Thermal State with Quadratic Interaction

We consider the perturbative construction, proposed in Fredenhagen and Lindner (Commun Math Phys 332:895, 2014), for a thermal state Ω β,λV{f} for the theory of a real scalar Klein–Gordon field ϕ with interacting potential V{ f}. Here, f is a space-time cut-off of the interaction V, and λ is a perturbative parameter. We assume that V is quadratic in the field ϕ and we compute the adiabatic limit f→ 1 of the state Ω β,λV{f} . The limit is shown to exist; moreover, the perturbative series in λ sums up to the thermal state for the corresponding (free) theory with potential V. In addition, we exploit the same methods to address a similar computation for the non-equilibrium steady state (NESS) Ruelle (J Stat Phys 98:57–75, 2000) recently constructed in Drago et al. (Commun Math Phys 357:267, 2018).