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

Abstract

It has been shown earlier [Phys. Rev. D 72, 085006 (2005)PRVDAQ1550-799810.1103/PhysRevD.72.085006, Phys. Rev. D 73, 065010 (2006)PRVDAQ1550-799810.1103/PhysRevD.73.065010] 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 (σ=0) of the real time formalism (as well as for its conjugate σ=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)CLPNAB0366-6824, Int. J. Mod. Phys. B 10, 1755 (1996)IJPBEV0217-979210.1142/S0217979296000817, Thermo Field Dynamics and Condensed States (North-Holland, Amsterdam, 1982), Phys. Rev. D 93, 125028 (2016)PRVDAQ2470-001010.1103/PhysRevD.93.125028]. We show how the (2×2) Bogoliubov transformation matrix naturally leads to the scalar thermal operator for σ=0, 1 while it fails for any other value 0