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Modern Physics Letters A

TIME-SPACE DEPENDENT FORMULATION IN THERMOFIELD DYNAMICS

Kouji Nakamura, Hiroomi Umezawa, Y. Yamanaka

Published Dec 14, 1992• 12 citations
Publisher

Abstract

The recently developed formulation of time-dependent non-equilibrium thermofield dynamics (TFD) is extended to spatially inhomogeneous field systems in this paper. This inhomogeneous formalism is constructed from introducing momentum mixing thermal Bogoliubov transformation. The momentum mixing number density in the Bogoliubov matrix is related to the number density distribution defined at each space-time point which is to be observed.

DOI
10.1142/s0217732392003001

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Modern Physics Letters A•Dec 7, 1992•16 citations
The recently developed formulation of time-dependent non-equilibrium thermofield dynamics (TFD) is extended to spatially inhomogeneous field systems in this paper. This inhomogeneous formalism is constructed from introducing momentum mixing thermal Bogoliubov transformation. The momentum mixing number density in the Bogoliubov matrix is related to the number density distribution defined at each space-time point which is to be observed.
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.
Making use of the thermo field dynamics (TFD) we formulate a calculable method for time-dependent nonequilibrium systems in a time representation (t-representation) rather than in the k0-Fourier representation. The corrected one-body propagator in the t-representation has the form of B−1 (diagonal matrix) B (B being a thermal Bogoliubov matrix). The number parameter in B here is the observed number (the Heisenberg number) with a fluctuation. With the usual definition of the on-shell self-energy a self-consistent renormalization condition leads to a kinetic equation for the number parameter. This equation turns out to be the Boltzmann equation, from which the entropy law follows.
Thermofield dynamics (TFD) is a powerful framework for accounting for thermal effects in a wave function setting and has been extensively used in physics and quantum optics. TFD relies on a duplicated state space and creates a correlated two-mode thermal state via a Bogoliubov transformation acting on the vacuum state. However, a very useful variant of TFD uses the vacuum state as the initial condition and transfers the Bogoliubov transformation into the propagator. This variant, referred to here as the inverse Bogoliubov transformation (iBT) variant, has recently been applied to vibronic coupling problems and coupled-oscillator Hamiltonians in a chemistry context, where the method is combined with efficient tensor network methods for high-dimensional quantum propagation. In the iBT-TFD representation, the mode expectation values are clearly defined and easy to calculate, but the thermalized reduced particle distributions, such as the reduced 1-particle densities or Wigner distributions, are highly non-trivial due to the Bogoliubov back-transformation of the original thermal TFD wave function. Here, we derive formal expressions for the reduced 1-particle density matrix (1-RDM) that use the correlations between the real and tilde modes encoded in the associated reduced 2-particle density matrix. We apply this formalism to define the 1-RDM and the Wigner distributions in the special case of a thermal harmonic oscillator. Moreover, we discuss several approximate schemes that can be extended to higher-dimensional distributions. These methods are demonstrated for the thermal reduced 1-particle density of an anharmonic oscillator.
By making use of time-dependent Bogoliubov transformations, we develop a calculation technique for time-dependent non-equilibrium systems of quantum fields in a time-representation (t-representation). The corrected one-body propagator in the t-representation turns out to have the form B−1 (diagonal matrix) B (B being a thermal Bogoliubov matrix). Applying the usual on-shell concept to the diagonal matrix part of the self-energy, we formulate a self-consistent renormalization scheme. This renormalization determines the vacuum and leads to a kinetic equation for the number density parameter, which reduces to the Boltzmann equation in the lowest approximation. This gives us the increasing entropy in time (the second law of thermodynamics).