Virial Theorem for Non-relativistic Quantum Fields in D Spatial Dimensions

TL;DR

This paper derives a generalized virial theorem for non-relativistic quantum fields in D dimensions, incorporating potential quantum anomalies and the effects of microscopic scales on thermodynamics, using path-integral and scaling methods.

Contribution

It introduces a generalized virial theorem for quantum fields in arbitrary dimensions, accounting for potential Jacobian effects and quantum anomalies, extending previous formulations.

Findings

Derived virial theorem in D dimensions with potential anomalies.

Identified the role of Jacobian J in the virial theorem.

Recast the theorem to include effects of microscopic scales.

Abstract

The virial theorem for non-relativistic complex fields in $D$ spatial dimensions and with arbitrary many-body potential is derived, using path-integral methods and scaling arguments recently developed to analyze quantum anomalies in low-dimensional systems. The potential appearance of a Jacobian $J$ due to a change of variables in the path-integral expression for the partition function of the system is pointed out, although in order to make contact with the literature most of the analysis deals with the $J=1$ case. The virial theorem is recast into a form that displays the effect of microscopic scales on the thermodynamics of the system. From the point of view of this paper the case usually considered, $J=1$, is not natural, and the generalization to the case $J\neq 1$ is briefly presented.