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An answer to a question (What physical processes other than scattering are accounted for by QFT? How do they fit into the general formalism?) about quantum field theory asserts

"we don't know any examples of interacting QFTs in 4-dimensions."

I would like to know if this is true, and if it is, then what exactly is the issue.

I have read an introduction to QFT by Gerardus (Gerard) 't Hooft in which he avoids the well-known renormalization difficulties in the first instance by doing the whole thing on a lattice. Then, if I understood correctly, the continuum limit of the lattice is taken at the end and I thought it was all ok. So please could someone clarify exactly what it means to say there are no known examples of interacting QFTs (if that is so). And as a follow-up, does the lattice approach succeed for all practical purposes?

This paper: https://arxiv.org/abs/1912.07973 may be relevant too (it seems to offer a counter-example to the claim this question is about).

Qmechanic
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Andrew Steane
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1 Answers1

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The Wightman axioms formalise what we mean by QFT, and to my knowledge there are no interacting examples mathematically proven to satisfy the Wightman axioms in 3+1 dimensions. There's a few things to point out:

  • You can equivalently talk about the Osterwalder-Schrader axioms for Euclidean $n$-point functions, which formalise what a Euclidean-time QFT is. There is a rigorous reconstruction theorem, that tells you that you can Wick Rotate between the two formalisms (the Osterwalder-Schrader reconstruction theorem).
  • In textbooks, the first example of an 'interacting QFT' in 3+1d is the scalar $\lambda \phi^4$ theory. Indeed we can do perturbative calculations and things look like they may work - but actually the theory is nonperturbatively quantum trivial. One way to formally say this is that if you put the theory on a lattice and try to take the continuum limit, you find that you are forced to take $\lambda \to 0$. There were first indications of this numerically, by Lüscher; but recently there is a formal mathematical proof of this fact.
  • The other main example we have is Quantum Yang-Mills theory: establishing it's formal existence is a millenium problem.
  • The lattice is a regulator of any continuum QFT, which is formally well-defined, and makes all correlation functions finite (you have both a UV-cutoff due to the lattice spacing, and an IR-cutoff due to the finite volume of the lattice). Note that by putting the theory on a lattice, we have broken Lorentz symmetry to a hypercubic subgroup, and we only recover Loretnz symmetry in the continuum limit. We can put QCD on the lattice for example, and numerically we seem to be able to take the continuum limit perfectly well. As of 2023, we have no proof that the continuum limit actually exists in any formal sense though. (Might find this question I asked a while ago interesting)

For all practical purposes, the continuum limit of lattice-QCD is very well studied numerically, and seems to be perfectly consistent. (But there's no proof that things don't suddenly go crazy below some critical lattice spacing..)

QCD_IS_GOOD
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