Thermal state with quadratic interaction

We consider the perturbative construction, proposed in [37], for a thermal state $\Omega_{\beta,\lambda V\{f\}}$ for the theory of a real scalar Klein-Gordon field $\phi$ with interacting potential $V\{f\}$. Here $f$ is a spacetime cut-off of the interaction $V$ and $\lambda$ is a perturbative parameter. We assume that $V$ is quadratic in the field $\phi$ and we compute the adiabatic limit $f\to 1$ of the state $\Omega_{\beta,\lambda V\{f\}}$. The limit is shown to exist, moreover, the perturbative series in $\lambda$ 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) [59] recently constructed in [25].