Quantum Hamilton-Jacobi Theory, Spectral Path Integrals and Exact-WKB Analysis

TL;DR

This paper introduces a novel quantum Hamilton-Jacobi framework for path integrals, utilizing spectral analysis and exact-WKB methods to provide a new perspective on quantum dynamics and spectral sums.

Contribution

It develops a quantum Hamilton-Jacobi formalism for path integrals, connecting phase space methods with spectral analysis and generalizing Gutzwiller's sum.

Findings

Formulation of spectral path integrals in quantum mechanics.

Derivation of quantum Hamilton's characteristic functions via asymptotic analysis.

Generalization of Gutzwiller's sum for quantum period lattices.

Abstract

We propose a new way to perform path integrals in quantum mechanics by using a quantum version of Hamilton-Jacobi theory. In classical mechanics, Hamilton-Jacobi theory is a powerful formalism, however, its utility is not explored in quantum theory beyond approximation schemes. The canonical transformation enables one to set the new Hamiltonian to constant or zero, but keeps the information about solution in Hamilton's characteristic function. To benefit from this in quantum theory, one must work with a formulation in which classical Hamiltonian is used. This uniquely points to phase space path integral. However, the main variable in HJ-formalism is energy, not time. Thus, we are led to consider Fourier transform of path integral, spectral path integral, $\tilde Z(E)$. The evaluation of path integral reduces to determining the quantum Hamilton's characteristic functions (which can be achieved via an asymptotic analysis), and a discrete sum over the quantum period lattice, generalizing Gutzwiller's sum.