Quantum Mechanics on Manifolds

Abstract

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.

34 pages