Perturbative construction of equilibrium states for interacting fermionic field theories Semiclassical Maxwell equation and the Debye screening length

In this paper, we aim to extend to interacting massive and massless fermionic theories the recent perturbative construction of equilibrium states developed within the framework of perturbative algebraic quantum field theory on Lorentzian spacetime. We analyze the case of interactions which depend on time by a smooth switch-on function and on space by a suitably bounded function that multiplies an interaction Lagrangian density constructed with the field of the theory. The construction is achieved by first considering the case of compact support and, in a second step, by removing the space cutoff with a suitable limit (adiabatic limit). As an application, we consider a Dirac field interacting with a classical stationary background electromagnetic potential, and we compute at first perturbative order (linear response) the expectation value of the conserved current on the equilibrium state for the interacting theory. The resulting expectation value is written as a convolution, in the space coordinates, between the electromagnetic potential and an integral kernel which, at vanishing conjugate momentum, gives the inverse of the square Debye screening length at finite temperature. The corresponding Debye screening effect is visible in the backreaction treated semiclassically of this current on the classical background electromagnetic potential sourced by a classical external current.