In my talk I will report on recent joint work with Kasia Rejzner and Nicola Pinamonti on the massive Sine-Gordon model in the ultraviolet finite regime.

Within the realm of perturbative Algebraic QFT, the observables of theory that are obtained as a suitable combination of the S−matrices of the interaction Lagrangian, whose support is restricted to compact spacetime regions. These S−matrices are given as power series in the coupling constant. Adapting techniques like conditioning and inverse conditioning to spacetimes with Lorentzian signature, we prove that these power series’ converge when evaluated on a generic field configuration.
Adapting the cluster expansion technique known from constructive Quantum field theory to the Lorentzian case, we moreover prove that the correlation functions calculated in the interacting equilibrium state at finite temperature (KMS state) can be constructed in the adiabatic limit, where the interaction Lagrangian is supported everywhere in space.