The Schwinger Action Principle and the Feynman Path Integral for Quantum Mechanics in Curved Space

TL;DR

This paper explores how the Schwinger action principle can resolve ambiguities in defining the Feynman path integral in curved space, emphasizing the role of the Van Vleck-Morette determinant for consistency.

Contribution

It demonstrates that incorporating the Schwinger action principle constrains the path integral measure, leading to the inclusion of the Van Vleck-Morette determinant in curved space quantum mechanics.

Findings

The path integral measure ambiguity can be resolved using the Schwinger action principle.

The Van Vleck-Morette determinant naturally appears as a necessary factor.

The approach aligns with stochastic differential equation methods.

Abstract

The Feynman path integral approach to quantum mechanics is examined in the case where the configuration space is curved. It is shown how the ambiguity that is present in the choice of path integral measure may be resolved if, in addition to general covariance, the path integral is also required to be consistent with the Schwinger action principle. On this basis it is argued that in addition to the natural volume element associated with the curved space, there should be a factor of the Van Vleck-Morette determinant present. This agrees with the conclusion of an approach based on the link between the path integral and stochastic differential equations.