  In this paper we investigate the massive Sine-Gordon model in the ultraviolet
finite regime in thermal states over the two-dimensional Minkowski spacetime.
We combine recently developed methods of perturbative algebraic quantum field
theory with techniques developed in the realm of constructive quantum field
theory over Euclidean spacetimes to construct the correlation functions of the
equilibrium state of the Sine-Gordon theory in the adiabatic limit. First of
all, the observables of the Sine-Gordon theory are seen as functionals over the
free configurations and are obtained as a suitable combination of the
S-matrices of the interaction Lagrangian restricted to compact spacetime
regions over the free massive theory. These S-matrices are given as power
series in the coupling constant with values in the algebra of fields over the
free massive theory. 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. The latter
observation implies convergence in the strong operator topology in the GNS
representations of the considered states. In the second part of the paper,
adapting the cluster expansion technique to the Lorentzian case, we prove that
the correlation functions of the interacting equilibrium state at finite
temperature (KMS state) can be constructed also in the adiabatic limit, where
the interaction Lagrangian is supported everywhere in space.
