Thermal equilibrium states in perturbative Algebraic Quantum Field Theory in relation to Thermal Field Theory

In the first part, we analyse the properties of an interacting, massive scalar field in an equilibrium state over Minkowski spacetime. We compare the known real- and imaginary-time formalisms of Thermal Field Theory with the recent construction by Fredenhagen and Lindner of a KMS state for perturbative interacting theories in the context of perturbative Algebraic Quantum Field Theory, in the adiabatic limit. In particular, we show that the construction of Fredenhagen and Lindner reduces to the real-time formalism only if the cocycle which intertwines between the free and interacting dynamics can be neglected. Furthermore, the Fredenhagen and Lindner construction reduces to the ordinary imaginary-time formalism if one considers the expectation value of translation invariant observables. We thus conclude that a complete description of thermal equilibrium for interacting scalar fields is generally obtained only by means of the state constructed by Fredenhagen and Lindner, which combines both formalisms of Thermal Field Theory. We also discuss the properties of the expansion of the Fredenhagen and Lindner construction in terms of Feynman diagrams in the adiabatic limit. We finally provide examples showing that the real- and the imaginary-time formalisms fail to describe thermal equilibrium already at first or second order in perturbation theory. The results presented in this part are summarized in (BDP19). In the second part, we discuss the so-called secular effects, characterized by the appearance of polynomial divergences in the large time limit of truncated perturbative expansions of expectation values in Quantum Field Theory. We show that, although such effect is an artifact of perturbation theory, and thus may not be obtained via exactly solving the dynamical equation if possible, they do not represent the breakdown of perturbation theory itself. Instead, we show that the polynomial divergences follow from a bad choice of state, and we present examples of states which produce expectation values whose perturbative expansion does not present secular effects. In particular, we point that it is possible to obtain non time-divergent perturbative expressions from thermal equilibrium states for the interacting theory. This last part is based on a research project which, by the time this thesis was written, had not been concluded yet.