Path Integral Solution of Linear Second Order Partial Differential Equations I. The General Construction

TL;DR

This paper introduces a general path integral framework for solving a broad class of linear second order partial differential equations with various boundary conditions, extending known solutions for specific equations like Schrödinger and diffusion.

Contribution

It develops a unified path integral construction for elliptic, parabolic, and hyperbolic PDEs with boundary conditions, based on Cartier/DeWitt-Morette's functional integration framework.

Findings

Constructed elementary kernels for Dirichlet and Neumann boundary conditions.

Unified path integral solution applicable to elliptic, parabolic, and hyperbolic PDEs.

Extended the path integral approach beyond unbounded Schrödinger/diffusion equations.

Abstract

A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schr\"{o}dinger/diffusion equation in unbounded space. The construction is based on a framework for functional integration introduced by Cartier/DeWitt-Morette.