Pair production due to an electric field in 1+1 dimensions and the validity of the semiclassical approximation

Solutions to the backreaction equation in 1+1-dimensional semiclassical electrodynamics are obtained and analyzed when considering a time-varying homogeneous electric field initially generated by a classical electric current, coupled to either a quantized scalar field or a quantized spin-$\frac{1}{2}$ field. Particle production by way of the Schwinger effect leads to backreaction effects that modulate the electric field strength. Details of the particle production process are investigated along with the transfer of energy between the electric field and the particles. The validity of the semiclassical approximation is also investigated using a criterion previously implemented for chaotic inflation and, in an earlier form, semiclassical gravity. The criterion states that the semiclassical approximation will break down if any linearized gauge-invariant quantity constructed from solutions to the linear response equation, with finite nonsingular data, grows rapidly for some period of time. Approximations to homogeneous solutions of the linear response equation are computed and it is found that the criterion is violated when the maximum value, $E_{\rm max}$, obtained by the electric field is of the order of the critical scale for the Schwinger effect, $E_{\rm max} \sim E_{\rm crit}\equiv m^2/q$, where $m$ is the mass of the quantized field and $q$ is its electric charge. For these approximate solutions the criterion appears to be satisfied in the extreme limits $\frac{qE_{\rm max}}{m^2} \ll 1$ and $\frac{qE_{\rm max}}{m^2} \gg 1$.