Equivariant Localization on Simply Connected Phase Spaces: Applications to Quantum Mechanics, Group Theory and Spin Systems

Abstract

When the phase space M of a dynamical system is compact, the condition that the Hamiltonian vector field V generate a global isometry of some Riemannian geometry on M automatically implies that its orbits must be closed circles (see ahead Section 5.2). This feature is usually essential for the finite-dimensional localization theorems, but within the loop space localization framework, where the arguments for localization are based on formal supersymmetry arguments on the infinite-dimensional manifold LM, the flows generated by V need not be closed and indeed many of the formal arguments of the last Chapter will still apply to non-compact group actions. For instance, if we wanted to apply the localization formalism to an n-dimensional potential problem, i.e. on the non-compact phase space M = ℝ2n , then we would expect to be allowed to use a Hamiltonian vector field which generates non-compact global isometries. As we have already emphasized, the underlying feature of quantum equivariant localization is the interpretation of an equivariant cohomological structure of the model as a supersymmetry among the physical, auxilliary or ghost variables. But as shown in Section 4.3, this structure is exhibited quite naturally by arbitrary phase space path integrals, so that, under the seemingly weak conditions outlined there, this formally results in the equivariant localization of these path integrals. This would in turn naively imply the exact computability of any phase space path integral.