Boundary Condition and the Auxiliary Phase in Feynman Path Integral

TL;DR

This paper investigates the role of auxiliary phase factors in Feynman path integrals for different boundary conditions, clarifying their nature and confirming their consistency with Schrödinger equation solutions.

Contribution

It provides a detailed analysis of boundary condition effects on phase factors in path integrals, extending understanding across Dirichlet, Neumann, Robin, and mixed types.

Findings

Phase factors depend on boundary conditions.

The derived propagator formulas match Schrödinger equation results.

Normalization factors are consistent with standard quantum mechanics.

Abstract

When employing Feynman path integrals to compute propagators in quantum physics, the concept of summing over the set of all paths is not always naive. In fact, an auxiliary phase often has to be included as a weight for each summand. In this article we discuss the nature of those phase factors for the various types of boundary conditions including all three of the Dirichlet, Neumann and Robin types, as well as their mixtures. We verify that for a free particle confined on a line segment, the resulting formula on the propagator matches those arising from the Schrodinger equation, with a trivial normalization factor.