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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Tilde substitution law in thermo field dynamics: Thermal state conditions

H. Matsumoto, Yoshimasa Nakano, Hiroomi Umezawa

Published Jan 15, 1985• 23 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.31.429

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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.
Thermofield Dynamics (TFD), a real time formalism for thermal quantum field theory, is formulated in terms of a path-integral approach following the Weinberg procedure.In order to assure precise rules for the development, we use a representation of Lie-algebras such that the TFD algebraic ingredients are derived, including the tilde conjugation rules.The association with the canonical formalism is obtained, and for the case of bosons, we introduce the Feynman diagrams and derive the n-point functions.
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.
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.