  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.
