Regularizing Feynman path integrals using the generalized Kontsevich-Vishik trace

TL;DR

This paper introduces a new regularization method for Feynman path integrals using the generalized Kontsevich-Vishik trace, enabling well-defined computations in both Euclidean and Minkowskian space-times through analytic continuation.

Contribution

It proposes a novel regularization of Feynman path integrals via the generalized Kontsevich-Vishik trace, extending classical trace concepts to Fourier Integral Operators for better mathematical rigor.

Findings

Regularized path integrals are consistent with lattice and Wick rotation formulations.

Computations reduce to integrals over spheres, simplifying evaluations.

Applicable to models like the massive Schwinger model and $$ theory.

Abstract

A fully regulated definition of Feynman's path integral is presented here. The proposed re-formulation of the path integral coincides with the familiar formulation whenever the path integral is well-defined. In particular, it is consistent with respect to lattice formulations and Wick rotations, i.e., it can be used in Euclidean and Minkowskian space-time. The path integral regularization is introduced through the generalized Kontsevich-Vishik trace, that is, the extension of the classical trace to Fourier Integral Operators. Physically, we are replacing the time-evolution semi-group by a holomorphic family of operator families such that the corresponding path integrals are well-defined in some half space of $\mathbb{C}$. The regularized path integral is, thus, defined through analytic continuation. This regularization can be performed by means of stationary phase approximation or computed analytically depending only on the Hamiltonian and the observable (i.e., known a priori). In either case, the computational effort to evaluate path integrals or expectations of observables reduces to the evaluation of integrals over spheres. Furthermore, computations can be performed directly in the continuum and applications (analytic computations and their implementations) to a number of models including the non-trivial cases of the massive Schwinger model and a $\varphi^4$ theory.