Generalizations of the thermal Bogoliubov transformation

TL;DR

This paper extends the thermal Bogoliubov transformation in thermo field dynamics by generalizing degrees of freedom and expanding the transformation matrix, applicable to both bosons and fermions.

Contribution

It introduces a generalized $ ilde{ ext{non-conserving}}$ representation and a larger $4 imes4$ matrix form, enhancing the theoretical framework of thermal transformations.

Findings

Extended the transformation to include non-conserving degrees of freedom.

Developed a $4\times4$ matrix form for mode mixing.

Applicable to both bosonic and fermionic systems.

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.