Quantum Mechanics on Manifolds

A definition of quantum mechanics on a manifold $M$ is proposed and a method to realize the definition is presented. This scheme is applicable to a homogeneous space $M = G / H$. The realization is a unitary representation of the transformation group $G$ on the space of vector bundle-valued functions. When $H \ne \{ e \}$, there exist a number of inequivalent realizations. As examples, quantum mechanics on a sphere $S^n$, a torus $T^n$ and a projective space $\RP$ are studied. In any case, it is shown that there are an infinite number of inequivalent realizations.