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

It has been shown earlier \cite{brandt,brandt1} 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 was shown to hold only in the imaginary time formalism and the closed time path ($\sigma=0$) of the real time formalism (as well as for its conjugate $\sigma=1$). 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 \cite{takahashi,umezawa,pushpa}. We show how the ($2\times 2$) Bogoliubov transformation matrix naturally leads to the scalar thermal operator for $\sigma=0,1$ while it fails for any other value $0<\sigma<1$. This analysis also suggests that a generalized scalar thermal operator description, in the mixed space, is possible even for $0<\sigma<1$. We also show the existence of a scalar thermal operator relation in the momentum space.