The Functional Integral for a Free Particle on a Half-Plane

TL;DR

This paper constructs the functional integral for a free particle confined to a half-plane, incorporating all self-adjoint boundary conditions, including non-local ones, and connects these to Green's functions via analytic continuation.

Contribution

It provides a comprehensive construction of the Brownian functional integral for all self-adjoint Hamiltonians with boundary conditions on a half-plane, including non-local boundary conditions.

Findings

Functional integral formulated for all boundary conditions

Non-local boundary conditions modeled by path jumps

Green's functions derived via analytic continuation

Abstract

A free non-relativistic particle moving in two dimensions on a half-plane can be described by self-adjoint Hamiltonians characterized by boundary conditions imposed on the systems. The most general boundary condition is parameterized in terms of the elements of an infinite-dimensional matrix. We construct the Brownian functional integral for each of these self-adjoint Hamiltonians. Non-local boundary conditions are implemented by allowing the paths striking the boundary to jump to other locations on the boundary. Analytic continuation in time results in the Green's functions of the Schrodinger equation satisfying the boundary condition characterizing the self-adjoint Hamiltonian.