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Communications in Theoretical Physics

Bogoliubov Matrices in Thermal Field Theory

Hong-Hua Xu

Published Oct 30, 1996• 1 citation
Publisher

Abstract

It is found that Bogoliubov matrices introduced in thermo field dynamics can be generally derived for an arbitrary value of the time-path parameter , by the closed-time path approach in a natural and simple way. Based on the generalized Bogoliubov matrices the meaning of widely believed -independence of thermal field theory is discussed.
DOI
10.1088/0253-6102/26/3/289

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We present a comprehensive review of the most fundamental and practical aspects of thermo-field dynamics (TFD), including some of the most recent developments in the field. To make TFD fully consistent, some suitable changes in the structure of the thermal doublets and the Bogoliubov transformation matrices have been made. A close comparison between TFD and the Schwinger-Keldysh closed time path formalism (SKF) is presented. We find that TFD and SKF are in many ways the same in form; in particular, the two approaches are identical in stationary situations. However, TFD and SKF are quite different in time-dependent nonequilibrium situations. The main source of this difference is that the time evolution of the density matrix itself is ignored in SKF while in TFD it is replaced by a time-dependent Bogoliubov transformation. In this sense TFD is a better candidate for time-dependent quantum field theory. Even in equilibrium situations, TFD has some remarkable advantages over the Matsubara approach and SKF, the most notable being the Feynman diagram recipes, which we will present. We will show that the calculations of two-point functions are simplified, instead of being complicated, by the matrix nature of the formalism. We will present some explicit calculations using TFD, including space-time inhomogeneous situations and the vacuum polarization in equilibrium relativistic QED.
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The thermal Bogoliubov transformation in thermo field dynamics is generalized in two respects. First, a generalization of the alpha\\alphaalpha--degree of freedom to tilde non--conserving representations is considered. Secondly, the usual 2times22\\times22times2 Bogoliubov matrix is extended to a 4times44\\times44times4 matrix including mixing of modes with non--trivial multiparticle correlations. The analysis is carried out for both bosons and fermions.
Density matrices are powerful mathematical tools for the description of closed and open quantum systems. Recently, methods for the direct computation of density matrix elements in scalar quantum field theory were developed based on thermo field dynamics (TFD) and the Schwinger-Keldysh formalism. In this article, we provide a more detailed discussion of these methods and derive expressions for density matrix elements of closed and open systems. At first, we look at closed systems by discussing general solutions to the Schrödinger-like form of the quantum Liouville equations in TFD, showing that the dynamical map is indeed divisible, deriving a path integral-based expression for the density matrix elements in Fock space, and explaining why perturbation theory enables us to use the last even in situations where all initial states in Fock space are occupied. Subsequently, we discuss open systems in the same manner after tracing out environmental degrees of freedom from the solutions for closed systems. We find that, even in a general basis, the dynamical map is not divisible, which renders the dynamics of open systems non-Markovian. Finally, we show how the resulting expressions for open systems can be used to obtain quantum master equations, and comment on the artificiality of time integrals over density matrices that usually appear in many other master equations in the literature but are absent in ours.