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

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 $J\mathrm{\ne }1$ is briefly presented.