Isospin precession in non-Abelian Aharonov-Bohm scattering

TL;DR

The paper demonstrates a pseudo-classical isospin treatment of the non-Abelian Aharonov–Bohm effect for SU(2) flux confined to a thin cylinder. It shows that isospin precession occurs when the incoming isospin is not aligned with the non-Abelian flux, and that the Wu–Yang phase factor acts as a classical S-matrix rotating isospin, with the external orbital motion remaining free. Flux quantization in two distinct even/odd series yields a pair of internal-symmetry branches, each producing three conserved charges and a doubled rotational symmetry; for non-integer flux, internal charges are lost and proton/neutron identities become gauge-covariant rather than conserved. The analysis bridges pseudo-classical dynamics, internal Noether charges, and van Holten’s algorithm, revealing spin-from-isospin structure and the conditions under which total angular momentum splits into independent external and internal contributions. The work suggests connections to laboratory realizations of non-Abelian gauge fields and deepens understanding of how non-Abelian holonomy governs internal dynamics and charge conservation. $

Abstract

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, $Φ_N=NΦ_0$ 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.

Figures (6)

• Figure 1: Schema of the Abelian Aharonov-Bohm experiment WuYang75.
• Figure 2: Scattering of a nucleon on a non-Abelian flux $\vec{\Phi}$, whose direction is perpendicular to the proton/neutron axis. The isopin is rotated by an angle ${\bf {\pm} \Delta\vartheta\,{\mathfrak{a}}}$ with the sign depending on the side the particles passes, i.e., the sign, ${\bf {L > 0}}$ or ${\bf {L < 0}}$, of the (3rd component) of the orbital angular momentum. The rotation of the isospin is generated by the Non-Abelian Phase Factor $\mathcal{F}(\Delta\vartheta)$ in \ref{['phirot']}.
• Figure 3: The "isospin clockwork" for non-Abelian flux $\Phi_a=-i{\pi}a\sigma_3$, ($a\neq0$). The incoming proton (at $L_\pm$) is aligned with ${\vec{p} = \sigma_1/2i}$ (at "6 o'clock"), ${\bf Q^p_{in}}$= ${{\vec{p}}}$. For ${\bf L > 0}$ the isospin rotates clockwise, but for ${\bf L < 0}$ it rotates counter-clockwise. For odd integer flux (${\bf a = 1}$, for example), drawn in red and blue, $\vec{Q}_{out}^{p} = -\vec{Q}_{in}^{p}$ ($R_{\pm}$ at "12 o'clock") on both sides. However when $a$ is not an integer e.g. for ${\bf a={{\frac{1}{2}}}}$ (mod even integers), drawn in magenta and cyan, the isospin is lagged behind. $\vec{Q}_{out}^{p}$ and ("9 o'clock") and $\vec{Q}_{out}^{p}$ ("3 o'clock") are both orthogonal to $\vec{Q}_{in}^{p}$ and are oppositely oriented in the two $L$-sectors. Similar behavior could be observed for any $\mathfrak{a}$.
• Figure 4: The "isospin clockwork" for protons and neutrons for $a=1+2k$ where $k$ is an integer. i.e., for flux $\Phi=-i(1+2k)\pi\sigma_3$. The incoming proton (in red and blue is aligned with $\vec{p}$, $Q_{in}^p= \sigma_1/2i$ ("6 o'clock") as in FIG.\ref{['clockwork']}. The incoming neutron (in purple and green) is along ${\hat{z}}$ (at "3 o'clock"), $Q_{in}^n ={\bm{\sigma}}_2/2i$ on both sides of the flux. For ${\bf L > 0}$ the isospin rotates clockwise for both $Q^p$ and $Q^n$ and for ${\bf L < 0}$ they rotate counter-clockwise. $Q^p$ and $Q^n$ turn rigidly, remaining orthogonal to each other. $\vec{Q}_{out}^{p,n} = -\vec{Q}_{in}^{p,n}$ ("12 o'clock" and "9 o'clock", respectively) on both sides, consistently with $S=-\mathds{I}$ in \ref{['II-II']}. The centers of the "shifting clocks" are at $\varphi = 0,\ \pi/2, \, \pi$.
• Figure 5: Isospin motion for ${\mathfrak{a}}=1$ for both protons \ref{['a=1Pcyclo']} and neutrons \ref{['a=1Ncyclo']}, in both respective half-planes. The trajectories follow cycloid segments, which all satisfy ${\vec{Q}}_{out} = -{\vec{Q}}_{in}$.

Theorems & Definitions (9)

• Theorem 1.1
• Definition 1.1
• Theorem 2.1
• Definition 3.1
• Theorem 3.1
• Theorem 5.1
• Theorem 6.1
• Theorem 6.2
• Theorem 7.1