S-matrix.

(3.3.18) - general formulas for Poincare generators (space-time translations and rotations) in any instant form interacting relativistic dynamics.

(3.3.20) - general formula for the boost generator in the instant form

(3.5.17) - expression of the boost interaction as an integral of the Hamiltonian density

(7.4.20) - another expression for the boost operator in a field theory.

The *interacting* quantum field and its transformations are not mentioned there.

Eugene.

 

[A. Neumaier]

I don't see any explicit proof of the covariant transformation law for interacting fields in the places you mentioned. Section 3.3 is titled "Symmetries of the S-matrix". There is not a word there about fields and their transformations.

This is because the transformations are more fundamental and must be present in any relativistic quantum theory, whether with or without fields. (Indeed, as long as one works in the Schroedinger representation, one can completely dispense with the fields; but they are nevertheless there, as shown by my construction below.)

Later he simply take this for granted and specializes it to quantum fields. This specialization is done first tentatively in Section 3.5 (see the middle of p.144), and further justified in Chapter 4 (see p.169); later it is assumed without further ado.

Note that the first few chapters are in the Schroedinger picture. The translation to the Heisenberg picture is as follows: For an arbitrary observable A_0 in the Schroedinger picture, the corresponding quantum field A(x) satisfies
A(x) = U(x) A_0 U(-x) ... (1)
A(Lambda x) = U(Lambda) A(x) U(Lambda^{-1}) ... (2)
where the translations U(x) and the Lorentz transforms U(Lambda) are the physical (interacting) ones. This transformation law appears for the special case of the interaction part of the energy density field in (3.5.11), but is a general property of the Heisenberg picture. (Proof: Take (1) as the definition of the field, and deduce (2) from (1) and the properties of arbitrary unitary representations of the Poincare group. The proof doesn't depend on whether the representation is given in the instant form or any other form.)

It seems that you are reading something between Weinberg's lines. I would like to see a more explicit proof.

I read the whole book and fully understand at least the first eight chapters. This is enough to read between the lines. I hope that the details above fill in what you missed.

The *interacting* quantum field and its transformations are not mentioned there.

I am surprised that you can't see them: On p. 144f, H_{curly}(x,t), the interaction part of the energy density is an interacting field since for free fields, it vanishes identically. Later chapters specialize the expression for H_{curly}(x,t) to those corresponding to Lagrangian field theories, expressing it in terms of the corresponding free field operators.
Section 3.5 discusses the needed properties of H_{curly}(x,t) for creating a good interacting representation of the Poincare group, resulting in the requirement of causal commutation rules (with caveats in the footnote for contact terms; cf. p. 277ff).

This is the reason why Chapter 5 bothers to construct free fields, since it is with their help that this condition can be satisfied if the interaction is represented as a sum of integrals of local products of free fields.