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DarMM
#241
Apr7-10, 03:44 AM
Sci Advisor
P: 303
Quote Quote by meopemuk View Post
I believe that people doing "rigorous QFT" are trying to solve these problems (e.g., time evolution and renormalization). I wish them well. However, in my personal (uneducated) opinion, they chose a wrong (formalistic) approach. I think one can also try an alternative approach which pays more attention to the physical meaning of theoretical constructions.
Yes, but remember that the rigorous approach is the only which has accomplished this goal nonperturbatively in any model. And they have done so in several models in two and three dimensions. They haven't accomplished four dimensions yet, but they have a better track record than approaches which have accomplished nothing nonperturbatively.

Wightman's axioms go beyond that and postulate that the same transformation law should be valid for interacting fields as well. As far as I know, there is no justification for this requirement.
No justification is a bit of a stretch. Let me list the theories where it is known to be true:
1. All pure scalar theories in 2D
2. All pure scalar theories in 3D
3. All Yukawa theories in 2D
4. All Yukawa theories in 3D
5. Yang-Mills in 2D
6. The Abelian Higgs-Model in 2D and 3D
7. The Gross-Neveu model in 2D and 3D
8. The Thirring model
and finally
9. All scalar theories in 4D.

The caveat on (9.) is that the only purely scalar theory which exists in 4D is probably the trivial one. However any field theory which exists has been proven to have this transformation property.
This list is basically every single theory we have constructed and understood nonperturbatively. So for every theory we have nonperturbative knowledge of, the transformation law holds.

The list of theories which exist nonperturbatively and don't obey the transformation law is an empty list. Hence I would say the assumption is justified, or at least far more justified than its negation.

The answer is given in Haag's theorem: If the "Lorentz" transformation condition is postulated for interacting fields, then there can be no interaction. So a theory having this postulate is simply inconsistent.
I don't know how many times I can repeat this, that is not what Haag's theorem says. Not even Shirokov, in the paper you quoted, mentions this. To transcribe what Haag's theorem says, again, into language you might understand:

Haag's theorem says that if the theory lives in the same Hilbert space as the free theory and obeys relativistic transformations and is translationally invariant, then it is free.

That is it says:
(Same Hilbert space) + (Normal transformation law) + (Translationally invariance) => Non-interacting

It does not say:
(Normal transformation law) => Non-interacting.

DarMM will tell you that there CAN be interaction, but in a different Hilbert space. That's something I can't comprehend.
It doesn't matter if you can't comprehend it or that I'm saying it. It is true and has been known to be true since 1969. I have even left references to papers which prove it in this thread, including in my two-part post above. It's perfectly fine if you can't imagine it, but it is true.