Conservation-law violation at high energy by anomalies

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 $j_{\mu }\mathrm{}\left(x\right)\mathrm{}$ conserved up to an axial-vector anomaly: ${\partial }^{\mu }{j}_{\mu }=\left(\frac{g^2}{32{\pi }^2}\right)N_{\mathrm{ij}}{F}_{\mu \nu }^i{\tilde{F}}^{j\mu \nu }$. It is shown explicitly that the time-dependent Yang-Mills field $A_{\mu }(\overrightarrow{\mathrm{x}},t)$ creates and destroys fermions in such a way that the total fermionic charge $\int j_0(\overrightarrow{\mathrm{x}},t)d^{3}x$ present in the final state differs from that in the initial state by precisely the amount predicted by the anomaly equation. If $A_{\mu }(\overrightarrow{\mathrm{x}},t)$ 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 $A_{\mu }$ contains physical radiation.