Virial Theorem for Nonrelativistic Quantum Fields in D Spatial Dimensions

Abstract

The virial theorem for nonrelativistic 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.

Chris L. Lin 1 and Carlos R. Ordoñez 1, 2

Academic Editor:Piero Nicolini

1, Department of Physics, University of Houston, Houston, TX 77204-5005, USA
2, Department of Science and Technology, Technological University of Panama, Campus Victor Levi, Panama City, Panama

Received 19 March 2015; Accepted 14 June 2015; 22 June 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³ .

1. Introduction

The virial theorem has been proven using a variety of methods. Recently, a path-integral derivation of the virial theorem has been developed in the context of quantum anomalies in nonrelativistic 2D systems, or more generally, systems with [figure omitted; refer to PDF] classical symmetry [1]. The path integral is most useful in isolating the anomaly contribution to the equation of state so obtained. This method is in fact quite general and applicable for nonrelativistic systems with an arbitrary 2-body potential [figure omitted; refer to PDF] in [figure omitted; refer to PDF] spatial dimensions, even when there are no quantum anomalies present. We present such derivation in this note, extending the original derivation, using also diagrammatic analysis, and recasting the virial theorem into a general equation that relates macroscopic thermodynamics variables to the microscopic physics. As it will be shown, there is generically a Jacobian term [figure omitted; refer to PDF] that may contribute to the virial theorem, regardless of the existence of a classical scaling symmetry. We will mainly concern ourselves here with the case [figure omitted; refer to PDF] (which we term "nonanomalous"). Conclusions and Comments end the paper.

2. Virial Theorem

The work in [1] was based partly on the work by Toyoda et al. [2-4]. They postulated that spatial scalings ¹ [figure omitted; refer to PDF] leave the particle number density invariant: [figure omitted; refer to PDF] Let us consider a nonrelativistic system whose microscopic physics is represented by a generic 2-body interaction ² [figure omitted; refer to PDF] Giving our system a macroscopic volume [figure omitted; refer to PDF] , temperature [figure omitted; refer to PDF] , and chemical potential [figure omitted; refer to PDF] and going into imaginary time gives for the partition function [figure omitted; refer to PDF] Now consider a new system with the same temperature and chemical potential but at volume [figure omitted; refer to PDF] : [figure omitted; refer to PDF] Substituting (1) into (5) gives [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the Jacobian for the transformation [figure omitted; refer to PDF] . As mentioned above, our emphasis will be in the nonanomalous case, and henceforth we assume [figure omitted; refer to PDF] (see however Conclusions and Comments). Then [figure omitted; refer to PDF] , where the superscript [figure omitted; refer to PDF] represents a microscopic system whose kinetic energy has a factor [figure omitted; refer to PDF] and whose potential is [figure omitted; refer to PDF] . Note that [figure omitted; refer to PDF] .

The pressures corresponding to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are equal, since the intensive variables [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the same, and they correspond to the same microscopic system. The argument we just made for the pressures being the same is valid in the thermodynamic limit, based on the principle that two intensive variables determine the third via an equation of state, for example, [figure omitted; refer to PDF] , for an ideal gas. However, in the next section we will also provide a diagrammatical proof that the two pressures are the same.

For now assume that the pressures are equal. Then using [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] Following [1], we set [figure omitted; refer to PDF] for infinitesimal [figure omitted; refer to PDF] : [figure omitted; refer to PDF] where we have defined [figure omitted; refer to PDF] . Cancelling the partition functions on both sides, noting that thermal expectation values for the fields at the same [figure omitted; refer to PDF] are independent of [figure omitted; refer to PDF] so that the [figure omitted; refer to PDF] integral pulls out a [figure omitted; refer to PDF] , and denoting the kinetic energy as [figure omitted; refer to PDF] [figure omitted; refer to PDF] which is the virial theorem in [figure omitted; refer to PDF] dimensions (see (3.30) and (2.6) in [3] and [4], resp.).

3. [figure omitted; refer to PDF] -Body

It is clear that this method can be generalized to the [figure omitted; refer to PDF] -body case. Since by (2) the scaling transformation preserves [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), an [figure omitted; refer to PDF] -body term transforms as [figure omitted; refer to PDF] Setting [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the center of mass of the [figure omitted; refer to PDF] 's gives [figure omitted; refer to PDF] For translationally invariant systems, we can ignore the derivative w.r.t. the center of mass.

4. Diagrammatic Proof of [figure omitted; refer to PDF]

To prove diagrammatically that the pressure [figure omitted; refer to PDF] corresponding to [figure omitted; refer to PDF] is equal to the pressure [figure omitted; refer to PDF] corresponding to [figure omitted; refer to PDF] , it suffices to show that [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the grand potential. By the cluster expansion, [figure omitted; refer to PDF] is given by the sum of connected vacuum graphs [5]. Using the Feynman rules, [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] expresses conservation of momentum of the vacuum and [figure omitted; refer to PDF] is the Feynman amplitude ³ which is independent of [figure omitted; refer to PDF] , since [figure omitted; refer to PDF] contains expressions like [figure omitted; refer to PDF] which in the continuum limit [figure omitted; refer to PDF] ⁴ . Taking [figure omitted; refer to PDF] , it is clear that [figure omitted; refer to PDF] , so [figure omitted; refer to PDF] in the continuum limit.

Alternatively, since [figure omitted; refer to PDF] , another way to show [figure omitted; refer to PDF] is to show that the grand potential [figure omitted; refer to PDF] of [figure omitted; refer to PDF] is larger by a factor of [figure omitted; refer to PDF] than [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] .

The grand potential [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF] By the cluster expansion, [figure omitted; refer to PDF] is given by the sum of connected vacuum graphs. [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have the same macroscopic parameters and only differ in that [figure omitted; refer to PDF] 's propagator is [figure omitted; refer to PDF] and that the potential is [figure omitted; refer to PDF] instead of [figure omitted; refer to PDF] . Fourier transforming equation (14) gives the relationship [figure omitted; refer to PDF] The Feynman rules for the theory say that each vertex contributes its Fourier transform [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the momentum flowing through the vertex, and each propagator contributes (13). For vacuum graphs, all momenta [figure omitted; refer to PDF] in the vertices and propagators are integrated over in loop momenta [figure omitted; refer to PDF] . Let us make the change of variables [figure omitted; refer to PDF] and relabel [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . This will cause [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in the loop integrals.

Therefore, [figure omitted; refer to PDF] is the same as [figure omitted; refer to PDF] , except for an overall scale factor of [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the number of vertices and [figure omitted; refer to PDF] is the number of loops. Topologically, for connected vacuum graphs of the 2-body potential, [figure omitted; refer to PDF] . So the overall scale factor becomes [figure omitted; refer to PDF] . Hence, [figure omitted; refer to PDF] , and therefore [figure omitted; refer to PDF] .

This generalizes to translationally invariant [figure omitted; refer to PDF] -body potentials and for spontaneous symmetry breaking. Suppose the interaction is of the form [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the number of fields in the interaction with spatial coordinate [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . For translationally invariant potentials [figure omitted; refer to PDF] So [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , ⁵ this again gives [figure omitted; refer to PDF] For a diagram with a mixture of vertices of different types, [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the number of vertices of type [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the number of lines coming out of each vertex: [figure omitted; refer to PDF]

5. Scale Equation

The virial equation (9) can be recast into a different form that illustrates the effect of microscopic scales on the thermodynamics of a system. A simple way to see this is to write the potential as ⁶ [figure omitted; refer to PDF] [figure omitted; refer to PDF] is a dimensionless function whose arguments are the ratios of the couplings [figure omitted; refer to PDF] of [figure omitted; refer to PDF] to their length dimension [figure omitted; refer to PDF] expressed in units of [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] provides units of energy) ⁷ . Denoting [figure omitted; refer to PDF] [figure omitted; refer to PDF] where the chain rule was used in line 2. Substituting this into (9) gives [figure omitted; refer to PDF] Rearranging [figure omitted; refer to PDF] On the LHS of (24) are macroscopic thermodynamic variables. The RHS is a measure of the microscopic physics of the system. In particular, if the potential has no scales [figure omitted; refer to PDF] and no anomalies (i.e., [figure omitted; refer to PDF] ), you get 0 on the RHS and (24) reduces to the equation of state for a nonrelativistic scale-invariant system [6].

6. Conclusion and Comments

The goal of this paper has been to highlight certain features in the derivation of the virial theorem for nonrelativistic systems, which display a potentially important omission due to the presence of the Jacobian needed in the path-integral derivation developed here. Indeed, while we set [figure omitted; refer to PDF] at the outset in order to make contact with the literature (specifically, Toyoda et al.'s work [2-4]), (6) shows that the natural procedure would be to not assume this and keep the contribution of the Jacobian, regardless of whether or not there is a classical scaling symmetry. Obviously, in the latter case, one has to keep the Jacobian in order to incorporate the quantum anomaly as was shown in [1]. The formal mathematical steps in the general case presented here are the same as in that paper, and (24) would become [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and we have also used the [figure omitted; refer to PDF] matrix notation of [7] ( [figure omitted; refer to PDF] includes both a matrix and functional trace).

As with the work in [1, 7], the key to assess the importance of the Jacobian term rests upon one's ability to compute its contribution in detail, which implies a careful regularization procedure and possibly also renormalization. The actual details will depend on the type of potentials considered. An interesting direction is the relativistic generalization of these ideas. Work on this is currently in progress [8].

Acknowledgment

Carlos R. Ordoñez wishes to thank the Technological University of Panama for its hospitality at different stages of this work.

Endnotes

1.: Toyoda et al. introduced an auxiliary external potential that has the effect of confining the system to a volume [figure omitted; refer to PDF] and, then, through a series of infinitesimal scalings and algebraic arguments derived what amounts to the equation of state, which they referred to as virial theorem. Unlike them, we are not using an external potential but simply consider a system with a large volume [figure omitted; refer to PDF] (so all the typical large-volume thermodynamical considerations apply), but, like them, we are also calling virial theorem the equation of state that will be derived in this paper.

2.: In this paper we set [figure omitted; refer to PDF] .

3.: [figure omitted; refer to PDF] is the T-matrix, and [figure omitted; refer to PDF] .

4.: For finite volume, momenta are discrete and summed over [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is a box of unit volume surrounding the discrete lattice point [figure omitted; refer to PDF] . In the limit of large [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is assumed not to vary much, so any point within [figure omitted; refer to PDF] not on the lattice would still contribute the same value of [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] .

5.: [figure omitted; refer to PDF] lines come out of each vertex, and each line coming out is [figure omitted; refer to PDF] of an internal line, so [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the number of internal lines. The number of loops is the number of independent momenta, [figure omitted; refer to PDF] . So [figure omitted; refer to PDF] .

6.: We are now restricting ourselves to radial potentials.

7.: As an example, consider [figure omitted; refer to PDF] , where the coupling [figure omitted; refer to PDF] has length dimension -4 and [figure omitted; refer to PDF] has length dimension -3. Then [figure omitted; refer to PDF] . The couplings [figure omitted; refer to PDF] and [figure omitted; refer to PDF] provide the characteristic length scales.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

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[8] C. L. Lin, C. R. Ordoñez In progress