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 $J\neq 1$ is briefly presented.


II. VIRIAL THEOREM
The work in [1] was based partly on the work by Toyoda et al. [2][3][4]. In this work, it was postulated that spatial scalings 1 leave the particle number density invariant: Consider a non-relativistic system whose microscopic physics is represented by a generic 2-body interaction 2 * Electronic address: cllin@uh.edu; Electronic address: cordonez@central.uh.edu 1 Toyoda et al. introduced an auxiliary external potential that has the effect of confining the system to a volume V , 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're not using an external potential but simply consider a system with a large volume V (so all the typical large-volume thermodynamical considerations apply), but like them, we're also calling virial theorem the equation of state that will be derived in this paper. 2 In this paper we seth = m = 1.
Giving our system a macroscopic volume V , temperature β −1 , and chemical potential µ, and going into imaginary time gives for the partition function: Now consider a new system with the same temperature and chemical potential, but at volume V ′ = λ D V : Substituting Eq. (1) into Eq. (5) gives: where J is the Jacobian for the transformation (ψ ′ * , ψ ′ ) → (ψ * , ψ). As mentioned above, our emphasis will be in the non-anomalous case, and henceforth we assume J = 1 (see however comments and conclusions). Then where the superscript λ represents a microscopic system whose kinetic energy has a factor 1 λ 2 and whose potential is Z[λ D V, β, µ] has the same pressure as Z[V, β, µ], since the intensive variables µ and β −1 are the same, and they correspond to the same microscopic system (the action in Eq. (5) is the same action as in Eq. (4) since the fields are dummy indices). 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 e.g., P = ρT 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 the pressures are equal. Then using Z = e βP V , we get: Following [1], we set λ = 1 + η for infinitesimal η: . Cancelling the partition functions on both sides, noting that thermal expectation values for the fields at the same τ are independent of τ so that the τ integral pulls out a β, and denoting the kinetic energy as KE: which is the virial theorem in D dimensions (Eqs. (3.30) and (2.6) in [3] and [4] respectively).

III. N-BODY
It is clear that this method can be generalized to the n-body case. Since by Eq. (2) the scaling transformation , an n-body term transforms as For translationally-invariant systems, we can ignore the potential term in the 1st line.

IV. DIAGRAMMATIC PROOF OF P=P'
To prove diagramatically that the pressure P ′ corresponding to Z[λ D V, β, µ] is equal to the pressure P corresponding to where Ω is the grand potential. By the cluster expansion, Ω is given by the sum of connected vacuum graphs [5]. Using the Feynman rules, where δ D (0) expresses conservation of momentum of the vacuum and M(β, µ) is the Feynman amplitude which is independent of V , since M contains expressions like ∆n1...∆nD The grand potential is given by: By the cluster expansion Ω λ is given by the sum of connected vacuum graphs. Z λ [V, β, µ] and Z[V, β, µ] have the same macroscopic parameters and only differ in that Z λ 's propagator is and that the potential is instead of V ( x − y). Fourier transforming Eq. (14) gives the relationship: The Feynman rules for the theory say that each vertex contributes its Fourier transform V λ k , where k is the momentum flowing through the vertex, and each propagator contributes Eq. (13). For vacuum graphs, all momentum k in the vertices and propagators are integrated over in loop momenta 2π) D and relabelk as k. This will cause ∆ λ (iω, k) = ∆ iω, k λ → ∆(iω, k) and Therefore, Ω λ is the same as Ω, except for an overall scale factor of 1 λ D ν λ D L , where ν is the number of vertices and L is the number of loops. Topologically, for connected vacuum graphs of the 2-body potential, L = ν + 1. So the overall scale factor becomes λ D . Hence Ω λ = λ D Ω, and therefore P ′ = P .
This generalizes to translationally-invariant n-body potentials, and for spontaneous symmetry breaking. Suppose the interaction is of the form: where m(i) is the number of fields in the interaction with spatial coordinate x i , and M = n i=1 m(i). For translationallyinvariant potentials So Since L = M 2 − 1 ν + 1, 3 this again gives: For a diagram with a mixture of vertices of different types, L = i Mi 2 − 1 ν i + 1, where ν i is the number of vertices of type i, and M i is the number of lines coming out of each vertex: (20)

V. SCALE EQUATION
The virial equation, Eq. (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 4 : 3 M lines come out of each vertex, and each line coming out is 1/2 of an internal line, so M ν 2 = I where I is the number of internal lines. The number of loops is the number of independent momenta, L = I − ν + 1. So L = M 2 − 1 ν + 1. 4 We are now restricting ourselves to radial potentials.
f is a dimensionless function whose arguments are the ratios of the couplings g i of V to their length dimension [g i ] expressed in units of | where the chain rule was used in line 2. Substituting this into Eq. (9) gives Rearranging: On the LHS of Eq. (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 [g i ] = 0 and no anomalies, you get 0 on the RHS, and Eq. (24) reduces to the equation of state for a non-relativistic scale-invariant system [6].

VI. CONCLUSION AND COMMENTS
The goal of this paper has been to highlight certain features in the derivation of the virial theorem for non-relativistic systems, which display a potentially important omission due to the presence of the Jacobian needed in the pathintegral derivation developed here. Indeed, while we set J = 1 at the outset in order to make contact with the literature (specifically, Toyoda's et al. work [2][3][4]), Eq. (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 Eq. (24) would become where I 2 = 1 0 0 1 ,θ s = − 1 + x · ∇ , and we have also used the 2 × 2 matrix notation of [7] (Tr includes both a matrix and functional trace).
As with the work in [1] and [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 of the type of potentials considered. An interesting direction is the relativistic generalization of these ideas. Work on this is currently in progress [8].