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Physical review. B, Condensed matter

Density matrix of interacting boson-fermion systems as functional integrals in the holomorphic representation

J. Florencio, Bernard Albert Goodman

Published Sep 15, 1986• 18 citations
Publisher

Abstract

We formulate the density matrix of many-body systems as functional integrals in the holomorphic representation, similar to the S-matrix formulation in field theory. Symmetry and antisymmetry requirements are built into the integration variables: c-numbers for bosons and Grassmann numbers for fermions. The integrals have well-defined normalization. They yield first-order differential equations for semiclassical extremal configurations and corrections to it, and do not require periodicity conditions on the individual integration paths, unlike the formulations found in the literature. We present some applications to simple boson and fermion systems.

DOI
10.1103/physrevb.34.3634

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We present a functional-integral method that is suitable for obtaining nonperturbative solutions to the partition function of fermion systems. The density matrix is cast as an expansion about the classical path contributions, and the quantum corrections can be calculated systematically by using retarded propagators in the space holomorphic functions. It is shown that the semiclassical expansion amounts to a high-temperature expansion, where the first nonvanishing quantum correction is of O(${\mathrm{\ensuremath{\beta}}}^{2}$). Each term of the semiclassical expansion may contain the coupling parameter in all orders of perturbation theory. The method can also be extended to boson systems.
Abstract Chapter 2 is rather descriptive and introduces the notion of functional (path and field) integrals, for boson (the holomorphic formalism) as well as fermion systems (this necessitates the introduction of Grassmann integration) as they are used in physics. Prior to the second half of the twentieth century, the technical tools of theoretical physics were mainly differential or partial differential equations. However, when systematic investigations of large scale systems with quantum of statistical fluctuations began, new tools were required. This led to the development of functional integration. In this chapter, the role of Gaussian measures and Gaussian expectation values is emphasized, leading to Wick’s theorem, a tool for perturbative calculations. Functional integrals provide remarkable tools to study the classical limit and barrier penetration. They provide a simple bridge between non–relativistic quantum mechanics and quantum field theory.
This chapter introduces the holomorphic representation of quantum mechanics, because it allows a study of the properties of boson systems both from the point of view of evolution and of quantum statistical physics (in the so-called second quantization formalism). Path integrals then take the form of integrals over trajectories in phase space in a complex parametrization. A parallel formalism, based on integration over anti-commuting type or Grassmann variables, makes it possible to handle fermions in a way quite analogous to bosons. The corresponding path integral representation of the statistical operator is then derived. The holomorphic formalism is specially well adapted to a study of the harmonic oscillator and, more generally, of perturbed harmonic oscillators. As an illustration, the formalism is applied to the Bose-Einstein condensation. This chapter also considers complex integrals and Wick's theorem, kernel of operators, general Hamiltonians, partition function, and the quantum Bose gas.
We present a unified formulation for quantum statistical physics based on the representation of the density matrix as a functional integral. For quantum statistical (thermal) field theory, the stochastic variable of the statistical theory is a boundary field configuration. We explore the properties of an effective theory for such boundary configurations and apply it to the computation of the partition function of an interacting one-dimensional quantum-mechanical system at finite temperature. Plots of free energy and specific heat show excellent agreement with more involved semiclassical results. The method of calculation provides an alternative to the usual sum over periodic trajectories: it sums over paths with coincident end points and includes nonvanishing boundary terms. An appropriately modified expansion into modified Matsubara modes is presented.