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Does there exist a mathematical rigorous theory of the Feynman-Path-Integral in Quantum Mechanics or Quantum Field Theory?

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Unambgiously "yes" for ordinary quantum mechanics, "depends on what you mean" for quantum field theory. A good reference is "Quantum Physics - A functional integral point of view" by Glimm and Jaffe.

The path integral in ordinary quantum mechanics can be made fully rigorous without any reservations in terms of measures on abstract Wiener spaces.

Glimm and Jaffe rigorously construct an analogous theory of quantum fields based on the Osterwalder-Schrader axioms, the Euclidean siblings of the Wightman axioms, which, by virtue of the OS reconstruction theorem, also rigorously define Lorentzian quantum fields obeying the Wightman axioms.

Unfortunately, Glimm and Jaffe's construction depends heavily on the spacetime dimension because the convergence properties of various integrals get worse as the dimension increases. There exist rigorous constructions for arbitrary scalar fields in 1+1 dimensions with arbitrary polynomial interaction terms, and some constructions for interacting theories in 2+1 dimensions.

Proving the rigorous existence of interacting quantum field theories in 3+1 dimensions, in particular of theories like the Standard Model, is an open question known as the Yang-Mills millenium problem.

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  • $\begingroup$ You say that it is a solved problem in QM, but isn't the Wick rotation from Euclidean to Minkowski still somewhat problematic? At least, that is the impression I got. $\endgroup$ May 1, 2023 at 3:08
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    $\begingroup$ @naturallyInconsistent I am not aware of any problems in the cases where the path integral is well-defined. The Osterwalder-Schrader reconstruction theorem (see also chapter 19 in Glimm/Jaffe) guarantees analytic continuation of the Euclidean fields and n-point functions produces valid Wightman fields and Wightman functions. (But note that, in general, you can't think of this analytic continuation as a "rotation". The rigorous meaning of "Wick rotation" is really just analytic continuation + the reconstruction theorem. $\endgroup$
    – ACuriousMind
    May 1, 2023 at 10:21
  • $\begingroup$ Yes, that Wick isn't a rotation but is rather analytic continuation is clear to me. But that can only be done if you have an exact integral. If you have a numerical integral, then the analytical continuation will lose too much information. $\endgroup$ May 2, 2023 at 1:41

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