4d Constructive Quantum Field Theory

Question

As a follow up to my previous question (How does Constructive Quantum Field Theory work?), I was wondering what difficulties physicists have had constructing 4d axiomatic qfts. Why has CQFT's success in 2 and 3d spaces not been extended to 4 dimensions? Once again, any level of answer is okay, but technical is preferable.

This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user Jimbo

Comments

@abdelmalek The papers aren't easy by the standards of math papers, but the $\phi_3^4$ model is easy by the standards of QFTs. I don't think you guys are actually disagreeing.

This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user user1504

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user1504=AJ I presume? I am afraid all this adds to the widespread confusion about constructive QFT. As it stands the conclusion of this discussion might look like: "super-renormalizable=easy according to physics standards, constructive QFT can only do super-renormalizable, ergo constructive QFT can safely be abandoned as a mathematical approach to QFT which has a chance at proving something of nontrivial physical interest". Unfortunately, almost everything in between quotes is wrong.

This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user Abdelmalek Abdesselam

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I guess I might have confused user1504 with another "user..."

This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user Abdelmalek Abdesselam

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@abdelmalek Dismissing constructive QFT because it succeeded in the 'easy' case is as silly as dismissing the Langlands Program because class field theory was 'easy'. (We'd never have gotten to Fermat's Last Theorem with that attitude!) I'm under the impression that there's still a lot of life in constructive field theory, but that a lot of the action has been on statistical physics problems where reflection positivity doesn't limit the supply of examples.

This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user user1504

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I agree. Even if one insists on keeping reflection positivity one has some reasonable examples to work with, but one has to accept fractional Laplacians as free propagators. In particular by tuning the exponent so one is only barely super-renormalizable one can get arbitrarily close to the boundary of the "easy" region.

This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user Abdelmalek Abdesselam

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@John: I know what "super-renormalizable" means but I am not sure what S. Carnahan meant by "degenerate".

This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user Abdelmalek Abdesselam

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He's just saying that this theory is too easy.

This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user John Baez

Answer 1

Modern constructive field theory is based on rigorous implementations of the renormalization group (RG) approach. To get an idea of what this is about see this short introductory paper. The RG is an infinite dimensional dynamical system and constructing a QFT essentially means constructing an orbit which typically joins two fixed points. So first you need a fixed point (for instance the massless Gaussian field) and you need it to have an unstable manifold which is not entirely made of Gaussian measures (trivial QFTs). In 4d the only fixed point we have at our disposal is the Gaussian one and at least at the level of perturbation theory one has strong indications that for models like phi-four and even much more complicated generalizations, the corresponding unstable manifold is Gaussian. The only models in 4d known not to suffer from this problem are non-Abelian gauge theories and their construction (in infinite volume) is a difficult question (one of the 7 Clay Millennium Problems).

The main technical obstacles for having good candidates to even consider constructing are stability (being in the region of positive coupling constant) and Osterwalder-Schrader positivity. In 4d one should be able to construct a phi-four model with fractional propagator $1/p^{\alpha}$ with $\alpha$ slightly bigger than 2 (the standard propagator). There are partial rigorous results in this direction by Brydges, Dimock and Hurd: "A non-Gaussian fixed point for $\varphi^4$ in $4−\varepsilon$ dimensions". Unfortunately, such a model would most likely not satisfy OS positivity.

This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user Abdelmalek Abdesselam

Answer 2

In 2d spacetime you have log-Sobolev inequalities that control the strength of the quantum field's potential energy in terms of its kinetic energy. Most of the success in 2d constructive quantum field theory is based on these; try this book for details:

• John Baez, Irving Segal and Zhengfang Zhou, Introduction to Algebraic and Construtive Quantum Field Theory.

In higher-dimensional spacetimes these inequalities don't apply, so we need more sophisticated methods.

Essentially, as we go to higher and higher dimensions it's possible for a field to undergo larger and larger fluctuations without much cost in kinetic energy (or, alternatively, action). Understanding Sobolev inequalities and how they work in different dimensions is a good way to start getting a feeling for this. The increased difficulty in higher dimensions due to this effect shows up in all work on analysis, not just quantum field theory. For example, the quantum mechanics of atoms and molecules (Schrödinger's equation with Coulomb interaction) would be badly behaved if there were an extra dimension of space.

This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user John Baez

Answer 3

... what difficulties physicists have had constructing 4d axiomatic qfts?

I have my own take on this subject. Physicists are stubbornly trying to axiomatize (dogmatize) wrong guesses of interaction. R. Feynman and P. Dirac insisted on better guessing rather than on dogmatizing wrong guesses. So did J. Schwinger. Modern RG things are nothing but tentatives to freeze wrong guesses and have the same ill defined QFT. As famously said J. Schwinger:

"This way of putting the matter can hardly fail to raise the question whether we have to proceed in this tortuous manner of introducing physically extraneous hypotheses, only to delete these at the end in order to get physically meaningful results. Clearly there would be a great improvement, both conceptually and computationally, if we could identify and remove the speculative hypotheses that are implicit in the unrenormalized equations, thereby working much more at the phenomenological level. ...

I continue to hope that it has great appeal to the true physicist (Where are you?)."