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

TL;DR

This paper provides an exact solution for Dirac electrons in a broad class of electric fields depending on light cone time, resolving the $p^+ = 0$ ambiguity and revealing new phenomena like instantaneous pair creation and nonanalytic current behavior.

Contribution

It introduces the first exact mode function solutions for arbitrary light cone dependent electric fields, explicitly resolving the $p^+ = 0$ ambiguity in QED.

Findings

Explicit resolution of the $p^+ = 0$ ambiguity.

Pair creation occurs instantaneously on the light cone.

Expectation values of currents depend nonanalytically on the background field.

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 $(I + \gamma^0 \gamma^1) \psi$ at $x^+ = 0$ but also $(I - \gamma^0 \gamma^1) \psi$ 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.