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I am having trouble finding any clear information on path integrals. I am trying to find the difference between a normal path integral, a Feynman path integral and a Wiener path integral and the conditions on each to make a well-defined and calculable integral. Please can someone explain.

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    $\begingroup$ A normal path integral specifies a particular path in space, whereas the other two types integrate over many possible paths at once. There is not yet any rigorous mathematical framework for the Feynman path integral. The Wiener integral assigns a measure to the space of continuous functions and Lebesgue-integrates across that space using that measure. $\endgroup$ – probably_someone Jul 27 '17 at 9:58
  • $\begingroup$ @probably_someone Thanks for your comment. By 'path integral' I actually mean 'functional integral' rather then 'line integral'. For Feynman I am aware there is no such rigorous framework but I am yet to find any (even non-mathematically rigorous) definition of what path integrals count as Feynman path integrals. $\endgroup$ – Quantum spaghettification Jul 27 '17 at 10:03
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  1. A (Feynman) path integral is what physicists call any of the functional integrals that pop up in the functional integral (or path integral) approach to quantum mechanics and quantum field theory. I will not exclude the possibility that some might use the qualifer "Feynman" to done some specific integral, but for most physicists a "path integral" and a "Feynman path integral" are the same. Note that you shouldn't confuse this with the Feynman parameter trick, where the integrals appearing might be called "Feynman integrals" by some. The path integral is, in general, not known to be mathematically well-defined, in particular only very rarely in spacetime dimensions greater than 3. For scalar fields (and also some fermionic fields) with polynomial interactions in two dimensions it is known - see the work by Glimm and Jaffe - how to set up a well-defined path integral using the Wightman/Osterwalder-Schrader axioms.

  2. A Wiener integral is either the integral with respect to the Wiener measure on the classical Wiener space of continuous paths, or the more general notion of the Paley-Wiener integral. It is mathematically well-defined. The relation to path integrals comes about because the quantum mechanical version of the path integral (that of "0+1 dimensional QFT" if you like) is a (conditional) Wiener integral, and therefore mathematically well-defined.

If you are interested in a detailed exploration of the rigorous mathematics of the path integrals that appear in physics, "Quantum physics - A functional integral point of view" by Glimm and Jaffe is an excellent, if somewhat dense, resource.

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  • $\begingroup$ Hi thanks for your answer. From what I have seen (Feynman) path integrals (and actually sticking with the 1d case here) are usually given by: $\int \mathscr{D}[x(\tau)] \exp(-\int L[x(\tau)] )$. Surely a more general form is $\int \mathscr{D}[x(\tau)] F[x(\tau)]$ where $F[x(\tau)]$ is an arbitrary functional. Is there any restriction on $F[x(\tau)]$ such that this is well-defined? $\endgroup$ – Quantum spaghettification Jul 27 '17 at 10:58
  • $\begingroup$ @Quantumspaghettification The measure "$\mathscr{D}[x(t)]$" is ill-defined on its own, it's just an abuse of notation physicists use because it suggests the correct formal properties. The "actual" measure includes the "free part" from the $\exp(-\int L)$, so what we are really doing is integrating $\exp(-\int L_\text{interaction})$ against that ("Gaußian") measure. If you want to know what the Wiener measure actually looks like from a rigorous standpoint, I really cannot do better than chapter 3 of Glimm and Jaffe. $\endgroup$ – ACuriousMind Jul 27 '17 at 11:12
  • $\begingroup$ Great thanks, I have just read the relivent sections and now understand the concepts of the Wiener measure and Gaußian measure. You say "we are really doing is integrating $\exp\left(−\int L_{\text{interaction}}\right)$ against that ("Gaußian") measure". In line with my question above could we integrate $F[x(\tau)]$ against the Gaußian measure or does it have to take this exponential form? $\endgroup$ – Quantum spaghettification Jul 27 '17 at 11:32

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