Virial Theorem for Nonrelativistic Quantum Fields inDSpatial Dimensions

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 = 1 is briefly presented.

Virial expansion for the Tan contact and Beth-Uhlenbeck formula from two-dimensional SO(2,1) anomalies

“…In this paper, we show that δb 2 is indeed produced entirely by the anomaly. We use a path-integral approach inspired by the work of [10][11][12][13][14][15][16]. In the process, we will describe the virial expansion of the Tan contact (which in 2D is interpreted as the anomaly [17]), as well as a procedure to compute δb n , n ≥ 2, using the Hubbard-Stratonovich (HS) representation of the partition function.…”

Dilational symmetry-breaking in thermodynamics

“…While the emphasis of this paper is on structural aspects of the thermodynamical traces 2E − DP and E − DP (nonrelativistic and relativistic cases respectively), we have included in section IV a few examples that connect our results with the literature and illustrate how to apply the techniques developed here. In our recent work [9][10][11]20] we have aimed at offering a different approach and new perspectives to the study of the type of systems presented here. We hope to use the insight gained to apply these techniques and concepts to other problems and systems [21].…”

“…In previous work [20] we derived the virial theorem via path integrals, and then used the virial theorem to derive Eq. ( 5).…”

Relationship between Fujikawa’s Method and the Background Field Method for the Scale Anomaly

“…The reason for embarking on this comparison is that a framework for applying Fujikawa's method to nonrelativistic, classically scale-invariant systems was undertaken recently [5][6][7]. While the quantum effective action is a standard tool in nonrelativistic physics (e.g., see [8,9]), Fujikawa's method is not.…”