Path integrals on curved manifolds

Abstract

A general framework for treating path integrals on curved manifolds is presented. We also show how to perform general coordinate and space-time transformations in path integrals. The main result is that one has to subtract a quantum correctionΔV∼h2 from the classical Lagrangian ℒ, i.e. the correct effective Lagrangian to be used in the path integral is ℒeff = ℒ−ΔV. A general prescription for calculating the quantum correction ΔV is given. It is based on a canonical approach using Weyl-ordering and the Hamiltonian path integral defined by the midpoint prescription. The general framework is illustrated by several examples: Thed-dimensional rotator, i.e. the motion on the sphereSd−1, the path integral ind-dimensional polar coordinates, the exact treatment of the hydrogen atom inR2 andR3 by performing a Kustaanheimo-Stiefel transformation, the Langer transformation and the path integral for the Morse potential.