Realistic interacting QFT construction

Question

May I ask is it true that all the interacting 4-dimension QFT's couldn't be constructed and defined consistently and rigorously? If we are able to rigorously construct a lower dimension QFT, what are the usage of those algebraic QFT's since real QFT's are in 4 dimensions and have interaction. Any experts can clarify?

Answer 1

We are not able, at least for the moment, to define in a rigorous mathematical fashion a meaningful interacting QFT in $3+1$ dimensions that is coherent with the perturbative theory utilized by physicists (in more precise words, that satisfies the Wightman axioms).

On the contrary, some rigorous interacting QFTs can be defined in lower (spatial) dimensions. Take for example the $\phi^4$ model: it is not well-defined in $3+1$ dimensions, but it is in $1+1$ and $2+1$ (the latter for sure on finite volume, I am not sure about the infinite volume limit) as proved by Glimm and Jaffe in the sixties. This simplified models, even if they may not be physically interesting, has been analyzed in the hope that the same tools may be utilized in the meaningful theories. Unluckily, this has not been the case so far (anyways, there is still ongoing work on the subject, especially concerning the Yang-Mills theory).

However there is no rigorous "no-go theorem" that says that the interacting quantum field theories in $3+1$ dimensions cannot be constructed, but it may be possible that for some model the limitations are more fundamental and not only related to the lack of mathematical tools (see the comment by A.A. below).

Answer 2

The tools and techniques for constructing QFTs are the same whatever the dimension. However you cannot prove conjectures which are false, no matter how powerful your tools are. The issue with 3+1 is that the class of models for which the conjecture "yeah it can be constructed" is in all likelihood true is much more narrow than in 1+1 and 2+1. Basically the only candidate is Yang-Mills theory which is a rather difficult beast. Yet there has been work on the finite volume construction by Balaban, Federbush as well as Magnen-Rivasseau-Seneor.

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Edit: Just to clarify, Wightman axioms are just a definition of what a QFT is. Namely, it is a precise formulation of what the end goal of "constructing a QFT" is, but it does not tell you how to get there. Besides, these axioms are not quite appropriate for $YM_4$, not because it is in 4 dimensions but because it is a gauge theory. Nevertheless, there is a precise formulation of the $YM_4$ end goal in the $1M problem description by Jaffe and Witten (see also the follow up by Douglas, all available here).

As for methods being the same, essentially you need to construct a probability measure on a space of distributions as a weak limit of well defined measures as you remove the UV and IR cut-offs, then you get the QFT in Minkowski space using the Osterwalder-Schrader Theorem. The difficult part is the construction of the limit probability measure. The methods for doing that are the rigorous renormalization group theory or its variant called the multiscale cluster expansion method. These methods work for scalar models in $1+1$, $2+1$ as well as $YM_4$ as treated by Balaban and the other authors I mentioned above.

Answer 3

QFTs in spacetime of dimension $<4$ have their use in real applications - 3D theories to quantum surfaces, and 2D theories to quantum wires. There much of the exceptional behavior of lower-dimensional QFTs can be observed.

A famous example is the fractional Hall effect with anyonic (rather than Bose or Fermi) statistics.