Quantum mechanics on homogeneous spaces

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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 Rn. This method gives a physically justified and mathematically well−defined method of connecting the free Hamiltonian of a quantum system in Rn with an operator proportional to the Laplace−Beltrami operator on M (with the Riemannian structure inherited from Rn) which is proposed to be the free Hamiltonian on M.