Complete Keldysh theory and its limiting cases

An investigation of the foundations of strong-field theories of the Keldysh type is undertaken. It is found first that the Volkov solution, upon which the Keldysh approximation depends, exists unambiguously only in the relativistic case. Accordingly, a fully relativistic theory of atomic photoionization is formulated from first principles. A strong-field approximation (SFA) is expressed, based only on the proposition that the photoionizing field dominates the atomic potential in the final state. The nonrelativistic limit of the SFA gives exactly one of the known Keldysh-type theories, but the original Keldysh theory itself is found to neglect some fraction of the strong-field effects. It is shown on a variety of grounds that ${\mathbf{A}}^2$ terms must be fully retained in Keldysh-type theories. Important in this matter are the proper application of asymptotic conditions, the interchangeability of A⋅p and ${\mathbf{A}}^2$ terms, and the consistent reduction to the nonrelativistic limit.