Group quantization of parametrized systems II. Pasting Hilbert spaces

TL;DR

This paper extends group quantization methods to parametrized systems without global transversal surfaces, demonstrating how to paste Hilbert spaces and verify equivalence with Dirac's operator constraint approach.

Contribution

It introduces a novel approach to pasting Hilbert spaces in group quantization for systems lacking global transversals, validated through a solvable toy model.

Findings

Successfully constructed quantum mechanics for the toy system.

Demonstrated equivalence with Dirac's operator constraint method.

Provided a framework for pasting Hilbert spaces in complex systems.

Abstract

The method of group quantization described in the preceeding paper I is extended so that it becomes applicable to some parametrized systems that do not admit a global transversal surface. A simple completely solvable toy system is studied that admits a pair of maximal transversal surfaces intersecting all orbits. The corresponding two quantum mechanics are constructed. The similarity of the canonical group actions in the classical phase spaces on the one hand and in the quantum Hilbert spaces on the other hand suggests how the two Hilbert spaces are to be pasted together. The resulting quantum theory is checked to be equivalent to that constructed directly by means of Dirac's operator constraint method. The complete system of partial Hamiltonians for any of the two transversal surfaces is chosen and the quantum Schr\"{o}dinger or Heisenberg pictures of time evolution are constructed.