Fermion number creation via electromagnetism

Abstract

Let an external current, whose support is confined to the space-like slab |x ⁰| < T in two-dimensional spacetime, build up a localized charge density which vanishes for times |x ⁰| > T. We show that the zero mass Dirac quantum field reacts to this current by a c-number shift of the fermion number, i.e. Q _out=Q _in+ΔQ, with $\Delta Q{\text{ = }}\smallint _{{\text{ - }}\infty }^\infty {\text{d}}x^0 \smallint _{{\text{ - }}\infty }^{x^0 } {\text{d}}\xi q(\xi )/\pi$, where q(x ⁰) denotes the total external charge. For the shift of the axial charge we obtain an extension of existing results.