One-loop Energy-momentum Tensor in Qed with Electric-like Background

We have obtained nonperturbative one-loop expressions for the mean energy-momentum tensor and current density of Dirac's field on a constant electric-like background. One of the goals of this calculation is to give a consistent description of back-reaction in such a theory. Two cases of initial states are considered: the vacuum state and the thermal equilibrium state. First, we perform calculations for the vacuum initial state. In the obtained expressions, we separate the contributions due to particle creation and vacuum polarization. The latter contributions are related to the Heisenberg–Euler Lagrangian. Then, we study the case of the thermal initial state. Here, we separate the contributions due to particle creation, vacuum polarization, and the contributions due to the work of the external field on the particles at the initial state. All these contributions are studied in detail, in different regimes of weak and strong fields and low and high temperatures. The obtained results allow us to establish restrictions on the electric field and its duration under which QED with a strong constant electric field is consistent. Under such restrictions, one can neglect the back-reaction of particles created by the electric field. Some of the obtained results generalize the calculations of Heisenberg–Euler for energy density to the case of arbitrary strong electric fields.