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

PATH INTEGRALS IN PHASE SPACE

Jean Zinn‐Justin

Oxford University Press eBooksNov 18, 20041 citation

\\epsilon \\to 0

limits of

1+ \\epsilon

-dimensional functional integrals.