Solutions of the Klein-Gordon and Dirac equations for a particle in a constant electric field and a plane electromagnetic wave propagating along the field
Abstract:
A complete set of solutions is found to the Klein–Gordon and Dirac equations for the case of
a constant (in space and time) electric field along which a plane electromagnetic wave propagates. The solutions are labeled by the numbers $p_1$, $p_2$, $p_3$, which become the conserved three-momentum when the field of the wave is switched off. These solutions are related by an integral transformation to previously obtained solutions labeled by conserved components of the momentum: $p_1$, $p_2$, $p_-=p_0-p_3$. In contrast to these last solutions, the $\psi_{p_3}$-solutions arc everywhere finite and can be explicitly classified with respect to the sign of the “frequency” as $x_0\to\pm\infty$. It is also shown that the solutions $\psi_{p_-}$ too can be classified with respect to the sign of the “frequency”. This means that they can be used in the usual manner to describe matrix elements. Propagators are obtained in the Fock–Schwinger and Feynman representations. It is shown that not only the total but also the differential probabilities of pair creation by the field are independent of the field of the
wave if they are expressed in terms of Lorentz and gauge-invariant quantities.
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