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According to this paper, "The gauge integral [a.k.a. Henstock-Kurzweil integral] provides the only formal framework that is close to the original development of the Feynman path integral", and also "if one requires that a mathematical formalisation remains close to the original treatment in physics, then there seems to be no choice other than the gauge integral for the formalisation of Feynman’s path integral."

Is there any merit to these claims? I've been under the impression that there are a lot of ways to make path integrals rigorous in various settings, maybe some more successful than others, but I haven't heard of this one before.

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I found arXiv:1712.00046 helpful as an explanation of the method of path integration using the Henstock-Kurzweil integral; the key merit seems to be that one can avoid going to imaginary time; this keeps the mathematical treatment closer to the physical picture, and might be accepted as a justification of the quote in the OP.

In the real time formulation the path integral is written as the expectation value of a potential functional on a space of continuous paths with respect to a complex probability distribution. This calls for the replacement of a probability theory based on countably additive positive measures by one based on finitely additive generalized Riemann sums, which is the essence of the Henstock-Kurzweil approach to integration.

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    $\begingroup$ I guess I don't see the difference between what's done in the paper you cite and the "sequential approach" described here. I thought the point of gauge integrals is that one uses a "gauge" to control oscillatory singularities, but it seemed to me that the path integral singularities are all out at infinity. So essentially they are simply integrating over a box and letting the box go to infinity. Am I missing something? $\endgroup$
    – Nik Weaver
    Sep 18 '18 at 13:13
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    $\begingroup$ The way I understand it, is that the sequential approach you mention gives a formal expression for the path integral, equation 2 in this note, which needs to be evaluated in a way that converges as we take the limit of an infinite number of time slices. Analytic continuation to imaginary time is one way, while equation 34 in arXiv:1712.00046 provides an alternative way. $\endgroup$ Sep 18 '18 at 14:14
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There's actually a full book length treatment by P. Muldowney called A General Theory of Integration in Function Spaces, including Weiner and Feynman Integration published in 1987.

I've only come across this recently and like you I'm curious as to whether the Henstock integral does actually solve the problem of path integration a la Feynman. It's well known that there is no Lesbegue measure that can do so. However, the Henstock integral is more general than the Lesbegue and so it's a definite possibility. Of course there are other ways in which the path integral is formalised, say with Gromov-Witten invariants and/or TQFTs, but there is an attraction to rigorously defining the integral exactly as Feynman did. In fact, one supposes that it must be capable of rigorous definition in this manner under certain hypotheses about the function space - otherwise Feynman couldn't have derived his formula no matter how physically heuristic it was.

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