Analytical solution for Klein-Gordon equation and action function of the solution for Dirac equation in counter-propagating laser waves

TL;DR

This paper develops an analytical solution for the Klein-Gordon and Dirac equations in a complex laser field configuration, expanding the tools available for nonperturbative QED calculations in strong, counter-propagating laser fields.

Contribution

It introduces a new method to derive analytical solutions for these equations in a two-laser field, under specific dynamical conditions, with advantages of Lorentz covariance and structural clarity.

Findings

Derived the analytical solution for Klein-Gordon equation in counter-propagating laser fields.

Obtained the action function for the Dirac equation in this configuration.

Demonstrated the solution's validity and applicability range.

Abstract

Nonperturbative calculation of QED processes participated by a strong electromagnetic field, especially provided by strong laser facilities at present and in the near future, generally resorts to the Furry picture with the usage of analytical solutions of the particle dynamical equation, such as the Klein-Gordon equation and Dirac equation. However only for limited field configurations such as a plane-wave field could the equations be solved analytically. Studies have shown significant interests in QED processes in a strong field composed of two counter-propagating laser waves, but the exact solutions in such a field is out of reach. In this paper, inspired by the observation of the structure of the solutions in a plane-wave field, we develop a new method and obtain the analytical solution for the Klein-Gordon equation and equivalently the action function of the solution for the Dirac equation in this field, under a largest dynamical parameter condition that there exists an inertial frame in which the particle free momentum is far larger than the other field dynamical parameters. The applicable range of the new solution is demonstrated and its validity is proven clearly. The result has the advantage of Lorentz covariance, clear structure and close similarity to the solution in a plane-wave field, and thus favors convenient application.