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arXiv:hep-th/0203222 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

SU(1,1) thermal group of bosonic strings andD-branes

M. C. B. Abdalla, A. L. Gadelha, Ion V. Vancea

Published Sep 19, 2002• 33 citations
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Abstract

All possible Bogoliubov operators that generate the thermal transformations in thermo field dynamics form an $\mathrm{SU}(1,1)$ group. We discuss this construction in the bosonic string theory. In particular, the transformation of the Fock space and string operators generated by the most general $\mathrm{SU}(1,1)$ unitary Bogoliubov transformation and the entropy of the corresponding thermal string are computed. Also, we construct the thermal D-brane generated by the $\mathrm{SU}(1,1)$ transformation in a constant Kalb-Ramond field and compute its entropy.

DOI
10.1103/physrevd.66.065005

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All possible Bogoliubov operators that generate the thermal transformations in thermo field dynamics form an SU(1,1)\mathrm{SU}(1,1)SU(1,1) group. We discuss this construction in the bosonic string theory. In particular, the transformation of the Fock space and string operators generated by the most general SU(1,1)\mathrm{SU}(1,1)SU(1,1) unitary Bogoliubov transformation and the entropy of the corresponding thermal string are computed. Also, we construct the thermal D-brane generated by the SU(1,1)\mathrm{SU}(1,1)SU(1,1) transformation in a constant Kalb-Ramond field and compute its entropy.
In this work we explain the construction of the thermal vacuum for the bosonic string, as well that of the thermal boundary state interpreted as a Dp-brane at finite temperature. In both case we calculate the respective entropy using the entropy operator of the Thermo Field Dynamics theory. We show that the contribution of the thermal string entropy is explicitly present in the Dp-brane entropy. Furthermore, we show that the Thermo Field approach is suitable to introduce temperature in boundary states.
In this paper we review the construction of the thermal bosonic string and DDD-brane in the framework of the Thermo Field Dynamics (TFD). We briefly recall the well-known light-cone quantization of the bosonic string in the conformal gauge in flat space-time. Then we give a bird's eye view of the fundamental concepts of the TFD. Also, we present the thermalization of the bosonic string and the construction of the thermal D-brane boundary state. Finally, we show the calculation of the entropy of the thermal open string states with all boundary conditions and the entropy of the thermal D-brane state.
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.