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

TL;DR

This paper investigates the origin of the thermal operator in real time formalism, linking it to Bogoliubov transformations, and explores its applicability across different formalisms and potential generalizations.

Contribution

It demonstrates how the Bogoliubov transformation matrix explains the thermal operator in real time formalism and proposes a generalized scalar thermal operator for all $\sigma$ values.

Findings

The scalar thermal operator arises naturally from the Bogoliubov transformation matrix.

The thermal operator relation holds for $\sigma=0,1$ but not for intermediate values.

A potential generalized thermal operator exists for all $\sigma$ in mixed space.

Abstract

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.