  In this paper 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 with results obtained
in ordinary thermal field theory, by means of real time and Matsubara
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. In order to cope with this smearing, the Matsubara propagator
needs to be modified. We obtain an expansion of the correlation functions of
the interacting equilibrium state 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. 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 $\lambda\phi^4$ at
finite temperature theory are expected.
