The canonical formalism and path integrals in curved spaces

Abstract

The authors discuss the canonical and path integral quantisation of a particle moving on a curved manifold. They advance a simple prescription to quantise the system within the canonical formalism. An effective Hamiltonian Heff is defined as the classical Hamiltonian plus a quantum potential term of order h(cross)2. The Hamilton operator is obtained by Weyl ordering the expression that results when one replaces coordinates and momenta by their corresponding Hermitian operators in the effective Hamiltonian. They construct the phase space path integral representation for the propagator in terms of Heff. When the propagator is expressed as a Lagrangian path integral with an invariant measure an additional correction is introduced.