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Generalizations of the thermal Bogoliubov transformation

Elmfors, P., Hiroomi Umezawa

Published Jan 1, 1993• 11 citations
OpenAlex

Abstract

The thermal Bogoliubov transformation in thermo field dynamics is generalized in two respects. First, a generalization of the $\\alpha$--degree of freedom to tilde non--conserving representations is considered. Secondly, the usual $2\\times2$ Bogoliubov matrix is extended to a $4\\times4$ matrix including mixing of modes with non--trivial multiparticle correlations. The analysis is carried out for both bosons and fermions.

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The thermal Bogoliubov transformation in thermo field dynamics is generalized in two respects. First, a generalization of the alpha\\alphaalpha--degree of freedom to tilde non--conserving representations is considered. Secondly, the usual 2times22\\times22times2 Bogoliubov matrix is extended to a 4times44\\times44times4 matrix including mixing of modes with non--trivial multiparticle correlations. The analysis is carried out for both bosons and fermions.
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.
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.
It has been shown earlier [Phys. Rev. D 72, 085006 (2005), Phys. Rev. D 73, 065010 (2006)] that, in the mixed space, there is an unexpected simple relation between any finite temperature graph and its zero temperature counterpart through a multiplicative scalar operator (termed thermal operator) which carries the entire temperature dependence. This holds only in the imaginary time formalism and the closed time path ($\ensuremath{\sigma}=0$) of the real time formalism (as well as for its conjugate $\ensuremath{\sigma}=1$). We study the origin of this operator from the more fundamental Bogoliubov transformation which acts, in the momentum space, on the doubled space of fields in the real time formalisms [Collective Phenomena 2, 55 (1975), Int. J. Mod. Phys. B 10, 1755 (1996), Thermo Field Dynamics and Condensed States (North-Holland, Amsterdam, 1982), Phys. Rev. D 93, 125028 (2016)]. We show how the ($2\ifmmode\times\else\texttimes\fi{}2$) Bogoliubov transformation matrix naturally leads to the scalar thermal operator for $\ensuremath{\sigma}=0$, 1 while it fails for any other value $0<\ensuremath{\sigma}<1$. This analysis also suggests that a generalized scalar thermal operator description, in the mixed space, is possible even for $0<\ensuremath{\sigma}<1$. We also show the existence of a scalar thermal operator relation in the momentum space.
Based on recent progress in the application of thermo field dynamics (TFD) [1] to thermal many-body systems, several authors [2, 3, 4] have considered a consistent thermal boson expansion (TBE). In this regard two points of view have been taken [2]. The first (path I) consists of a bosonization of the original degrees of freedom of the system, substituting for these ideal boson images. The thermalization is then achieved by doubling those newly introduced bosons as according to TDF prescription. The second possibility (path II) proceeds on the other hand via a thermalization of the system by doubling the original degrees of freedom and a subsequent bosonization of the entire new system. The two paths do not lead to the same results. Moreover, when applied to the Lipkin model, a closer look shows that the choice of path I implies that