Conservation-law violation at high energy by anomalies

Abstract

The time evolution of a quantum Fermi field is investigated in the background of a Minkowski-space, Yang-Mills field configuration with nonvanishing topological charge. The Fermi system is assumed to possess a current ${j}_{\ensuremath{\mu}}(x)$ conserved up to an axial-vector anomaly: ${\ensuremath{\partial}}^{\ensuremath{\mu}}{j}_{\ensuremath{\mu}}=(\frac{{g}^{2}}{32{\ensuremath{\pi}}^{2}}){N}_{\mathrm{ij}}{F}_{\ensuremath{\mu}\ensuremath{\nu}}^{i}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}}^{j\ensuremath{\mu}\ensuremath{\nu}}$. It is shown explicitly that the time-dependent Yang-Mills field ${A}_{\ensuremath{\mu}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}},t)$ creates and destroys fermions in such a way that the total fermionic charge $\ensuremath{\int}{j}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}},t){d}^{3}x$ present in the final state differs from that in the initial state by precisely the amount predicted by the anomaly equation. If ${A}_{\ensuremath{\mu}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}},t)$ approaches a gauge transformation sufficiently rapidly for large $t$, this change in charge can be identified with the number of zero crossings present in the energy spectrum of the time-dependent Dirac Hamiltonian. Finally, it is demonstrated that the change in the charge carried by the fermions will differ from that predicted by the axial-vector anomaly if the large-time limit of ${A}_{\ensuremath{\mu}}$ contains physical radiation.