The Axial Anomaly in D=3+1 Light-Cone QED

Abstract

We consider $(3+1)$-dimensional, Dirac electrons of arbitrary mass, propagating in the presence of electric and magnetic fields which are both parallel to the $x^3$ axis. The magnetic field is constant in space and time whereas the electric field depends arbitrarily upon the light-cone time parameter $x^+ = (x^0 + x^3)/\sqrt{2}$. We present an explicit solution to the Heisenberg equations for the electron field operator in this background. The electric field results in the creation of electron-positron pairs. We compute the expectation values of the vector and axial vector currents in the presence of a state which is free vacuum at $x^+ = 0$. Both current conservation and the standard result for the axial vector anomaly are verified for the first time ever in $(3+1)$-dimensional light-cone QED. An interesting feature of our operator solution is the fact that it depends in an essential way upon operators from the characteristic at $x^- = -L$, in addition to the usual dependence upon operators at $x^+ = 0$. This dependence survives even in the limit of infinite $L$. Ignoring the $x^-$ operators leads to a progressive loss of unitarity, to the violation of current conservation, to the loss of renormalizability, and to an incorrect result for the axial vector anomaly.