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

Abstract

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.

Classical and quantum particle dynamics in univariate background fields

“…A different approach is given in [28,29], but only for a specific field shape (monochromatic, and circular polarization, as above). The idea is again to replace (15) with a soluble first-order equation.…”

“…The choice of this equation is motivated by taking its derivative, and showing that it reproduces (15) up to terms small in some parameter. In [28,29] the parameter is…”

“…We present now an alternative method which combines the perturbative approach above with that in [28,29]: we again look for an "effective" first-order equation, but related to an expansion in the small parameter ϵ. Observe that dividing (16) by 2k:p makes the coefficients dimensionless, and the equation becomes…”

“…Either one can look for general approximations, which hold for all field shapes as the Volkov solution for plane waves does [8], or one can look for approximations specific to a particular field shape. In the case of k 2 > 0 both the former [16] and latter [17,28,29] have recently begun to attract attention. Here we will mainly focus on the case k 2 < 0.…”

Effective photon mass and exact translating quantum relativistic structures

“…Examples are provided by the analysis of the Zitterbewegung (trembling motion) of Klein-Gordon particles in extremely small spatial scales, and its simulation by classical systems, 16 the KGE as a model for the Weibel instability in relativistic quantum plasmas, 17 the description of standing EM solitons in degenerate relativistic plasmas, 18 the KGE as the starting point for the wave kinetics of relativistic quantum plasmas, 19 the KGE in the presence of a strong rotating electric field and the QED cascade, 20 the Klein-Gordon-Maxwell multistream model for quantum plasmas, 21 the negative energy waves and quantum relativistic Buneman instabilities, 22 the separation of variables of the KGE in a curved space-time in open cosmological universes, 23 the resolution of the KGE equation in the presence of Kratzer 24 and Coulombtype 25 potentials, the KGE with a short-range separable potential and interacting with an intense plane-wave EM field, 26 electrostatic one-dimensional propagating nonlinear structures and pseudo-relativistic effects on solitons in quantum semiconductor plasma, 27 the square-root KGE, 28 hot nonlinear quantum mechanics, 29 a quantum-mechanical free-electron laser model based on the single electron KGE, 30 and the inverse bremsstrahlung in relativistic quantum plasmas. Very often, the treatment of charged particle dynamics described by the Klein-Gordon or Dirac equations assumes a circularly polarized electromagnetic (CPEM) wave, [31][32][33][34][35][36][37][38][39][40] mainly due to the analytical simplicity.…”