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A Stochastic Approach to the Definition of the Path Integral Measure

Timur Obolenskiy

TL;DR

The paper tackles the long-standing problem of a rigorous path integral measure in Lorentzian quantum mechanics by restricting the path space to a tubular neighborhood around a classical trajectory and formulating fluctuations as a Gaussian $L^2$-flux process on a covariant bundle. It develops a complete geometric and probabilistic framework: a tubular normal-bundle geometry, a separable $L^2$ flux space, a covariant stochastic dynamics, and a Gaussian measure whose Radon–Nikodym derivative yields a well-defined stochastic path integral $I_oldsymbol{μ}(O)=\mathbb{E}_oldsymbol{μ}[O(X) e^{(i/\\hbar)S(X)}]$, along with a semigroup structure and a Riemann product convergence. The work proves the stochastic construction is equivalent to the Euclidean theory via Wick rotation and to the Feynman–Kac theorem, enabling transfer of rigorous Euclidean results to a Lorentzian, tube-confined setting. This approach highlights the role of normal and longitudinal fluctuations and offers a pathway to extensions to gauge theories, topologies, and more general path-space geometries while addressing convergence and measure-definition issues inherent in traditional path integrals.

Abstract

We to define a Stochastic 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. 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 equivalence of the Stochastic Path Integral to the Euclidean Path Integral theory and the Feynman-Kac theorem.

A Stochastic Approach to the Definition of the Path Integral Measure

TL;DR

The paper tackles the long-standing problem of a rigorous path integral measure in Lorentzian quantum mechanics by restricting the path space to a tubular neighborhood around a classical trajectory and formulating fluctuations as a Gaussian -flux process on a covariant bundle. It develops a complete geometric and probabilistic framework: a tubular normal-bundle geometry, a separable flux space, a covariant stochastic dynamics, and a Gaussian measure whose Radon–Nikodym derivative yields a well-defined stochastic path integral , along with a semigroup structure and a Riemann product convergence. The work proves the stochastic construction is equivalent to the Euclidean theory via Wick rotation and to the Feynman–Kac theorem, enabling transfer of rigorous Euclidean results to a Lorentzian, tube-confined setting. This approach highlights the role of normal and longitudinal fluctuations and offers a pathway to extensions to gauge theories, topologies, and more general path-space geometries while addressing convergence and measure-definition issues inherent in traditional path integrals.

Abstract

We to define a Stochastic Path Integral in Lorentzian time by restricting the relevant domain of integration on over a Riemannian configuration manifold and considering the dynamics of a particle evolving between to fixed endpoints with a referential non-degenerate classical trajectory. Through fibration, we reduce the infinite-dimensional space under consideration to an -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 equivalence of the Stochastic Path Integral to the Euclidean Path Integral theory and the Feynman-Kac theorem.

Paper Structure

This paper contains 16 sections, 10 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Redefining the Path Space
  3. The Classical View
  4. Hamilton's Principle and Boundedness
  5. The Quantum-Mechanical View
  6. On Relevant Trajectories
  7. Stochastic Processes on Path Space and Fibration
  8. Geometry of the $\eta$-Tube and Bundle structure
  9. Stochastic Dynamics in $\mathcal{T}_\eta$
  10. The Gaussian Expectation as a Stochastic Path Integral
  11. Formulating the Measure and the Stochastic Path Integral
  12. Semigroup structure and Riemann Product
  13. Equivalence of the Stochastic Path Integral and Euclidean Theory
  14. The Stochastic Path Integral and Wick Rotation
  15. Stochastic Path Integral and the Feynman-Kac theorem
  16. ...and 1 more sections

Key Result

Lemma 1

$\forall E \in \sigma$ where $\sigma$ is the non-degenerate part of the energy spectrum for a physical system with the Lagrangian $L(q,\dot{q}), \quad \exists\gamma_E \in \mathcal{P}, \space \gamma_E:[0,1] \rightarrow M, \space \gamma_E(0)=A, \space \gamma(1)=B.$ with $\gamma_E$ unique.

Theorems & Definitions (13)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Definition 1: Flux space
  • Lemma 4
  • Definition 2: Covariant Sobolev flux space
  • Definition 3
  • Theorem 2: Fubini theorem
  • Theorem 3
  • ...and 3 more