Operator description for thermal quantum field theories on an arbitrary path in the real time formalism

Abstract

We develop an operator description, much like thermofield dynamics, for quantum field theories on a real time path with an arbitrary parameter $σ\,(0\leqσ\leqβ)$. We point out new features which arise when $σ\neq \fracβ{2}$ in that the Hilbert space develops a natural, modified inner product different from the standard Dirac inner product. We construct the Bogoliubov transformation which connects the doubled vacuum state at zero temperature to the thermal vacuum in this case. We obtain the thermal Green's function (propagator) for the real massive Klein-Gordon theory as an expectation value in this thermal vacuum (with a modified inner product). The factorization of the thermal Green's function follows from this analysis. We also discuss, in the main text as well as in two appendices, various other interesting features which arise in such a description.