Axial anomaly inD=3+1light-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 ${x}^{\ensuremath{-}}=\ensuremath{-}L,$ in addition to the usual dependence upon operators at ${x}^{+}=0.$ This dependence survives even in the limit of infinite L. Ignoring the ${x}^{\ensuremath{-}}$ 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.