Bogoliubov transformation and the thermal operator representation in the real time formalism

Abstract

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.