Recursive Generation of The Semi-Classical Expansion in Arbitrary Dimension

TL;DR

This paper introduces a recursive method to generate semi-classical expansions of the quantum action in arbitrary dimensions, utilizing spectral information from propagator singularities and complex analysis techniques.

Contribution

It presents a novel recursive procedure based on small time propagator expansion that generalizes semi-classical calculations to higher dimensions.

Findings

Method successfully applied to simple quantum mechanics examples.

Spectral analysis via propagator singularities enables dimensional generalization.

Provides a systematic approach for semi-classical expansion in arbitrary dimensions.

Abstract

We present a recursive procedure, which is based on the small time expansion of the propagator, in order to generate a semi-classical expansion of the \textit{quantum action} for a quantum mechanical potential in arbitrary dimensions. In the method we use the spectral information emerges from the singularities of the propagator on the complex $t$ plane, which are handled by the $i\ve$ prescription and basic complex analysis. This feature allows for generalization to higher dimensions. We illustrate the procedure by providing simple examples in non-relativistic quantum mechanics.