Semiclassical linear-stability analysis of homogeneous electric-field modes coupled to a scalar quantum field.

Abstract

We examine the problem of a homogeneous, time-dependent electric field $E(t)$ in the presence of a scalar quantum field in its vacuum state, using the semiclassical Maxwell equations. We obtain the Laplace transform of $E(t)$ as a function of initial conditions in the linear approximation. For the case we study, this approximation is valid for a certain length of time which is increased indefinitely as the initial values of the electric field and other perturbation quantities are made vanishingly small. We find the existence of unphysical instabilities (exponentially growing modes) with an imaginary frequency which is shown to be identical to the homogeneous Landau ghost mode at the one-loop level in scalar QED. We show that this mode may be suppressed, however, by a certain restriction on the initial data, which then leads to asymptotically well-behaved solutions for $E(t)$. For $t>>{m}^{\ensuremath{-}1}$, these solutions approach a constant value which differs from $E(0)$ only by a small relative amount of $O(\ensuremath{\alpha})$ in general.