Nontrivial generalizations of the Schwinger pair production result

We present new, nontrivial generalizations of the recent Tomaras-Tsamis-Woodard extension of the original Schwinger formula for charged pair production in a constant electric field. That extension generalized the Schwinger result to electric fields E-3(x(+/-)) dependent upon one or the other light-cone coordinates, x(+) or x(-), x(+/-)=x(3)+/-x(0); the present work generalizes their result to electric fields E-3(x(+),x(-)) dependent upon both coordinates. Displayed in the form of a final, functional integral, or equivalent linkage operation, our result does not appear to be exactly calculable in the general case; and we give a simple, approximate example when E-3(x(+),x(-)) is a slowly varying function of its variables. We extend this result to the more general case where E can point in a varying direction, and where an arbitrary magnetic field (B) over right arrow is present; both extensions can be cast into the form of Gaussian-weighted functional integrals over well defined factors, which are amenable to approximations depending on the nature and variations of the fields.