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 $\\epsilon \\to 0$ limits of $1+ \\epsilon$ -dimensional functional integrals.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 $\\epsilon \\to 0$ limits of $1+ \\epsilon$-dimensional functional integrals.

Jan 1, 199929 citations

Canonical and covariant path integrals

Allen C. Hirshfeld

Physics Letters AJul 1, 197818 citations

METHOD OF PATH INTEGRALS

WORLD SCIENTIFIC eBooksJan 1, 19810 citations

\\epsilon \\to 0

limits of

1+ \\epsilon

-dimensional functional integrals.