Search for periodic vacuum in QED$_2$

TL;DR

This paper investigates whether the vacuum state of QED in two dimensions favors a periodic electric field, finding that it does not, based on a detailed analysis of the energy density and effective action at one-loop order.

Contribution

The study introduces a projected effective action that depends on a non-zero background field and analyzes the vacuum energy with a non-perturbative approach.

Findings

Vacuum does not favor a periodic electric mean field in QED$_2$

Energy density depends on the periodic vacuum expectation value of the vector potential

Vacuum energy differs from the negative of the effective action in this setting

Abstract

It is shown that the vacuum of QED$_2$ in Minkowski spacetime does not favour a periodic electric mean field. The projected effective action exhibiting a genuine dependence on the non-vanishing background field has been introduced. The functional dependence of the energy density of the vacuum on the assumed periodic vacuum expectation value of the vector potential is determined from the component $T^{00}$ of the energy-momentum tensor at one-loop order. Treating the background field non-perturbatively, the energy of the vacuum in the presence of a periodic mean field is found not be equal to the negative of the effective action.