Virial Theorem for Nonrelativistic Quantum Fields in Spatial Dimensions
The virial theorem for nonrelativistic complex fields in 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 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 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, , is not natural, and the generalization to the case is briefly presented.
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 classical symmetry . 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 in 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 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 (which we term “nonanomalous”). Conclusions and Comments end the paper.
2. Virial Theorem
The work in  was based partly on the work by Toyoda et al. [2–4]. They postulated that spatial scalings1leave the particle number density invariant:Let us consider a nonrelativistic system whose microscopic physics is represented by a generic 2-body interaction2Giving our system a macroscopic volume , temperature , and chemical potential and going into imaginary time gives for the partition functionNow consider a new system with the same temperature and chemical potential but at volume :Substituting (1) into (5) giveswhere is the Jacobian for the transformation . As mentioned above, our emphasis will be in the nonanomalous case, and henceforth we assume (see however Conclusions and Comments). Then , where the superscript represents a microscopic system whose kinetic energy has a factor and whose potential is . Note that .
The pressures corresponding to and are equal, since the intensive variables and 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, , 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 , we getFollowing , we set for infinitesimal :where we have defined . Cancelling the partition functions on both sides, noting that thermal expectation values for the fields at the same are independent of so that the integral pulls out a , and denoting the kinetic energy as which is the virial theorem in dimensions (see (3.30) and (2.6) in  and , resp.).
It is clear that this method can be generalized to the -body case. Since by (2) the scaling transformation preserves (), an -body term transforms asSetting where and is the center of mass of the ’s givesFor translationally invariant systems, we can ignore the derivative w.r.t. the center of mass.
4. Diagrammatic Proof of
To prove diagrammatically that the pressure corresponding to is equal to the pressure corresponding to , it suffices to show that , where is the grand potential. By the cluster expansion, is given by the sum of connected vacuum graphs . Using the Feynman rules, , where expresses conservation of momentum of the vacuum and is the Feynman amplitude3 which is independent of , since contains expressions like which in the continuum limit 4. Taking , it is clear that , so in the continuum limit.
Alternatively, since , another way to show is to show that the grand potential of is larger by a factor of than . Then .
The grand potential is given byBy the cluster expansion, is given by the sum of connected vacuum graphs. and have the same macroscopic parameters and only differ in that ’s propagator isand that the potential isinstead of . Fourier transforming equation (14) gives the relationshipThe Feynman rules for the theory say that each vertex contributes its Fourier transform , where is the momentum flowing through the vertex, and each propagator contributes (13). For vacuum graphs, all momenta in the vertices and propagators are integrated over in loop momenta . Let us make the change of variables and relabel as . This will cause and in the loop integrals.
Therefore, is the same as , except for an overall scale factor of , where is the number of vertices and is the number of loops. Topologically, for connected vacuum graphs of the 2-body potential, . So the overall scale factor becomes . Hence, , and therefore .
This generalizes to translationally invariant -body potentials and for spontaneous symmetry breaking. Suppose the interaction is of the formwhere is the number of fields in the interaction with spatial coordinate , and . For translationally invariant potentialsSoSince ,5 this again givesFor a diagram with a mixture of vertices of different types, , where is the number of vertices of type and is the number of lines coming out of each vertex:
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 as6 is a dimensionless function whose arguments are the ratios of the couplings of to their length dimension expressed in units of ( provides units of energy)7. Denoting where the chain rule was used in line 2. Substituting this into (9) givesRearrangingOn 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 and no anomalies (i.e., ), you get 0 on the RHS and (24) reduces to the equation of state for a nonrelativistic scale-invariant system .
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 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 . The formal mathematical steps in the general case presented here are the same as in that paper, and (24) would becomewhere , , and we have also used the matrix notation of  ( 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 .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Carlos R. Ordóñez wishes to thank the Technological University of Panama for its hospitality at different stages of this work.
- Toyoda et al. introduced an auxiliary external potential that has the effect of confining the system to a volume 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 (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.
- In this paper we set .
- is the T-matrix, and .
- For finite volume, momenta are discrete and summed over . is a box of unit volume surrounding the discrete lattice point . In the limit of large , is assumed not to vary much, so any point within not on the lattice would still contribute the same value of . Then .
- lines come out of each vertex, and each line coming out is of an internal line, so where is the number of internal lines. The number of loops is the number of independent momenta, . So .
- We are now restricting ourselves to radial potentials.
- As an example, consider , where the coupling has length dimension −4 and has length dimension −3. Then . The couplings and provide the characteristic length scales.
S. Weinberg, The Quantum Theory of Fields, vol. 1-2, Cambridge University Press, Cambridge, UK, 2005.
C. L. Lin and C. R. Ordóñez, In progress.