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In QFT, many mathematical issues arise. Setting aside renormalization, these deal with rigorous constructions of objects underlying QFT:

i) In the canonical quantization approach, the main issue comes from trying to multiply (operator-valued) distributions. My understanding is that mathematicians have formalized some settings in which this makes sense, but you have to be very careful

ii) In the path integral approach, the main issue comes from defining the path integral (both in terms of defining a sensible measure on paths, as well as making the integral well-defined despite the presence of oscillatory integrals).

My main question is: Are the two issues (one for the canonical approach and the other for the path integral approach) related? If so, intuitively (from a purely mathematical perspective) how is the problem of defining products of distributions related to the problem of defining path integrals?

I'm particularly curious as to whether there's some intuition to be gained from the usual proof of the equivalence (between the two approaches) in non-relativistic QM (which begin's with Schrodinger's equation, inserts of bunch of intermediate states, and removes operators one by one).

In the case of non-relativistic QM, my understanding is that the canonical approach can be made fully rigorous, while the path integral approach isn't quite so (one can use Wick rotation to compute the integral using the Weiner measure, then rotate back using some analytic continuation argument, but I have read that this is only justified in some cases). Given this, do we expect that the issues of rigor aren't quite the same in the QFT land as well?

Disclaimer:

  • Apologies for any inaccuracies in my characterization of anything, as I'm still a beginner grasping with many aspects of QFT
  • I am aware that decades of work have gone into formalizing QFT rigorously, and have addressed many of of i) and ii) for different variations of QFTs. What I'd like to understand here in particular is, if these approaches have given insight into how the different issues I outlined above are related.

EDIT: I edited the original question to focus on just one question. Originally, I also asked about renormalization (which is what one of the comments addresses)

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    $\begingroup$ We know from Causal Perturbation theory that the renormalisation infinities come from multiplying operator valued distributions in a too naïve way. That is, it is possible to start directly with renormalised quantities so that no infinity appears anywhere in the perturbation series, because you would have carefully handled every multiplication of operator valued distributions. The resulting QFT will not have counterterms, which is great. But the perturbation series with every term finite, could still diverge by summing, even if we deal with the IR divergence somehow. $\endgroup$ Aug 18, 2023 at 4:40
  • $\begingroup$ @naturallyInconsistent thank you for the pointer! (for reference, I've edited the question, however, to not deal with renormalization. I will ask a separate question later.) $\endgroup$
    – Sam Park
    Aug 19, 2023 at 1:42

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Yes. One can show that the appropriate space on which the path integral is defined is the space of distributions on Euclidean space. Moreover, even for a free field in d>=2, the field configuration is is singular with probability 1. The best case is d=1, in which the field (at this point just a free brownian particle) is continuous but differentiable nowhere with probability 1.

Because of this, the construction of the path integral suffers from the same UV problems resulting from the multiplication of distributions. Familiar renormalization operations such as wick ordering have analogous operations on the path integral side. A standard reference for these questions is Glimm and Jaffe.

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