  Analytic continuation of quantum statistical physics from imaginary to real
time is analyzed. Adiabatic vanishing of interactions at real time infinities
gives origin to singularities at complex times. This undermines the hypothesis
of decoupling of interactions at $t \rightarrow \infty$. Hence an interacting
thermal vacuum is a necessary component of the exact real-time formalism.
Consequences for Thermo-field dynamics are discussed.
