# 4d Constructive Quantum Field Theory

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.

• 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
• 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. – 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
• @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. – user1504 Feb 17 '16 at 19:17

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.