The Topological Particle and Morse Theory

TL;DR

This paper presents a rigorous approach to the topological particle using Morse theory and BRST quantization, employing stochastic calculus to connect quantum evolution with manifold topology through critical points.

Contribution

It introduces a novel stochastic calculus method to analyze the topological particle, providing explicit expressions for evolution operators and linking quantum mechanics with Morse theory.

Findings

Explicit matrix elements for the BRST Hamiltonian evolution operator.

Representation of manifold cohomology via critical points of Morse functions.

Rigorous tunnelling results in curved space using Brownian paths.

Abstract

Canonical BRST quantization of the topological particle defined by a Morse function h is described. Stochastic calculus, using Brownian paths which implement the WKB method in a new way providing rigorous tunnelling results even in curved space, is used to give an explicit and simple expression for the matrix elements of the evolution operator for the BRST Hamiltonian. These matrix elements lead to a representation of the manifold cohomology in terms of critical points of h along lines developed by Witten.