A novel solution to the Klein–Gordon equation in the presence of a strong rotating electric field

The Klein–Gordon equation in the presence of a strong electric field, taking the form of the Mathieu equation, is studied. A novel analytical solution is derived for particles whose asymptotic energy is much lower or much higher than the electromagnetic field amplitude. The condition for which the new solution recovers the familiar Volkov wavefunction naturally follows. When not satisfied, significant deviation from the Volkov wavefunction is demonstrated. The new condition is shown to differ by orders of magnitudes from the commonly used one. As this equation describes (neglecting spin effects) the emission processes and the particle motion in Quantum Electrodynamics (QED) cascades, our results suggest that the standard theoretical approach towards this phenomenon should be revised.