  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.
