A gauge model for quantum mechanics on a stratified space

TL;DR

This paper develops a quantum gauge theory on a stratified space, explicitly constructing the associated costratified Hilbert space and analyzing tunneling, energy states, and expectation values in the context of a single plaquette.

Contribution

It introduces a novel quantization procedure on stratified Kähler spaces and constructs a costratified Hilbert space reflecting the space's stratification, specifically applied to quantum gauge theory on a plaquette.

Findings

Constructed a costratified Hilbert space for the gauge theory.

Analyzed tunneling probabilities between strata.

Determined energy eigenstates and expectation values in different coupling regimes.

Abstract

In the Hamiltonian approach on a single spatial plaquette, we construct a quantum (lattice) gauge theory which incorporates the classical singularities. The reduced phase space is a stratified K\"ahler space, and we make explicit the requisite singular holomorphic quantization procedure on this space. On the quantum level, this procedure furnishes a costratified Hilbert space, that is, a Hilbert space together with a system which consists of the subspaces associated with the strata of the reduced phase space and of the corresponding orthoprojectors. The costratified Hilbert space structure reflects the stratification of the reduced phase space. For the special case where the structure group is $\mathrm{SU}(2)$, we discuss the tunneling probabilities between the strata, determine the energy eigenstates and study the corresponding expectation values of the orthoprojectors onto the subspaces associated with the strata in the strong and weak coupling approximations.