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

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.

arXiv comment9 pages, version to be published in Physical Review D

References in corpus (11)

• An alternative construction of the positive inner product in non-Hermitian quantum mechanics
• An alternative construction of the positive inner product for pseudo-Hermitian Hamiltonians: Examples
• Thermal Operator Representation of Finite Temperature Graphs
• Thermal Operator Representation of Finite Temperature Graphs II
• Factorization of finite temperature graphs in thermal QED
• Operator description for thermal quantum field theories on an arbitrary path in the real time formalism
• Forward scattering amplitudes and the thermal operator representation
• Thermal Operator Representation of Finite-Temperature Amplitudes in the Presence of Chemical Potential
• Thermal Operator and Cutting Rules at Finite Temperature and Chemical Potential
• Hard thermal effective action in QCD through the thermal operator
• Thermal Operator and Dispersion Relation in QED at Finite Temperature and Chemical Potential

Cited by in corpus (1)

• Real-time thermal photon-photon interactions in the mixed space