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In the perturbative treatment of interacting quantum field theories, if theinteraction Lagrangian changes adiabatically in a finite interval of time,secular growths may appear in the truncated perturbative series also when theinteraction Lagrangian density is returned to be constant. If this happens, theperturbative approach does not furnish reliable results in the evaluation ofscattering amplitudes or expectation values. In this paper we show that theseeffects can be avoided for adiabatically switched-on interactions, if thespatial support of the interaction is compact and if the background state issuitably chosen. We start considering equilibrium background states and showthat, when thermalisation occurs (interaction Lagrangian of spatial compactsupport), secular effects are avoided. Furthermore, no secular effects pop upif the limit where the Lagrangian is supported everywhere in space is takenafter thermalisation (large time limit), in contrast to the reversed order.This result is generalized showing that if the interaction Lagrangian isspatially compact, secular growths are avoided for generic background stateswhich are only invariant under time translation and to states whose explicitdependence of time is not too strong. Finally, as an example, we apply thepresented theorems to study a complex scalar and a Dirac field in a classicalexternal electromagnetic potential, on a background KMS state, to manifest thata spatially compact supported interaction does not give rise to seculargrowths.

Secular growths and their relation to equilibrium states in perturbative QFT