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 $E_3{(x}_{\pm })$ dependent upon one or the other light-cone coordinates, $x_{+}$ or $x_{-},$ $x_{\pm }{=x}_3\pm x_0;$ 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 $\overrightarrow{E}$ can point in a varying direction, and where an arbitrary magnetic field $\overrightarrow{B}$ 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.