Existence of real time quantum path integrals

TL;DR

This paper demonstrates a non-perturbative, real-time formulation of quantum path integrals for theories with analytic classical actions, using complex thimbles and eigenflow to ensure convergence without Wick rotation.

Contribution

It introduces a novel method to define real-time path integrals non-perturbatively by deforming integration contours onto complex thimbles and replacing gradient flow with eigenflow in infinite dimensions.

Findings

Path integrals are absolutely convergent on complex thimbles.

Eigenflow effectively identifies relevant thimbles in infinite dimensions.

The approach is promising for quantum gravity and theories lacking Euclidean formulations.

Abstract

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.