Resolving problems with the continuum limit in coherent-state path integrals

TL;DR

This paper addresses the continuum limit issues in bosonic thermal coherent-state path integrals by constructing exact discrete versions, analyzing different orderings, and establishing the symmetric ordering as the correct choice through a renormalization approach.

Contribution

It provides a rigorous justification for using symmetric ordering in coherent-state path integrals for all Hamiltonians, resolving longstanding ambiguities.

Findings

Symmetric ordering is validated as the correct continuum limit.

Exact discrete path integrals are constructed for different orderings.

A renormalization procedure justifies the symmetric ordering for all Hamiltonians.

Abstract

The paper solves the problem of continuum limit in bosonic thermal coherent-state path integrals. For this purpose, exact discrete versions of the path integral are constructed for three different orderings of the Hamiltonian: normal, anti-normal and symmetric (Weyl order). Subsequently, their different continuum versions are checked on the harmonic oscillator, to choose the symmetric ordering as a possibly correct choice for all Hamiltonians. Spotted mathematical subtleties in the simple case serve as a clue to the general solution. Finally, a general justification for the symmetric order is provided by deriving the continuum path integral starting from the exact discrete case by a renormalization procedure in the imaginary time frequency domain. While the role of Weyl order has already been found, the paper provides the missing proof of its suitability for every Hamiltonian and simplifies the previously established construction by referring only to creation and annihilation operators (without position and momentum operators).