Construction of KMS states in perturbative QFT and renormalized Hamiltonian dynamics (Q462875)

In statistical quantum field theory, a system in thermal equilibrium can be described by a state satisfying the Kubo-Martin-Schwinger (KMS) condition. Unfortunately, when working with infinite-dimensional Hilbert spaces, we run into a lot of troubles like the occurrence of infrared divergences, spontaneous symmetry breaking, and phase transitions. Surprisingly, in Euclidean field theory at non-zero temperature the divergence problem is absent. However, the Osterwalder-Schrader reconstruction theorem is no longer valid here. Several approaches to solve this problem may be found in the literature. In their present paper, Fredenhagen and Lindner choose a new approach to the problem which is based on perturbative algebraic quantum field theory. It provides a state-independent description of local observables, and so the problem of quantum field theory at some non-zero temperature is reduced to the construction of a KMS state on local observable algebras. This enables them to apply standard methods of quantum statistical mechanics to construct a KMS state of an interacting field theoretical model as a perturbation of some KMS state of the free theory. The notorious infrared divergences are absent due to locality. The new approach may be interpreted as an extension of the Schwinger-Keldysh formalism.