Path Integral for Separable Hamiltonians of Liouville-type

TL;DR

This paper reviews a path integral approach for Liouville-type separable Hamiltonians, emphasizing gauge choices, operator ordering, and providing an exact Green's function for the hydrogen atom, linking quantum gravity concepts.

Contribution

It introduces a path integral formulation based on Jacobi's principle for Liouville-type Hamiltonians, highlighting gauge invariance and operator ordering effects, with an explicit example for the hydrogen atom.

Findings

Exact path integral for hydrogen atom Green's function in parabolic coordinates

Demonstration of gauge independence in the path integral formulation

Connection between path integral and one-dimensional quantum gravity

Abstract

A general path integral analysis of the separable Hamiltonian of Liouville-type is reviewed. The basic dynamical principle used is the Jacobi's principle of least action for given energy which is reparametrization invariant, and thus the gauge freedom naturally appears. The choice of gauge in path integral corresponds to the separation of variables in operator formalism. The gauge independence and the operator ordering are closely related. The path integral in this formulation sums over orbits in space instead of space-time. An exact path integral of the Green's function for the hydrogen atom in parabolic coordinates is ilustrated as an example, which is also interpreted as one-dimensional quantum gravity with a quantized cosmological constant.