The geometry of real reducible polarizations in quantum mechanics

TL;DR

This paper generalizes geometric quantization for real polarizations by introducing connections, leading to a modified scheme that produces energy operators consistent with physics literature, especially on Riemannian manifolds.

Contribution

It extends geometric quantization to include arbitrary connections on polarization bundles, enabling new quantum structures on Liouville symplectic manifolds.

Findings

Energy operator without scalar curvature term

Consistent with standard physics expressions

Applicable to quantum mechanics on Riemannian manifolds

Abstract

The formulation of Geometric Quantization contains several axioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introducing an arbitrary connection on the polarization bundle. The existence of reducible quantum structures leads to considering the class of Liouville symplectic manifolds. Our main application of this modified geometric quantization scheme is to Quantum Mechanics on Riemannian manifolds. With this method we obtain an energy operator without the scalar curvature term that appears in the standard formulation, thus agreeing with the usual expression found in the Physics literature.