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arXiv:quant-ph/0102127 · Modern Physics Letters A

A PHYSICAL EXPLANATION FOR THE TILDE SYSTEM IN THERMO FIELD DYNAMICS

Dong Mi, He‐Shan Song, Ying An

Published Mar 28, 2001• 4 citations
arXivHTMLPDF

Abstract

For a two-body quantum system, any pure state can be represented by a biorthogonal expression by means of Schmidt decomposition. Using this in the composite system which includes a thermodynamic system and its surroundings, it is found that the tilde system in thermo field dynamics is just the surroundings of the real system.
DOI
10.1142/s0217732301003607

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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.
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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.
According to the thermo field dynamics proposed by Takahashi–Umezawa, the Hamiltonian h = a†a - ã†ã and h|n, ñ〉 = 0 of the original optical field excitation state |n〉 and its tilde state |ñ〉 in the Hilbert space cannot embody the eigenvalue spectrum of h. To avoid the weakness, new thermo excitation states are successfully constructed by introducing appropriate thermo excitation operators. It is noted that they are entangled states. Applications of the new thermo excitation state in constructing new thermo squeezed state and phase state are also presented.