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

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.

26 pages, plain TeX; UBCTP 93-011

References in corpus (5)

• Two Dimensional Gauge Theories Revisited
• Cohomological Partition Functions for a Class of Bosonic Theories
• Exact Path Integrals by Equivariant Cohomology
• On Quantum Integrability and the Lefschetz Number
• Index Theorems and Loop Space Geometry

Cited by in corpus (3)

• Localization and Diagonalization: A review of functional integral techniques for low-dimensional gauge theories and topological field theories
• Perturbative and global anomalies in bosonic analogues of integer quantum Hall and topological insulator phases
• Conformal Motions and the Duistermaat-Heckman Integration Formula