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.

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 phifour 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 nonAbelian 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 OsterwalderSchrader positivity. In 4d one should be able to construct a phifour 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 nonGaussian fixed point for $\varphi^4$ in $4−\varepsilon$ dimensions". Unfortunately, such a model would most likely not satisfy OS positivity. 


In 2d spacetime you have logSobolev 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 higherdimensional 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. 

