Worldline instantons and the fluctuation prefactor

In a previous paper [G. V. Dunne and C. Schubert, Phys. Rev. D 72, 105004 (2005).], it was shown that the worldline expression for the nonperturbative imaginary part of the QED effective action can be approximated by the contribution of a special closed classical path in Euclidean spacetime, known as a worldline instanton. Here we extend this formalism to compute also the prefactor arising from quantum fluctuations about this classical closed path. We present a direct numerical approach for determining this prefactor, and we find a simple explicit formula for the prefactor in the cases where the inhomogeneous electric field is a function of just one spacetime coordinate. We find excellent agreement between our semiclassical approximation, conventional WKB, and recent numerical results using numerical worldline loops.

Figure 1

The dotted line plots the ratio of our semiclassical worldline instanton expression (4.7) to the weak-field limit of the corresponding locally constant field approximation (4.8). The dashed line is the same ratio using a numerical integration of the exact expression, derived from Nikishov’s exact result in 49 (see also 36). The circles represent the numerical worldline results of Gies and Klingmüller [39], which were evaluated for $\frac{eE}{m^2}=1$. Note that the agreement is excellent, even though it is far from the weak-field limit.