Thermal state with quadratic interaction

TL;DR

This paper constructs and analyzes a thermal state for a scalar field with quadratic interaction, demonstrating the existence of the adiabatic limit and linking the perturbed state to the free theory's thermal state.

Contribution

It provides a perturbative construction of thermal states with quadratic interactions and proves the existence of the adiabatic limit, connecting it to the free theory's thermal state.

Findings

Adiabatic limit of the perturbed thermal state exists.

Perturbative series sums to the free theory's thermal state.

Methods extend to non-equilibrium steady states (NESS).

Abstract

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].