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Chinese Physics Letters

η-ξ vacuum and Thermal Green's functions

Xu Feng, Gui Yuan-Xing

Published Aug 1, 1988• 2 citations
Publisher

Abstract

In the previous paper, a new vacuum state was defined. As claimed in that paper, this vacuum state is a thermal state in the ordinary sense. In this paper, we derive the Green's function of this vacuum and prove that it is exactly the ordinary finite temperature thermal Green function. So we demonstrate that this state is exactly the thermally equilibrium state.

DOI
10.1088/0256-307x/5/8/012

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In the previous paper, a new vacuum state was defined. As claimed in that paper, this vacuum state is a thermal state in the ordinary sense. In this paper, we derive the Green's function of this vacuum and prove that it is exactly the ordinary finite temperature thermal Green function. So we demonstrate that this state is exactly the thermally equilibrium state.
In the quantum field theory with the vacuum being the ground state the Green functions are the vacuum expectation values of the chronological, retarded or advanced products of the field operators (Bjoken & Drell, 1964; Itzykson & Zuber, 1985; Peskin & Schroeder, 1995). They are the generalized functions of the real time variables ti (and also other spatial coordinates). For the application of the Green function technique to the study of the timeindependent phenomena in equilibrium many-body systems at a finite temperature, the Matsubara imaginary time Green functions were introduced and widely used (Abrikosov et al., 1975; Bruuns & Flensberg, 2004; Haken, 1976). They are the mean values over a statistical ensemble at a finite temperature of the chronological products of the imaginary timedependent operators. Both these types of Green functions are inadequate for the application to the study of the time-dependent phenomena in the many-body systems with a finite density and at a finite temperature, in particular the non-equilibrium systems. For the application to the study of the time-dependent dynamical processes in non-equilibrium many-body systems Keldysh (Keldysh, 1965) has introduced a more general class of timedependent Green functions at finite temperature and density. They are the mean values of the time-ordered products of quantum operators in the Heisenberg picture over statistical ensembles of many-body systems with finite densities and at finite temperatures (which may be non-vanishing). The simplest example is the two-point Green function
We study the dynamical equation of the time-ordered Green's function at finite temperature. We show that the time-ordered Green's function obeys a conventional Dyson equation only at equilibrium and in the limit of zero temperature. In all other cases, i.e. finite temperature at equilibrium or non-equilibrium, the time-ordered Green's function obeys instead a modified Dyson equation. The derivation of this result is obtained from the general formalism of the non-equilibrium Green's functions on the Keldysh time-loop contour. At equilibrium, our result is fully consistent with the Matsubara temperature Green's function formalism and also justifies rigorously the correction terms introduced in an ad hoc way with Hedin and Lundqvist. Our results show that one should use the appropriate dynamical equation for the time-ordered Green's function when working beyond the equilibrium zero-temperature limit.
Following the spirit of thermo field dynamics initiated by Takahashi and Umezawa, we employ the technique of integration within an ordered product of operators to derive the thermal vacuum state (TVS) for the Hamiltonian H of the two-coupled-oscillator model. The ensemble averages of the system are derived conveniently by using the TVS. In addition, the entropy for this system is discussed based on the relation between the generalized Hellmann—Feynman theorem and the entroy variation in the context of the TVS.
The Gaussian wavefunctional approach is developed in thermofield dynamics. We construct the thermal vacuum wavefunctional, its creation as well as annihilation operators, and accordingly the thermo-particle excited states. For a ( D + 1)-dimensional scalar field system with an arbitrary potential whose Fourier representation exists in the sense of tempered distributions, we calculate the finite-temperature Gaussian effective potential (FTGEP), one- and two-thermo-particle-state energies. The zero-temperature limit of each of them is just the corresponding result in quantum field theory, and the FTGEP can lead to the same one for some concrete models as calculated by the imaginary-time Green function.