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# Exact solutions of the bound Dirac and Klein Gordon equations in non co propagating electromagnetic plane waves

preprint

posted on 2023-11-30, 06:07 authored by A. HartinA new class of exact solutions of the bound Dirac and bound Klein Gordon equations in non co propagating plane waves is found. The solutions are based on the physical principle of maintaining local gauge invariance in the Furry picture Lagrangian when N external fields can undergo independent gauge transformations. The solutions can be expressed in terms of the Hamilton Jacobi action and a gauge invariant effective particle momentum in the ensemble of external fields. Rotations of the effective particle momentum, which preserve local gauge invariance, are introduced into the action using matrix calculus. The set of such rotations provides the class of new solutions constituting a family of Volkov like solutions for one external field. When applied to two or more non co propagating external fields, the rotational symmetry provides counter terms which decouple the fields. The bound state equations of motion become solvable for any number of non co propagating external fields. Through angular spectral decomposition, which represents electromagnetic fields of any form as a spatial Fourier series of non co propagating plane waves, the new solutions described here can be applied to strong field physics problems in any external electromagnetic field.