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A new class of exact solutions of the bound Dirac and bound Klein Gordonequations in non co propagating plane waves is found. The solutions are basedon the physical principle of maintaining local gauge invariance in the Furrypicture Lagrangian when N external fields can undergo independent gaugetransformations. The solutions can be expressed in terms of the Hamilton Jacobiaction and a gauge invariant effective particle momentum in the ensemble ofexternal fields. Rotations of the effective particle momentum, which preservelocal gauge invariance, are introduced into the action using matrix calculus.The set of such rotations provides the class of new solutions constituting afamily of Volkov like solutions for one external field. When applied to two ormore non co propagating external fields, the rotational symmetry providescounter terms which decouple the fields. The bound state equations of motionbecome solvable for any number of non co propagating external fields. Throughangular spectral decomposition, which represents electromagnetic fields of anyform as a spatial Fourier series of non co propagating plane waves, the newsolutions described here can be applied to strong field physics problems in anyexternal electromagnetic field.

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