Phase Space Isometries and Equivariant Localization of Path Integrals in Two Dimensions

TL;DR

This paper explores the application of equivariant localization formulas to path integrals on two-dimensional symplectic manifolds, revealing that harmonic oscillators are central to the Hamiltonians in maximally symmetric cases.

Contribution

It characterizes the class of Hamiltonians suitable for equivariant localization on two-dimensional phase spaces, including generalized harmonic oscillators and Darboux Hamiltonians, with implications for quantum geometry.

Findings

Maximally symmetric phase spaces admit only harmonic oscillator Hamiltonians.

Non-homogeneous phase spaces allow more Hamiltonian possibilities but introduce ambiguities.

General formulas for Darboux and generalized coherent state Hamiltonians are provided.

Abstract

By considering the most general metric which can occur on a contractable two dimensional symplectic manifold, we find the most general Hamiltonians on a two dimensional phase space to which equivariant localization formulas for the associated path integrals can be applied. We show that in the case of a maximally symmetric phase space the only applicable Hamiltonians are essentially harmonic oscillators, while for non-homogeneous phase spaces the possibilities are more numerous but ambiguities in the path integrals occur. In the latter case we give general formulas for the Darboux Hamiltonians, as well as the Hamiltonians which result naturally from a generalized coherent state formulation of the quantum theory which shows that again the Hamiltonians so obtained are just generalized versions of harmonic oscillators. Our analysis and results describe the quantum geometry of some two dimensional systems.