Existence of real time quantum path integrals

Job Feldbrugge*, Neil Turok

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

Many interesting physical theories have analytic classical actions. We show how Feynman's path integral may be defined non-perturbatively, for such theories, without a Wick rotation to imaginary time. We start by introducing a class of smooth regulators which render interference integrals absolutely convergent and thus unambiguous. The analyticity of the regulators allows us to use Cauchy's theorem to deform the integration domain onto a set of relevant, complex "thimbles" (or generalized steepest descent contours) each associated with a classical saddle. The regulator can then be removed to obtain an exact, non-perturbative representation. We show why the usual method of gradient flow, used to identify relevant saddles and steepest descent "thimbles" for finite-dimensional oscillatory integrals, fails in the infinite-dimensional case. For the troublesome high frequency modes, we replace it with a method we call "eigenflow" which we employ to identify the infinite-dimensional, complex "eigenthimble" over which the real time path integral is absolutely convergent. We then bound the path integral over high frequency modes by the corresponding Wiener measure for a free particle. Using the dominated convergence theorem we infer that the interacting path integral defines a good measure. While the real time path integral is more intricate than its Euclidean counterpart, it is superior in several respects. It seems particularly well-suited to theories such as quantum gravity where the classical theory is well developed but the Euclidean path integral does not exist.
Original languageEnglish
Article number169315
Pages (from-to)1-31
Number of pages31
JournalAnnals of Physics
Early online date20 Apr 2023
Publication statusPublished - 1 Jul 2023

Keywords / Materials (for Non-textual outputs)

  • Real time path integral
  • Functional integration
  • Picard–Lefschetz theory Interference


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