A Stochastic Approach to the Definition of the Path Integral Measure

TL;DR

This paper introduces a rigorous stochastic framework for defining Lorentzian path integrals by restricting the domain, reducing to flux spaces, and relating the formulation to Euclidean path integrals via the Feynman-Kac theorem.

Contribution

It presents a novel stochastic approach to define Lorentzian path integrals using Gaussian measures and fibration techniques, establishing a rigorous connection to Euclidean path integrals.

Findings

Defined Lorentzian path integral as an expectation over Gaussian measure.

Reduced infinite-dimensional space to L^2 flux spaces via fibration.

Proved correspondence between stochastic and Euclidean path integrals using Feynman-Kac theorem.

Abstract

We to define a Path Integral in Lorentzian time by restricting the relevant domain of integration on $C([0,1],M)$ over a Riemannian configuration manifold $(M,g)$ and considering the dynamics of a particle evolving between to fixed endpoints with a referential non-degenerate classical trajectory, formulating a framework around a quadratic Lagrangian. Through fibration, we reduce the infinite-dimensional space under consideration to an $L^2$-isometric flux spaces in which we consider a stochastic process associated to a Gaussian measure. The Path Integral is subsequently defined as an expectation value with respect to the Gaussian measure, allowing us to rigorously formulate the former as a functional integral. We prove mathematical correspondence between the Stochastic Path Integral and the Euclidean Path Integral theory formulated rigorously under the Feynman-Kac theorem.