Equilibrium States in Thermal Field Theory and in Algebraic Quantum Field Theory

We compare the construction of equilibrium states at finite temperature for self-interacting massive scalar quantum field theories on Minkowski spacetime proposed by Fredenhagen and Lindner (Commun Math Phys 332:895, 2014) with results obtained in ordinary thermal field theory, by means of real-time and Matsubara (or imaginary time) formalisms. In the construction of this state, even if the adiabatic limit is considered, the interaction Lagrangian is multiplied by a smooth time cut-off. In this way the interaction starts adiabatically and the correlation functions are free from divergences. The corresponding interaction Hamiltonian is a local interacting field smeared over the interval of time where the chosen cut-off is not constant. We observe that, in order to cope with this smearing, the Matsubara propagator, which is used to expand the relative partition function between the free and interacting equilibrium states, needs to be modified. We thus obtain an expansion of the correlation functions of the equilibrium state for the interacting field as a sum over certain type of graphs with mixed edges, some of them correspond to modified Matsubara propagators and others to propagators of the real-time formalism. An integration over the adiabatic time cut-off is present in every vertex. However, at every order in perturbation theory, the final result does not depend on the particular form of the cut-off function. The obtained graphical expansion contains in it both the real-time formalism and the Matsubara formalism as particular cases. For special interaction Lagrangians, the real-time formalism is recovered in the limit where the adiabatic start of the interaction occurs at past infinity. At least formally, the combinatorics of the Matsubara formalism is obtained in the limit where the switch on is realised with an Heaviside step function and the field observables have no time dependence. Finally, we show that a particular factorisation which is used to derive the ordinary real-time formalism holds only in special cases and we present a counterexample. We conclude with the analysis of certain correlation functions, and we notice that corrections to the self-energy in a λϕ4 at finite temperature theory are expected.