Recursive generation of the semiclassical expansion in arbitrary dimension

Cihan Pazarbaşı

doi:10.1103/physrevd.103.085011

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.

Highlights

One of the main practical tasks of a theoretical physicist is constructing and solving differential or integral equations
We propose a method that can be used for practical calculations, and only infinities we encounter will be related to the physical spectrum
We investigated the recursive nature of the derivative expansion of the quantum action and showed how to implement it in practical calculations for quantum anharmonic oscillators in arbitrary dimensions

Summary

INTRODUCTION

One of the main practical tasks of a theoretical physicist is constructing and solving differential or integral equations. The corresponding differential equation for the quantum corrections were generated3 [7,8,9,10] Another method is based on the recursive nature of the holomorphic anomaly equations of topological string theory [11,12], which is related to the one-dimensional quantum mechanics in the NekrasovShatashvili limit [13]. All of these approaches utilize the recursive dependence of the quantum corrections to the classical term, i.e., ħ0 order. V, we finish the paper with a discussion of our analysis and an outlook to future work

SPECTRAL PROBLEM

EXPANSION IN D DIMENSIONS

AN EXAMPLE

A comment on quantization conditions

DISCUSSION AND OUTLOOK

Leading order