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Journal of Mathematical Physics

A derivation of the virial expansion with application to Euclidean quantum field theory

Ralph Menikoff, David H. Sharp

Published Jan 1, 1978• 16 citations
Publisher

Abstract

In this paper we give a derivation of the virial expansion and some of its generalizations. Our derivation is based on the generating functional which defines a representation of the density operator ρ (x) in a nonrelativistic local current algebra. The virial expansion results from solving a functional differential equation for this quantity. We exploit the well-known analogy between quantum field theory and classical statistical mechanics to explore the use of the virial expansion in Euclidean quantum field theory. Specifically, we show that the virial expansion can be used to derive Feynman’s rules and to provide a perturbation expansion about a static ultralocal model. The latter is worked out in detail in the case of a free neutral scalar model, and outlined in the case of a λφ4 model.

DOI
10.1063/1.523531

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In this paper we give a derivation of the virial expansion and some of its generalizations. Our derivation is based on the generating functional which defines a representation of the density operator ρ (x) in a nonrelativistic local current algebra. The virial expansion results from solving a functional differential equation for this quantity. We exploit the well-known analogy between quantum field theory and classical statistical mechanics to explore the use of the virial expansion in Euclidean quantum field theory. Specifically, we show that the virial expansion can be used to derive Feynman’s rules and to provide a perturbation expansion about a static ultralocal model. The latter is worked out in detail in the case of a free neutral scalar model, and outlined in the case of a λφ4 model.
Representations will be given for the nonrelativistic local current algebra consisting of ρ(x), the particle number density, and J(x), the flux density of particles. These representations correspond to the N/V limit of (i) a free Bose gas, (ii) Bose gas in an external potential, (iii) free Fermi gas, and (iv) Bose gas (in one dimension) with a two-body interaction potential V(x)=2/x2. In each case the generating functional L(f), the ground state expectation value of exp[iρ(f)], determining the representation will be given. It will also be shown the generating functional satisfies a functional equation of the form [∇ − i∇f(x)](1/i) [δ/δf(x)]L (f) = A(x, (1/i)(δ/δf))L(f) and that the Hamiltonian written in terms of ρ and J has the form H = (1/8)∫dxK̃(x)† [1/ρ(x)] K̃(x) with K̃(x)=∇ρ(x) +2iJ(x)−A(x,ρ).
Abstract Functional integrals are basic tools to study first quantum mechanics (QM), and quantum field theory (QFT). The path integral formulation of QM is well suited to the study of systems with an arbitrary number of degrees of freedom. It makes a smooth transition between nonrelativistic QM and QFT possible. The Euclidean functional integral also emphasizes the deep connection between QFT and the statistical physics of systems with short-range interactions near a continuous phase transition. The path integral representation of the matrix elements of the quantum statistical operator e-β H for Hamiltonians of the simple separable form p2/2m +V(q) is derived. To the path integral corresponds a functional measure and expectation values called correlation functions, which are generalized moments, and related to quantum observables, after an analytic continuation in time. The path integral corresponding to the Euclidean action of a harmonic oscillator, to which is added a time-dependent external force, is calculated explicitly. The result is used to generate Gaussian correlation functions and also to reduce the evaluation of path integrals to perturbation theory. The path integral also provides a convenient tool to derive semi-classical approximations.
Local currents are used to describe nonrelativistic many-body quantum mechanics in the thermodynamic limit. The problem of determining a representation of the local currents corresponding to a given Hamiltonian is studied. We formulate the dynamics in such a way that one solves simultaneously for the ground state and the representation of the local currents. This leads to two coupled functional equations relating the generating functional to a functional which describes the ground state. Together these functionals determine a representation of the local currents in which the Hamiltonian is a well-defined operator. The functional equations are equivalent to a set of integro–differential equations for expansion coefficients of the two functionals.
An expansion for quantum statistical mechanics is derived that gives classical statistical mechanics as the leading term. Each quantum correction comes from successively larger permutation loops, which arise from the factorization of the symmetrization of the wave function with respect to localized particle interchange. Explicit application of the theory yields the full fugacity expansion for the quantum ideal gas, and the second fugacity coefficient for interacting quantum particles, which agree with known results. Compared to the Lee-Yang virial cluster expansion, the present expansion is expected to be more rapidly converging and the individual terms appear to be simpler to evaluate. The results obtained in this paper are intended for practical computer simulation algorithms for terrestrial condensed matter quantum systems.
The virial expansion provides a non-perturbative view into the thermodynamics of quantum many-body systems in dilute regimes. While powerful, the expansion is challenging as calculating its coefficients at each order n requires analyzing (if not solving) the quantum n-body problem. In this work, we present a comprehensive review of automated algebra methods, which we developed to calculate high-order virial coefficients. The methods are computational but non-stochastic, thus avoiding statistical effects; they are also for the most part analytic, not numerical, and amenable to massively parallel computer architectures. We show formalism and results for coefficients characterizing the thermodynamics (pressure, density, energy, static susceptibilities) of homogeneous and harmonically trapped systems and explain how to generalize them to other observables such as the momentum distribution, Tan contact, and the structure factor.