Isospin precession in non-Abelian Aharonov-Bohm scattering

The concept of pseudoclassical isospin is illustrated by the non-Abelian Aharonov-Bohm effect proposed by Wu and Yang in 1975. The spatial motion is free however the isospin precesses when the enclosed magnetic flux and the incoming particle's isosopin are not parallel. The non-Abelian phase factor $\mathfrak{F}$ of Wu and Yang acts on the isospin as an S-matrix. The scattering becomes side-independent when the enclosed flux is quantized, ${\Phi}_N=N\Phi_0$ with $N$ an integer. The gauge group $SU(2)$ is an internal symmetry and generates conserved charges only when the flux is quantized, which then splits into two series: for $N=2k$ $SU(2)$ acts trivially but for $N=1+2k$ the implementation is twisted. The orbital and the internal angular momenta are separately conserved. The double rotational symmetry is broken to $SO(2)\times SO(2)$ when $N$ odd. For unquantized flux there are no internal symmetries, the charge is not conserved and protons can be turned into neutrons.