Quantizing the Cotangent Bundle and the Development of Quantum Mechanics in Curved Spaces

This work explores the formulation of quantum mechanics on the cotangent bundle, where the interplay between geometry and phase space reveals new structure. Nonlinear connections are employed to partition the Hamilton space into position and momentum components, generating a canonical metrical connection and tensor fields that respect the decomposition. The position and momentum representations are adapted to this new formalism, and the Fourier transform is modified to be compatible with the manifold's curvature. In doing this, we find that conjugate symmetry cannot hold in both position and momentum representations in curved space, and so a new form of Parseval's relation must be considered. Then, in generating a formal correspondence between the quantum and classical operators, the cotangent bundle is geometrically quantized. The Hilbert space realization of operators is generated by Hamilton flows from the classical observables. Gauge transforms are used to create globally defined operators at the expense of keeping track of phases, a result corresponding to Berry's phase. A complex line bundle then is naturally the object which the quantum operators act on. We show the existence of this line bundle by a form of Weil's integrality condition that uses the de Rham cohomology. In doing so, we see that phase space is quantizable in all Hamilton Spaces. This lays the groundwork for quantum theory in curved spaces, and generates a consistent theory on the classical and quantum level.