Sum-Over-Histories Quantization of Relativistic Particle

TL;DR

This paper explores the sum-over-histories quantization approach for relativistic particles in curved space, analyzing the propagator's behavior, boundary conditions, and interactions with sources, highlighting the influence of measure choices and boundary effects.

Contribution

It provides a detailed derivation of the propagator's boundary conditions and the form of the Laplace-like operator in curved manifolds with boundaries, including distributional boundary terms.

Findings

The propagator satisfies a Schrödinger-like wave equation with a measure-dependent Laplace operator.

Boundary conditions for the propagator are derived from boundary measurement considerations.

Distributional boundary terms in the operator induce specific boundary conditions for the propagator.

Abstract

Sum-over-histories quantization of particle-like theory in curved space is discussed. It is reviewed that the propagator satisfies the Schrodinger equation respective wave equation with a Laplace-like operator. The exact dependence of the operator on the choice of measure is shown. Next, modifications needed for a manifold with a boundary are introduced, and the exact form of the equation for the propagator is derived. It is shown that the Laplace-like operator contains some distributional terms localized on the boundary. These terms induce proper boundary conditions for the propagator. This choice of boundary conditions is explained as a consequence of a measurement of particles on the boundary. The interaction with sources inside of the domain and sources on the boundary is also discussed.