Hubbard-Stratonovich Transformation: Successes, Failure, and Cure

TL;DR

This paper reviews the successes and limitations of the Hubbard-Stratonovich Transformation in many-body theory and proposes a variational approach using classical fields to overcome its failures, achieving faster convergence.

Contribution

It introduces a variational perturbation method with classical fields to improve upon the Hubbard-Stratonovich Transformation's handling of collective phenomena.

Findings

HST is successful in some cases but fails with competing collective channels.

Variational perturbation theory with classical fields converges exponentially fast.

The new approach addresses the limitations of the traditional HST.

Abstract

We recall the successes of the Hubbard-Stratonovich Transformation (HST) of many-body theory, point out its failure to cope with competing channels of collective phenomena and show how to overcome this by Variational Perturbation Theory. That yields exponentially fast converging results, thanks to the help of a variety of {\it collective classical fields}, rather than a fluctuating {\it collective quantum field} as suggested by the HST.