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4d Constructive Quantum Field Theory

+ 7 like - 0 dislike
113 views

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
asked May 26, 2014 in Theoretical Physics by Jimbo (35 points) [ no revision ]
retagged Feb 21, 2016
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Just how much success has happened in 3d? I thought the only known results were highly degenerate.

This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user S. Carnahan
sciencedirect.com/science/article/pii/0003491676902232 demonstrates a self-interacting (quartic interaction) 3d scalar field. I am not sure what other 3d results there are, but this one appears to work nicely.

This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user Jimbo
Word on the street is that one super-renormalizable scalar theory is not enough success to sustain a vibrant research community.

This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user S. Carnahan
@Scott: in what sense is phi 4 in 3d degenerate for you?

This post imported from StackExchange MathOverflow at 2016-02-21 14:42 (UTC), posted by SE-user Abdelmalek Abdesselam
It's super-renormalizable. Very roughly, this means you only have to worry about finitely many divergent Feynman diagrams.

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

2 Answers

+ 8 like - 0 dislike

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
answered Jun 19, 2014 by Abdelmalek Abdesselam (190 points) [ no revision ]
+ 5 like - 0 dislike

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
answered Feb 17, 2016 by John Baez (365 points) [ no revision ]

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