Worldline instantons and the Sauter-Schwinger effect

The Sauter-Schwinger effect, that is, electron-positron pair creation from the vacuum due to a strong electric field, was predicted only shortly after the foundations of quantum mechanics were first laid out. But even today, many years later, although the existence of the effect follows straightforwardly from a relativistic quantum description of the electron and can be motivated quite intuitively by a tunneling picture, our understanding is far from complete.

First of all, it has not yet been experimentally verified due to the extremely large field strength required to obtain a detectable pair creation probability. A theoretical treatment is difficult as well, since it is a nonperturbative effect so the usual toolkit of perturbation theory does not apply. In general, to calculate the pair production probability for a non-static or spatially inhomogeneous electric field, we need to solve the Dirac equation exactly for that background field.

Only recently progress has been made both in terms of computational methods, numerical and analytical, and in terms of insight on how more complicated field profiles affect the pair production rate. A method well suited to study the Sauter-Schwinger effect in inhomogeneous fields is the worldline instanton method. Worldline instantons provide a semiclassical approximation that is valid in many field configurations of interest. Although the computational problem of the worldline instanton method can be stated very succinctly for arbitrary background fields, actually finding instantons is still rather complicated for general fields.

This thesis has three main goals: 1. to give a complete, self-contained derivation of the worldline instanton method suitable for a reader new to the topic, 2. to introduce a completely general numerical scheme to robustly find instantons in arbitrary background fields, and 3. to build some intuition by analytically calculating results for several fields and using the numerical method to show how they qualitatively carry over to more complex and realistic scenarios that are inaccessible to other approaches. We will especially focus on the dynamically assisted Sauter-Schwinger effect here.

Apart from the worldline instanton method this thesis also introduces a scheme to numerically integrate the Riccati equation, an exact formulation of the pair production problem, with high precision, to obtain exact (up to tunable numerical errors) results for time dependent fields.


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