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Fortschritte der Physik/Progress of Physics

A Time-Dependent Nonequilibrium Calculational Scheme towards the Study of Temperature Fluctuations

Tim Evans, I. Hardman, Hiroomi Umezawa, Y. Yamanaka

Published Jan 1, 1993• 13 citations

Abstract

A new perturbative scheme for interacting nonequilibrium thermal quantum fields using thermo field dynamics is outlined by explicitly considering the temporal change of the thermal vacuum as it moves through many inequivalent state vector spaces. One is then naturally led to two sources of time dependence, one from the dynamics and the other from the change of thermal vacuum, which are taken care of by the Hamiltonian and the thermal generator, respectively. To obtain a practical scheme we restrict ourselves by the demand that a spectral representation for the full propagator exists. This leads to a time dependent temperature. The addition of a diagonalization condition for the quasi-particle Hamiltonian provides the master equation for the number density. We show that our formalism is equivalent to an extended form of the path-ordering method. This formalism is a first step towards the study of the origin of heat and temperature in high-energy heavy ion collisions.

DOI: 10.1002/prop.19930410203

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A new perturbative scheme for interacting nonequilibrium thermal quantum fields using thermo field dynamics is outlined by explicitly considering the temporal change of the thermal vacuum as it moves through many inequivalent state vector spaces. One is then naturally led to two sources of time dependence, one from the dynamics and the other from the change of thermal vacuum, which are taken care of by the Hamiltonian and the thermal generator, respectively. To obtain a practical scheme we restrict ourselves by the demand that a spectral representation for the full propagator exists. This leads to a time dependent temperature. The addition of a diagonalization condition for the quasi-particle Hamiltonian provides the master equation for the number density. We show that our formalism is equivalent to an extended form of the path-ordering method. This formalism is a first step towards the study of the origin of heat and temperature in high-energy heavy ion collisions.

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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.

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.

Perturbation theory in thermofield dynamics (a real time finite temperature quantum field theory) is formulated in the interaction representation and a diagrammatic expansion for the expectation value of time ordered products is derived solely on the basis of the thermal state condition and the algebraic properties of the operators. This new formulation has a much wider domain of applicability than those presented previously. The method is applicable even in the case of purely quantum mechanical systems. When certain bosonic operators, which commute with the unperturbed Hamiltonian, contribute to the renormalization of the thermal state vacuum the vacuum diagrams do not cancel in the usual way. It will be shown how these diagrams arise and how they may be computed.