Worldline instantons and pair production in inhomogenous fields

We show how to do semiclassical nonperturbative computations within the worldline approach to quantum field theory using “worldline instantons.” These worldline instantons are classical solutions to the Euclidean worldline loop equations of motion, and are closed spacetime loops parametrized by the proper time. Specifically, we compute the imaginary part of the one-loop effective action in scalar QED using “worldline instantons” for a wide class of inhomogeneous electric field backgrounds. We treat both time-dependent and space-dependent electric fields and note that temporal inhomogeneities tend to shrink the instanton loops, while spatial inhomogeneities tend to expand them. This corresponds to temporal inhomogeneities tending to enhance local pair production, with spatial inhomogeneities tending to suppress local pair production. We also show how the worldline instanton technique extends to spinor QED.

Figure 1

Parametric plot of the stationary worldline instanton paths in the $(x_3,x_4)$ plane for the case of a constant electric field of strength $E$. The paths are circular and the radius has been expressed in units of $\frac{m}{eE}$.

Figure 2

Parametric plot of the stationary worldline instanton paths (36) in the $(x_3,x_4)$ plane for the case of a time-dependent electric field $E(t)=E{\mathrm{sech}}^2(\omega t)$. The paths are shown for various values of the adiabaticity parameter $\gamma =\frac{m\omega }{eE}$ defined in (29), and $x_3$ and $x_4$ have been expressed in units of $\frac{m}{eE}$. Note that in the static limit, $\gamma \to 0$, the instanton paths reduce to the circular ones of the constant field case shown in Fig. 1.

Figure 3

Plot of the instanton action $S_0$, in units of $n\frac{m^2}{eE}$, in (38) for the time-dependent electric field $E(t)=E{\mathrm{sech}}^2(\omega t)$, plotted as a function of the adiabaticity parameter $\gamma$. Contrast this plot with the behavior in Fig. 7 for a spatial inhomogeneity of the same form.

Figure 4

Parametric plot of the stationary worldline instanton paths (48) in the $(x_3,x_4)$ plane for the case of a time-dependent electric field $E(t)=E\cos (\omega t)$. The paths are shown for various values of the adiabaticity parameter $\gamma =\frac{m\omega }{eE}$ defined in (29), and $x_3$ and $x_4$ have been expressed in units of $\frac{m}{eE}$. Note that in the static limit, $\gamma \to 0$, the instanton paths reduce to the circular ones of the constant field case shown in Fig. 1.

Figure 5

Plot of the instanton action $S_0$, in units of $n\frac{m^2}{eE}$, for the time-dependent electric field $E(t)=E\cos (\omega t)$, plotted as a function of the adiabaticity parameter $\gamma$. Contrast this plot with the behavior in Fig. 10 for a spatial inhomogeneity of the same form.

Figure 6

Instanton paths for the spatially inhomogeneous electric field $E(x)=E{\mathrm{sech}}^2(kx)$ for various values of the inhomogeneity parameter $\tilde{\gamma }$ defined in (60). As $\tilde{\gamma }\to 0$ we recover the circular paths of the constant field case, but as $\tilde{\gamma }\to 1$ the loops become infinitely large.

Figure 7

The stationary action $S_0$, in units of $n\frac{m^2}{eE}$, plotted as a function of the inhomogeneity parameter $\tilde{\gamma }$ defined in (60). Note that $S_0$ increases with $\tilde{\gamma }$ and diverges as $\tilde{\gamma }\to 1$. Contrast this plot with the behavior in Fig. 3 for a temporal inhomogeneity of the same form.

Figure 8

The ratio (63) of $\mathrm{Im}\Gamma$ for the spatially inhomogeneous electric field $E(x)=E{\mathrm{sech}}^2(kx)$ to the locally constant field result, in the single instanton loop approximation, plotted as a function of the inhomogeneity parameter $\tilde{\gamma }$ defined in (60).

Figure 9

Instanton paths for the spatially inhomogeneous electric field $E(x)=E\cos (kx)$ for various values of the inhomogeneity parameter $\tilde{\gamma }$ defined in (60). As $\tilde{\gamma }\to 0$ we recover the circular paths of the constant field case, but as $\tilde{\gamma }\to 1$ the loops become elongated in the $x_3$ direction.

Figure 10

The stationary action $S_0$, in units of $n\frac{m^2}{eE}$, plotted as a function of the inhomogeneity parameter $\tilde{\gamma }$ defined in (60). Note that $S_0$ increases with $\tilde{\gamma }$ and approaches 4 as $\tilde{\gamma }\to 1$. Contrast this plot with the behavior in Fig. 5 for a temporal inhomogeneity of the same form.

Figure 11

The ratio (63) of $\mathrm{Im}\Gamma$ for the spatially inhomogeneous electric field $E(x)=E\cos (kx)$ to the constant field result, in the single instanton loop approximation, plotted as a function of the inhomogeneity parameter $\tilde{\gamma }$ defined in (60).