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Physical review. B, Condensed matter

Finite-temperature real-time field theories for spin 1/2

Vitor Rocha Vieira

Published Apr 1, 1989• 6 citations
Publisher

Abstract

The finite-temperature real-time field theories for spin (1/2 are derived with use of a functional-integral approach, and their application to problems of equilibrium and nonequilibrium statistical mechanics discussed. These formalisms have a standard Wick's theorem and linked-graph theorem, leading to the use of Feynman diagrams. Similar to the problems with bosons or fermions, the calculation of equilibrium quantities requires the consideration of the imaginary-time branch, leading to a three-component formalism, instead of the usual two-component formalism, typical of real-time field theories.

DOI
10.1103/physrevb.39.7174

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The finite-temperature real-time field theories for spin (1/2 are derived with use of a functional-integral approach, and their application to problems of equilibrium and nonequilibrium statistical mechanics discussed. These formalisms have a standard Wick's theorem and linked-graph theorem, leading to the use of Feynman diagrams. Similar to the problems with bosons or fermions, the calculation of equilibrium quantities requires the consideration of the imaginary-time branch, leading to a three-component formalism, instead of the usual two-component formalism, typical of real-time field theories.
The main objective of this paper is to obtain an operator realization for the bosonization of fermions in 1 + 1 dimensions, at finite, non-zero temperature T. This is achieved in the framework of the real time formalism of Thermofield Dynamics. Formally the results parallel those of the T = 0 case. The well known two-dimensional Fermion-Boson correspondences at zero temperature are shown to hold also at finite temperature. In order to emphasize the usefulness of the operator realization for handling a large class of two-dimensional quantum field-theoretic problems, we contrast this global approach with the cumbersome calculation of the fermion-current two-point function in the imaginary-time formalism and real time formalisms. The calculations also illustrate the very different ways in which the transmutation from Fermi-Dirac to Bose-Einstein statistics is realized.
In this article, we present a concise and self-contained introduction to nonequilibrium statistical mechanics with quantum field theory by considering an ensemble of interacting identical bosons or fermions as an example. Readers are assumed to be familiar with the Matsubara formalism of equilibrium statistical mechanics such as Feynman diagrams, the proper self-energy, and Dyson's equation. The aims are threefold: (i) to explain the fundamentals of nonequilibrium quantum field theory as simple as possible on the basis of the knowledge of the equilibrium counterpart; (ii) to elucidate the hierarchy in describing nonequilibrium systems from Dyson's equation on the Keldysh contour to the Navier-Stokes equation in fluid mechanics via quantum transport equations and the Boltzmann equation; (iii) to derive an expression of nonequilibrium entropy that evolves with time. In stage (i), we introduce nonequilibrium Green's function and the self-energy uniquely on the round-trip Keldysh contour, thereby avoiding possible confusions that may arise from defining multiple Green's functions at the very beginning. We try to present the Feynman rules for the perturbation expansion as simple as possible. In particular, we focus on the self-consistent perturbation expansion with the Luttinger-Ward thermodynamic functional, i.e., Baym's Phi-derivable approximation, which has a crucial property for nonequilibrium systems of obeying various conservation laws automatically. We also show how the two-particle correlations can be calculated within the Phi-derivable approximation, i.e., an issue of how to handle the "Bogoliubov-Born-Green-Kirkwood-Yvons (BBGKY) hierarchy".
We present an operator realization for the bosonization of fermions in 1 + 1 dimensions, at finite, non-zero temperature T .This is achieved in the framework of the real time formalism of Thermofield Dynamics.The well known two-dimensional Fermion-Boson correspondences at zero temperature are shown to hold also at finite temperature.In order to emphasize the usefulness of the operator realization for handling a large class of two-dimensional quantum field-theoretic problems, we contrast this global approach with the cumbersome calculation of the fermioncurrent two-point function in the imaginary-time formalism and real time formalisms.
This book provides a comprehensive and pedagogical account of the various methods used in the quantum theory of finite systems, including molecular, atomic, nuclear, and particle phenomena. Covering both background material and advanced topics and including nearly 200 problems, Quantum Theory of Finite Systems has been designed to serve primarily as a text and will also prove useful as a reference in research.The first of the book's four parts introduces the basic mathematical apparatus: second quantization, canonical transformations, Wick theorems and the resulting diagram expansions, and oscillator models. The second part presents mean field approximations and the recently developed path integral methods for the quantization of collective modes. Part three develops perturbation theory in terms of both time-dependent Feynman diagrams and time-independent Goldstone diagrams. A fourth part discusses variational methods based on correlated wavefunctions, including spin correlations. The approximation schemes are formulated for fermions and bosons at eigher zero or non-zero temperature.Although the formalism developed applies to both finite and infinite systems, the book stresses those aspects of the theory that are specific to the description of finite systems. Thus special attention is given to mean field approximations, the ensuing broken symmetries, and the associated collective motions such as rotations. Conversely, some specific features of systems with infinite numbers of degrees of freedom (such as the thermodynamic limit, critical phenomena, and the elimination of ultraviolet divergencies) are deliberately omitted.Jean-Paul Blaizot and Georges Ripka are associated with the Centre d'Etudes Nucleaires de Saclay.
Wick's theorem is combined with a cluster composition to derive a generalized statistical Wick's theorem for a non-equilibrium system with initial correlations. This enables a diagrammatic time-dependent perturbation theory which is a simple extension of the usual one to be developed for time-path causal statistical Green functions and their matrix equivalents. This provides a complete non-equilibrium theory. Reduction of diagrams and equations of motion for the Green functions are also discussed.