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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.

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Just how much success has happened in 3d? I thought the only known results were highly degenerate. – S. Carnahan May 26 '14 at 23:17
1 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. – Jimbo May 26 '14 at 23:29
Word on the street is that one super-renormalizable scalar theory is not enough success to sustain a vibrant research community. – S. Carnahan May 26 '14 at 23:41
@Scott: in what sense is phi 4 in 3d degenerate for you? – Abdelmalek Abdesselam Jun 19 '14 at 16:10

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.

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