Resolving the $p^+ = 0$ Ambiguity in a Homogeneous Electric Background

Abstract

I present an exact solution for the Heisenberg picture, Dirac electron in the presence of an electric field which depends arbitrarily upon the light cone time parameter $x^+ = (t+x)/\sqrt{2}$. This is the largest class of background fields for which the mode functions have ever been obtained. The solution applies to electrons of any mass and in any spacetime dimension. The traditional ambiguity at $p^+ = 0$ is explicitly resolved. It turns out that the initial value operators include not only $(I + γ^0 γ^1) ψ$ at $x^+ = 0$ but also $(I - γ^0 γ^1) ψ$ at $x^- = -L$. Pair creation is a discrete and instantaneous event on the light cone, so one can compute the particle production rate in real time. In $D=1+1$ dimensions one can also see the anomaly. Another novel feature of the solution is that the expectation value of the current operators depends nonanalytically upon the background field. This seems to suggest a new, strong phase of QED.