Reparametrization Invariance of Path Integrals

TL;DR

This paper proves that perturbative one-dimensional path integrals in quantum mechanics are invariant under reparametrizations up to three loops, clarifying previous misconceptions and providing a rigorous definition via higher-dimensional limits.

Contribution

It demonstrates reparametrization invariance of path integrals up to three loops and resolves earlier issues by defining integrals as limits of higher-dimensional cases.

Findings

Reparametrization invariance holds up to three loops.

Earlier failures were due to improper definitions of path integrals.

A new definition via epsilon->0 limit of higher-dimensional integrals ensures invariance.

Abstract

We demonstrate the reparametrization invariance of perturbatively defined one-dimensional functional integrals up to the three-loop level for a path integral of a quantum-mechanical point particle in a box. We exhibit the origin of the failure of earlier authors to establish reparametrization invariance which led them to introduce, superfluously, a compensating potential depending on the connection of the coordinate system. We show that problems with invariance are absent by defining path integrals as the epsilon-> 0 -limit of 1+ epsilon -dimensional functional integrals.