# Hubbard-Stratonovich transformation and consistent ordering in the coherent state path integral: insights from stochastic calculus

## TL;DR

This paper clarifies foundational issues in the coherent state path integral by applying stochastic calculus, resolving controversies about its validity, operator ordering, and the calculation of functional determinants involving Hubbard-Stratonovich fields.

## Contribution

It demonstrates that the problems stem from misapplying standard calculus rules in path integrals and provides a stochastic calculus-based framework to establish correct foundations.

## Key findings

- Resolved controversies on path integral validity and operator ordering.
- Established proper calculation methods for functional determinants with Hubbard-Stratonovich fields.
- Linked path integral issues to stochastic calculus principles.

## Abstract

Recently, doubts have been cast on the validity of the continuous-time coherent state path integral. This has led to controversies regarding the correct way of performing calculations with path integrals, and to several alternative definitions of what should be their continuous limit. Furthermore, the issue of a supposedly proper ordering of the Hamiltonian operator, entangled with the continuous-time limit, has led to considerable confusion in the literature. Since coherent state path integrals are at the basis of the modern formulation of many-body quantum theory, it should be laid on solid foundations. Here, we show that the issues raised above are coming from the illegitimate use of the (standard) rules of calculus, which are not necessarily valid in path integrals. This is well known in the context of stochastic equations, in particular in their path integral formulation. This insight allows for solving these issues and addressing the correspondence between the various orderings at the level of the path integral. We also use this opportunity to address the proper calculation of a functional determinant in the presence of a Hubbard-Stratonovich field, which shares in the controversies.