Exact Path Integrals by Equivariant Cohomology

TL;DR

This paper explores the use of equivariant cohomology to compute path integrals exactly, demonstrating success in some quantum mechanics cases like the 1-d hydrogen atom, but also revealing ambiguities and limitations in the method.

Contribution

It applies equivariant localization techniques to quantum mechanics path integrals, showing both their potential and their limitations in explicit examples.

Findings

Path integral for 1-d hydrogen atom is localizable and computable.

Significant ambiguities exist in the localization approach.

The method fails in a large class of quantum mechanics examples.

Abstract

It is a common belief among field theorists that path integrals can be computed exactly only in a limited number of special cases, and that most of these cases are already known. However recent developments, which generalize the WKBJ method using equivariant cohomology, appear to contradict this folk wisdom. At the formal level, equivariant localization would seem to allow exact computation of phase space path integrals for an arbitrary partition function! To see how, and if, these methods really work in practice, we have applied them in explicit quantum mechanics examples. We show that the path integral for the 1-d hydrogen atom, which is not WKBJ exact, is localizable and computable using the more general formalism. We find however considerable ambiguities in this approach, which we can only partially resolve. In addition, we find a large class of quantum mechanics examples where the localization procedure breaks down completely.