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

摘要

Nonperturbative calculation of QED processes participated by a strongelectromagnetic field, especially provided by strong laser facilities atpresent and in the near future, generally resorts to the Furry picture with theusage of analytical solutions of the particle dynamical equation, such as theKlein-Gordon equation and Dirac equation. However only for limited fieldconfigurations such as a plane-wave field could the equations be solvedanalytically. Studies have shown significant interests in QED processes in astrong field composed of two counter-propagating laser waves, but the exactsolutions in such a field is out of reach. In this paper, inspired by theobservation of the structure of the solutions in a plane-wave field, we developa new method and obtain the analytical solution for the Klein-Gordon equationand equivalently the action function of the solution for the Dirac equation inthis field, under a largest dynamical parameter condition that there exists aninertial frame in which the particle free momentum is far larger than the otherfield dynamical parameters. The applicable range of the new solution isdemonstrated and its validity is proven clearly. The result has the advantageof Lorentz covariance, clear structure and close similarity to the solution ina plane-wave field, and thus favors convenient application.