Geometric quantization of non-relativistic and relativistic Hamiltonian mechanics

TL;DR

This paper demonstrates that both non-relativistic and relativistic Hamiltonian systems can be geometrically quantized within a unified framework by viewing them as Dirac constraint systems on the same cotangent bundle.

Contribution

It introduces a unified geometric quantization approach for non-relativistic and relativistic mechanics using Dirac constraint systems on cotangent bundles.

Findings

Compatible covariant quantizations for both systems are achieved.

The approach unifies non-relativistic and relativistic quantization methods.

Quantization is performed via vertical polarization on the cotangent bundle.

Abstract

We show that non-relativistic and relativistic mechanical systems on a configuration space Q can be seen as the conservative Dirac constraint systems with zero Hamiltonians on different subbundles of the same cotangent bundle T^*Q. The geometric quantization of this cotangent bundle under the vertical polarization leads to compatible covariant quantizations of non-relativistic and relativistic Hamiltonian mechanics.