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Journal of Physics A Mathematical and General

eta - xi spacetime and the tilde freedom in thermo field dynamics

Zuo Jian, Gui Yuan-Xing

Published Sep 7, 1995• 3 citations
Publisher

Abstract

The CM operations are introduced in the theory of eta - xi spacetime. They have the same properties as the tilde rules in thermo field dynamics (TFD); we can, therefore, identify them with the tilde operation. This leads to the conclusion that the tilde system in TFD can be regarded as the system in region II (mirror universe) of eta - xi spacetime.

DOI
10.1088/0305-4470/28/17/021

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The CM operations are introduced in the theory of eta - xi spacetime. They have the same properties as the tilde rules in thermo field dynamics (TFD); we can, therefore, identify them with the tilde operation. This leads to the conclusion that the tilde system in TFD can be regarded as the system in region II (mirror universe) of eta - xi spacetime.
The algebraic structure of Thermo Field Dynamics lies in the qqq-deformation of the algebra of creation and annihilation operators. Doubling of the degrees of freedom, tilde-conjugation rules, and Bogoliubov transformation for bosons and fermions are recognized as algebraic properties of hq(1)h_{q}(1)hq​(1) and of hq(1∣1)h_{q}(1|1)hq​(1∣1), respectively.
The tilde substitution law (which will be called in this paper the thermal state condition) plays an important role in determining the temperature-dependent ground state in thermo field dynamics (a real-time operator formalism of quantum field theory at finite temperature). Rules which define the phase conventions for the tilde substitution for any arbitrary Heisenberg operator are derived. These rules also lead to a clear correspondence between the perturbation series obtained in thermo field dynamics and the path-ordered formalism of the statistical average.
The algebraic structure of Thermo Field Dynamics for bosons can be fully incorporated in the q-deformation of the Weyl-Heisenberg algebra hq (1) . The doubling of the degrees of freedom, the set of the tilde-conjugation rules, the Bogoliubov transformation and its generator have a direct and simple interpretation in terms of operators and of properties of hq (1) . The notion of “thermal degree of freedom” introduced by Umezawa also finds a more specific formalization since the corresponding “thermal conjugate momentum” can be formally introduced, thus providing us with a set of canonical “thermal” variables.
A fundamental framework of thermo field dynamics (TFD), including non-equilibrium situations as well as equilibrium stuations, is formulated through a compact axiomatic form, in which no reference to reservoirs is needed, by considering quantum fields. The semi-free quantum fields in TFD are constructed in a way parallel to that of the usual quantum field theory without thermal degrees of freedom, i.e. in terms of orthonormalization relations among wave functions, the sum rule for wave functions, the divisor operator and the particle-antiparticle conjugation (C-conjugation).