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, $E_{\max }$, obtained by the electric field is of the order of the critical scale for the Schwinger effect, $E_{\max }\sim E_{\mathrm{crit}}\equiv m^2/q$, 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 $\frac{q{E}_{\max }}{m^2}\ll 1$ and $\frac{q{E}_{\max }}{m^2}\gg 1$.

Figure 1

Various quantities are plotted for solutions to the semiclassical backreaction equations for a quantized scalar field with the classical current profile $J_C=-E_0\delta (t)$. The solutions for ${\tilde{E}}_0=1$ are shown across the top row of panels and those for ${\tilde{E}}_0=5$ are shown across the bottom row. The mass of the scalar field is $\frac{m^2}{q^2}=10$ and thus $\frac{E_0}{q}=10$ and 50 respectively. In the left panels the electric field $\tilde{E}$ and the electric current $⟨{\tilde{J}}_Q{⟩}_{\mathrm{ren}}$ are plotted. For each of the middle panels the blue dashed curve corresponds to the energy density of the electric field ${\rho }_{\mathrm{elec}}$, the orange solid curve represents the energy density of the created particles $⟨\tilde{\rho }{⟩}_{\mathrm{ren}}$, and the straight yellow line is the total energy density of the system. The total particle number $⟨\tilde{N}⟩$ is plotted in the right panels.

Figure 2

Various quantities are plotted for solutions to the semiclassical backreaction equations for a quantized spin-$\frac{1}{2}$ field with the classical current profile $J_C=-E_0\delta (t)$. The structure of the figure, and also the initial conditions for the electric profile, are the same as in Fig. 1.

Figure 3

The time-dependent particle number is shown for individual modes when ${\tilde{E}}_0=1$ for the scalar field (top row) and spin-$\frac{1}{2}$ (bottom row) cases for the classical current profile $J_C=-E_0\delta (t)$. The mass of the scalar field is $\frac{m^2}{q^2}=10$ and thus $\frac{E_0}{q}=10$. The vector potential is plotted in the far right panels. For each row the first panel on the left shows the particle number for $\frac{k}{q}=-20$ and the middle panel shows the particle number for $\frac{k}{q}=-40$.

Figure 4

Electric profiles for $E_0/q=2$. In the left (top) panel we show the asymptotically constant profile. In the right (top) panel we show the Sauter pulse. In both bottom panels, the classical current generating the respective electric field profiles is plotted. For the asymptotically constant profile we choose an initial time $t_0=0$, while for the Sauter pulse the initial time has to be fixed as $t_0=-\infty$.

Figure 5

Results obtained from numerical solutions to the semiclassical backreaction equation for spin-$\frac{1}{2}$ fields and the asymptotically constant classical profile are shown for $E_0/q=1$. The masses are chosen so that $\frac{q{E}_0}{m^2}\le 1$. The electric field and the induced electric current $⟨J_Q⟩/q^2$ for each case are plotted in the left panels. Plots for the corresponding number of particles, $⟨N⟩$, are shown in the middle panel and plots of the quantity $R_Q$ appear in the right panel. For the latter, the values $E_{01}/q=1$ and $E_{02}/q=1+{10}^{-3}$ have been chosen for the two solutions that are subtracted. The values of $m^2/q^2$ for each case are shown along with the type of curve for that solution in the legend in the right panel.

Figure 6

Results obtained from numerical solutions to the semiclassical backreaction equation for spin-$\frac{1}{2}$ fields and the asymptotically constant classical profile are shown for $E_0/q=1$. The masses are chosen so that $\frac{q{E}_0}{m^2}\ge 1$. The structure of the figure is the same as in Fig. 5.

Figure 7

Results obtained from numerical solutions to the semiclassical backreaction equation for spin-$\frac{1}{2}$ fields and the asymptotically constant classical profile are shown for $E_0/q=10$. The masses are chosen so that $\frac{q{E}_0}{m^2}>1$. The structure of the figure is the same as in Fig. 5. Here, the values $E_{01}/q=10$ and $E_{02}/q=10+{10}^{-3}$ have been chosen for the representation of the function $R_Q$.

Figure 8

Results obtained from numerical solutions to the semiclassical backreaction equation for scalar fields and the asymptotically constant classical profile are shown for $E_0/q=10$. The masses are chosen so that $\frac{q{E}_0}{m^2}\ge 1$. The structure of the figure is the same as in Fig. 5. We have chosen $E_{01}/q=10$ and $E_{02}/q=10+{10}^{-3}$ to represent the function $R_Q$.

Figure 9

Results obtained from numerical solutions to the semiclassical backreaction equation for both spin-$\frac{1}{2}$ fields and scalar fields when the asymptotically constant classical profile is used are shown. For all of the plots the solid curve (blue) corresponds to the scalar field, the dashed curve (orange) corresponds to the spin-$\frac{1}{2}$ field, and the dotted curve (yellow), when shown, corresponds to the classical solution when no quantum fields are coupled to the electromagnetic field. In the upper tier $\frac{q{E}_0}{m^2}=1$ and $m^2/q^2=1$ while in the lower tier $\frac{q{E}_0}{m^2}=10$ and $m^2/q^2=1$. For each tier the left panels show plots of the electric field and the induced electric current $⟨J_Q⟩/q^2$, the middle panel shows plots of the number of particles $⟨N⟩$, and the right panel shows the quantity $R_Q$. For the latter the values $E_{01}/q=E_0/q$ and $E_{02}/q=E_0/q+{10}^{-3}$ have been chosen for the two solutions that are subtracted.

Figure 10

Results obtained from numerical solutions to the semiclassical backreaction equation for both spin-$\frac{1}{2}$ fields and scalar fields and the Sauter pulse classical profile are shown. The structure of the figure, including the initial values and the parameters of the fields, is the same as in Fig. 9.