Quantum mechanics on homogeneous spaces

Abstract

A complete description of quantum kinematics on a homogeneous G−space M is presented using imprimitivity systems for G based on M. The kinematics on M is considered (if possible and consistent with this quantization) as kinematics on a G−orbit equivalent to M in some Euclidean space R_n. This method gives a physically justified and mathematically well−defined method of connecting the free Hamiltonian of a quantum system in R_n with an operator proportional to the Laplace−Beltrami operator on M (with the Riemannian structure inherited from R_n) which is proposed to be the free Hamiltonian on M.