# Isospin precession in non-Abelian Aharonov-Bohm scattering

Peng-Ming Zhang, Peter Horvathy

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Peng-Ming Zhang, Peter Horvathy. Isospin precession in non-Abelian Aharonov-Bohm scattering. 2025. hal-04494374v2

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Distributed under a Creative Commons Attribution 4.0 International License
Isospin precession

in

non-Abelian Aharonov-Bohm scattering *

P.-M. Zhang¹†, and P. A. Horváthy²‡

¹ *School of Physics and Astronomy,*
*Sun Yat-sen University,*
*Zhuhai 519082, (China)*

² *Institut Denis-Poisson CNRS/UMR 7013 - Université de Tours - Université d'Orléans Parc de Grammont,*
*37200, Tours, (France)*

(Dated: May 16, 2025)

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, $\Phi_N = N\Phi_0$ 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.

<footnote>* Dedicated to the memory of Professor Tai-Tsun Wu (1933-2024).</footnote>

<footnote>† mailto:zhangpm5@mail.sysu.edu.cn</footnote>

<footnote>‡ mailto:horvathy@lmpt.univ-tours.fr</footnote>
Contents

<table><tr><td>I. Introduction: the vector potential in gauge theory</td><td>3</td></tr><tr><td>II. The Wu-Yang phase factor</td><td>6</td></tr><tr><td>III. (Pseudo)classical isospin</td><td>8</td></tr><tr><td>A. Internal charges and symmetries</td><td>9</td></tr><tr><td>B. Conserved quantities</td><td>10</td></tr><tr><td>IV. Isospin precession</td><td>13</td></tr><tr><td>V. Isospin scattering</td><td>16</td></tr><tr><td>VI. Internal charges</td><td>21</td></tr><tr><td>A. Implementability and internal symmetry</td><td>21</td></tr><tr><td>B. Conserved charges in the non-Abelian Aharonov-Bohm effect</td><td>24</td></tr><tr><td>C. Spin from isospin</td><td>28</td></tr><tr><td>VII. Conserved quantities by van Holten's algorithm</td><td>29</td></tr><tr><td>VIII. Conclusion and outlook</td><td>32</td></tr><tr><td>Acknowledgments</td><td>35</td></tr><tr><td>References</td><td>36</td></tr></table>
# I. INTRODUCTION: THE VECTOR POTENTIAL IN GAUGE THEORY

The vector potential has long been thought of as an auxiliary quantity until Aharonov and Bohm pointed out that it can produce an observable effect [1], shown schematically on FIG. 1: electrically charged classical particles move freely outside a cylinder carrying a

<figure><img src="image_2.png" /><figcaption>FIG. 1: Schema of the Abelian Aharonov-Bohm experiment [9].</figcaption></figure>

magnetic flux, however the curl-free vector potential $A = A_{\mu}dx^{\mu}$ allows for a quantum effect. The experimental confirmation of the Aharonov-Bohm prediction [2–8] led Wu and Yang to argue in their seminal paper [9] that:

(a) The field strength $F_{\mu\nu}$ under-describes (b) the phase $\oint A_{\mu}dx^{\mu}$ over-describes electromagnetism, What provides a complete description that is neither too much nor too little is,

$$ \mathfrak{F}^{AB} = \exp \left[ -i \oint A_{\mu} dx^{\mu} \right]. \qquad (I.1) $$

[...] Electromagnetism is the gauge-invariant manifestation of a nonintegrable phase factor.

Here we took unit electric charge; $\mathfrak{F}^{AB} = e^{-i\Phi}$ where $\Phi = \oint A_{\mu}dx^{\mu}$ is the magnetic flux enclosed by a loop which goes around the cylinder once counter-clockwise $^1$. Dropping the $z$ direction we choose the gauge where $A = A_{\mu}dx^{\mu} = \alpha d\vartheta$. Shifting $\alpha$ by an integer,

<footnote>$^1$ In geometric language, such a loop generates the first homotopy group $\pi_1 \simeq \mathbb{Z}$ of the space from which the cylinder has been removed [10–17]. All values of the $\mathfrak{F}$ span the holonomy group [18, 19].</footnote>
$\alpha \to \alpha + N$, shifts the magnetic flux, $\Phi \to \Phi + 2N\pi$, but yields identical Wu-Yang factors,

$$ \Phi^{AB} = -2\pi\alpha - 2N\pi \Rightarrow \mathcal{F}_{\alpha}^{AB} = e^{i2\pi\alpha}. \qquad (I.2) $$

Two vector potentials are therefore physically equivalent whenever their fluxes differ by an even multiple of $\pi$ [1–3], $\Phi^{(1)} - \Phi^{(2)} = 2N\pi$, $N = 0, \pm 1, \dots$ [5–8].

The non-Abelian generalisation of electromagnetism was proposed by Yang and Mills (YM) in 1954 [20]. Twenty years later, Wu and Yang [9] underlined the rôle of the non-Abelian generalisation of the phase factor (I.1),

$$ \mathcal{F}^{NA} = \exp[-\Phi] = \exp \left[ - \oint A_{\mu} dx^{\mu} \right]. \qquad (I.3) $$

This expression, which will hence be referred to as the Wu-Yang factor, looks as the Abelian expression (I.2) except for that $A_{\mu} = (A_{\mu}^{a})$ is the Lie algebra valued Yang-Mills potential, and path-ordering is understood $^2$.

To verify their proposal Wu and Yang suggested, by analogy with the electromagnetic case, an experimentum crucis. Quoting again from [9], sec. VII, p. 3855 :

...many theorists ... believe that SU(2), gauge fields do exist. However, so far there is no experimental proof, since conservation of isotopic spin only suggests, and does not require, the existence of an isotopic spin gauge field. What kind of experiment would be a definitive test of the existence of an isotopic spin gauge field? A generalized Bohm-Aharonov experiment would be. One constructs the cylinder of material for which the total $I_z$ spin is not zero, e.g., a cylinder made of heavy elements with a neutron excess. One spins the cylinder around its axis, ... setting up a "magnetic "flux inside the cylinder along the $I_z$ "direction" (see Fig.1).

...one imagines a rotating a cylinder which has an average $\langle \vec{I} \rangle$ which is [...] not in the $I_z$ direction [with] a magnetic flux would then be set up which is in a "direction" other than $I_z$. Scattering a beam of protons would then produce some neutrons as well as protons. This implies, of course, that there is electric charge transfer between the beam and the cylinder together with the gauge field around it.

<footnote>$^2$ Greek indices $\mu, \nu$ etc refer to space-time with $i, j$ etc space indices. Latin indices $a, b$ etc refer to the gauge group.</footnote>
In this paper we argue that while there is no classical electromagnetic Aharonov-Bohm effect, however in the non-Abelian case a “shadow” of the quantum effect, namely *isospin precession* [9, 22, 35–37] can be recovered at the (*pseudo*) *classical level*. The difference with the Abelian case comes from that a particle with non-Abelian structure carries an additional, internal degree of freedom represented by an *isospin vector* $\vec{Q} = (Q^a)$ [or equivalently, by a matrix in the Lie algebra, $su(2)$, of the gauge group] which couples the particle to the YM field. The clue is that, unlike the electric charge, the isospin has *non-trivial dynamics* rather than being a mere constant [55–64, 66]. Then our first result says:

**Theorem I.1.** *The isospin of a (pseudo) classical particle with internal structure in the non-Abelian Aharonov-Bohm field precesses around the non-Abelian flux in isospace.*

The original Wu-Yang proposal does not appear practical $^3$. Therefore the Wu-Yang proposal has long been view as a Gedankenexperiment and the study was focused at particles coupled to a pure SU(2) Yang-Mills field with no electromagnetic interaction $^4$. However, as emphasised by Yang and Mills already [20],

> "...when electro-magnetic interactions can be neglected, [...] the orientation of the isotopic spin is of no physical significance. The differentiation between a neutron and a proton is then a purely arbitrary process. However, [...] this arbitrariness is subject to the following limitation: [...] once one chooses what to call a proton, what a neutron at one space-time point, one is then not free to make any choices at other space-time points."

However the Wu-Yang suggestion that incoming protons could be turned into neutrons raises deep questions, as it leads to conflict with electric charge conservation [9, 22, 27–29] $^5$. When should we call a particle a proton or a neutron? Following Yang and Mills, we posit:

**Definition I.1.** *The definition of protons and neutrons should be gauge covariant. A proton must have charge 1 and a neutron must have charge 0.*

<footnote>$^3$ Zeilinger et al. [21] did actually scatter a neutron beam on a rotating rod of $^{138}$U, setting up an upper limit $10^{-15}$ on the strength of the non-Abelian effect when compared to its electromagnetic counterpart.</footnote>

<footnote>$^4$ Ignoring the electric charge is reminiscent of von Neumann’s “dry water”, though.</footnote>

<footnote>$^5$ *The question of the referee of [22] about charge conservation which turned out to be related to the so-called “color problem” [23–28] led subsequently to [29].</footnote>
Putting the Yang-Mills argument above into practice, we attach a reference frame to each point of spacetime such that satisfies the criteria above and then allows us to distinguish protons and neutrons.

Our paper is organized as follows. After outlining the general theory, we focus our attention at the pseudoclassical description of the Non-Abelian Aharonov-Bohm Effect.

Then in sec. VI we derive conserved charges, associated with internal symmetries. Remarkably, non-Abelian Aharonov-Bohm field admits two types of the latter: a trivial one and a “twisted” one, which arise when the enclosed flux is quantized in even resp. in odd integers, see eqn. (VI.9) [27–29].

Important conceptual and experimental progress has been made since those times, though [30–41]. The developments [42–51] which concern effective non-Abelian gauge fields change the assessment dramatically, opening the way to laboratory testing.

## II. THE WU-YANG PHASE FACTOR

Hence we limit our investigations to $G = \text{SU}(2)$ Yang-Mills fields. The Lie algebra $\mathfrak{G} = \text{su}(2)$ is generated by the Pauli matrices,

$$ \sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad (II.1) $$

whose commutation relations are $[\sigma_a, \sigma_b] = 2i\epsilon_{abc}\sigma_c$. The correspondance between 3-vectors $\vec{Q}$ and $\text{su}(2)$ matrices $Q$ is,

$$ \vec{Q} \equiv \begin{pmatrix} Q^1 \\ Q^2 \\ Q^3 \end{pmatrix} \Leftrightarrow Q^1 \frac{\sigma_1}{2i} + Q^2 \frac{\sigma_2}{2i} + Q^3 \frac{\sigma_3}{2i} = \frac{1}{2i} \begin{pmatrix} Q^3 & Q^1 - iQ^2 \\ Q^1 + iQ^2 & -Q^3 \end{pmatrix} \equiv Q. \qquad (II.2) $$

The vector and matrix forms will be used alternatively. The gauge group $G$ is implemented according to,

$$ g \cdot \vec{Q} \quad \text{i.e.,} \quad (g^{1/2})^{-1} Q g^{1/2}, \qquad (II.3) $$

where $g^{1/2}$ is square-root of the $\text{SU}(2)$ matrix $g = (g_{ab})$ and $g \cdot \vec{Q}$ means the action of $\text{SO}(3) = \text{SU}(2)/\mathbb{Z}_2$ matrix on the 3-vector $\vec{Q}$. The Yang-Mills vector potential and field strength [20] are given by, $A \equiv A_\mu dx^\mu$ and $F = \frac{1}{2}F_{\mu\nu}dx^\mu dx^\nu$, respectively, where $A_\mu = (A_\mu^a)$ and $F_{\mu\nu} = (F_{\mu\nu}^a)$ are $\mathfrak{G} = \text{su}(2)$ matrices. $A_\mu \to g^{-1}A_\mu g + g^{-1}\partial_\mu g$, $F_{\mu\nu} \to g^{-1}F_{\mu\nu}g$ under $\text{SU}(2)$-valued gauge transformations [20].
We focus henceforth our attention at the Non-Abelian Aharonov-Bohm Effect. A non-Abelian flux is confined to a thin cylinder outside of which $F_{\mu\nu} = 0$. Dropping the irrelevant $z$ coordinate, we consider a flat SU(2) Yang-Mills potential $A = A_\mu dx^\mu$, $DA = 0$, on the punctured plane $\mathbb{R}^2 \setminus \{0\}$. Such Yang-Mills fields are characterized by the non-Abelian generalization of the non-integrable phase factor (I.1), defined, for an arbitrary path $\gamma$, as

$$ \mathcal{F}^{NA}(\gamma) = \exp \left[ - \int_{\gamma} A_{\mu} dx^{\mu} \right] \in \mathrm{SU}(2), \qquad (II.4) $$

with path-ordering understood [9]. Was the gauge field strength non-zero, $F_{\mu\nu} \neq 0$, the result would depend on the path. However in the non-Abelian Aharonov-Bohm case the non-Abelian version of Stokes' theorem [52] implies that, owing to $DA = 0$, (II.4) is a homotopy invariant [29], i.e., it takes the same values when $\gamma$ is continuously deformed while keeping its end points fixed. For a generating loop of $\pi_1 \approx \mathbb{Z}$ of the punctured plane which starts and ends at some point $x_0$ (II.4) depends on the choice of $x_0$: choosing a different $x_0$ results by conjugation by a fixed element of the gauge group. Keeping this fact in mind, we shall write simply $\mathcal{F} = \exp[-\Phi] = \exp[-\oint A_\mu dx^\mu]$ as in (I.3). The suffix “NA” will be dropped henceforth and the non-Abelian character tacitly assumed.

In the gauge we shall call diagonal the non-Abelian vector potential is in polar coordinates $(r, \vartheta)$,

$$ A_r = 0, \quad A_\vartheta = a \frac{\sigma_3}{2i}. \qquad (II.5) $$

The generated non-Abelian flux<fnref n="6" /> and Wu-Yang factor are,

$$ \Phi = \oint A_i dx^i = -i\pi a \sigma_3, \qquad (II.6a) $$

$$ \mathcal{F} = \exp[-\Phi] = \begin{pmatrix} e^{i\pi a} & 0 \\ 0 & e^{-i\pi a} \end{pmatrix}, \qquad (II.6b) $$

respectively. It follows that gauge potentials which differ by an *even* integer,

$$ a \text{ and } a + 2k, \quad k = 0, \pm 1, \dots \qquad (II.7) $$

i.e., whose non-Abelian fluxes are related as

$$ \Phi \text{ and } \Phi - 2k\pi i \sigma_3, \qquad (II.8) $$

<footnote n="6">⁶ Our non-Abelian flux $\Phi$ is Wu-Yang's $I$ [9].</footnote>
respectively, define identical Wu-Yang factors and are therefore physically equivalent. For an arc between $\vartheta_0$ and $\vartheta_0 + \Delta\vartheta$

$$ \mathcal{F}(\varphi) = \exp \left[ -\Delta\vartheta \mathfrak{a} \frac{\sigma_3}{2i} \right], \qquad (II.9) $$

is a rotation by angle $\Delta\vartheta \mathfrak{a}$ around the 3rd axis in isospace. For a full circle we recover the Wu-Yang factor, $\mathcal{F}(2\pi) = \mathfrak{F}$ in (II.6). The non-Abelian Wu-Yang factor (II.6b) can be presented in terms of its Abelian counterparts (I.2),

$$ \mathcal{F}^{NA} \equiv \mathfrak{F} = \begin{pmatrix} (\mathfrak{F}^{AB}_{(\alpha/2)}) & 0 \\ 0 & (\mathfrak{F}^{AB}_{(-\alpha/2)}) \end{pmatrix}. \qquad (II.10) $$

Intuitively: “the number of components doubled but the exponents got halved, $2\alpha \to \mathfrak{a}$” when compared to (I.2).

The Non-Abelian Phase Factor $\mathfrak{F}$ transforms covariantly under a gauge transformation, $\mathfrak{F}(x_0) \to g^{-1}(x_0) \mathfrak{F}(x_0) g(x_0)$, and conversely: two curl-free vector potentials $A^{(1)}$ and $A^{(2)}$ can be shown to be gauge-equivalent if and only if their Wu-Yang factors are conjugate [29?],

$$ \mathfrak{F}^{(1)} = h^{-1} \mathfrak{F}^{(2)} h \qquad (II.11) $$

for some fixed $h \in SU(2)$. In conclusion, we confirm:

**Theorem II.1.** The gauge-equivalent flat YM potentials are in 1-1 correspondence with the Wu-Yang factors up to conjugation by some fixed element of the gauge group.

### III. (PSEUDO)CLASSICAL ISOSPIN

The additional concept we needed is that of classical isospin, represented by a 3-vector $\vec{Q}$ or alternatively, by an $SU(2)$ matrix, eqn. (II.2). Limiting ourselves at a non-relativistic study in flat space, the equations of motion of a classical isospin-carying particle put forward by Kerner [55], and by Wong [56], and further discussed in [57–68] are,

$$ \ddot{x}_{\mu} = -Q^{a} F_{\mu\nu}^{a} \dot{x}^{\nu}, \qquad (III.1a) $$

$$ D_t Q \equiv \dot{Q} + [A_{\nu} \dot{x}^{\nu}, Q] = 0, \qquad (III.1b) $$

where the “dot” means $d/dt$ with $t$ the non-relativistic time. The first of these is a generalized Lorentz-force equation, and the second says that the isospin is parallel transported
along the space-time trajectory. The latter is gauge invariant whereas that of the isospin transforms covariantly when gauge transformations are implemented in the adjoint representation,

$$Q \rightarrow g^{-1}Qg \quad \Rightarrow \quad D_tQ \rightarrow g^{-1}D_tQg. \qquad (III.2)$$

Moreover, deriving $|Q|^2 = Q^aQ^a$ and using the isospin equation (III.1b) implies that the length of the isospin is a constant of the motion,

$$|Q|^2 = |Q_0|^2 = \text{const}, \qquad (III.3)$$

where $Q_0$ is an arbitrary point in an adjoint orbit $\mathcal{O} = \{g^{-1}Q_0g | g \in G\}$ of the gauge group [62–69]. For $G = \text{SU}(2)$ the orbit is a 2-sphere $S^2$ of radius $|Q_0|$ in isospace.

## A. Internal charges and symmetries

As emphasised by Yang and Mills and recalled in the Introduction, the distinction between protons and neutrons is based on their charges. The requirements are satisfied by choosing, in each point, a “direction field” in the Lie algebra $\mathfrak{g} = \text{su}(2)$, represented by vector $\hat{\Psi}$ of unit length, $|\hat{\Psi}| = 1$, which should behave under a gauge transformation as a (normalized) adjoint Higgs field,

$$\hat{\Psi} \rightarrow g^{-1}\hat{\Psi}g. \qquad (III.4)$$

Once a choice has been made, the electric charge of our particle will be defined as the projection of the isospin onto $\hat{\Psi}$,

$$q = -2 \text{tr}(Q \cdot \hat{\Psi}) = Q^a \hat{\Psi}^a. \qquad (III.5)$$

Its gauge invariance follows from combining (III.4) with that of the isospin vector, (III.2). But is it dynamically conserved? Before studying the Non-Abelian Aharonov-Bohm Effect, let’s first consider a different but similar problem, namely that of motion in the field of a Wu-Yang monopole [70],

$$A_i^a = \epsilon_{iak} \frac{x_k}{r^2}. \qquad (III.6)$$

This “hedgehog” Ansatz identifies the internal isospace, $\text{su}(2)$, with the external 3-space. Then projecting the isospin onto the radial direction,

$$\hat{\Psi}^a = \frac{x_a}{r}. \qquad (III.7)$$
The auxiliary direction field $\hat{r} = \frac{x_a}{r}$ behaves as an adjoint Higgs field in the particular “hedgehog” gauge and then (III.5) allows us to recover the charge,

$$q = Q^a \frac{x_a}{r}, \qquad (III.8)$$

whose conservation follows from $D_\mu \hat{\Psi} = 0$.

The charge (III.5) generalized to non-Abelian monopoles is conserved outside the monopole core, where $\hat{\Psi}$ is the direction of the Higgs field, which for large $r$ becomes covariantly constant [52, 71, 72].

More generally, when a covariantly constant “direction field” $\hat{\Psi}$ does exist then we posit, consistently with what we said in the introduction:

**Definition III.1.** The particle is a proton if its internal charge (III.5) is 1, and it is a neutron if its charge is 0.

The charge-based proton/neutron distinction makes us wonder: when do we get an ordinarily, and not only covariantly, $D_t(\cdot) = 0$, conserved charge,

$$\dot{q} = 0? \qquad (III.9)$$

The parallel-transport equation (III.1b) implies at once that this is so when $\hat{\Psi}$ is covariantly constant,

$$D_{\mu} \hat{\Psi} = \partial_{\mu} \hat{\Psi} + [A_{\mu}, \hat{\Psi}] = 0. \qquad (III.10)$$

However diatomic molecules [35–37] provide us with an example where no covariantly constant direction fields and thus no conserved charges. Our investigations in sec. VI then show that this is also what happens in the Non-Abelian Aharonov-Bohm Effect — with the remarkable exception when the non-Abelian flux is quantized as in (VI.9).

## B. Conserved quantities

Conserved quantities are usually derived from space-time symmetries by Noether’s theorem [73, 74]. Below we argue that the charge (III.5) is generated by an internal symmetry [27, 28]. We first recall some general facts [66–68, 74–78]. A vectorfield $X$ is a symmetry for a gauge field if the variation of the vector potential can be compensated by a gauge transformation,

$$L_X A = DW \quad \text{or equivalently} \quad F(X, \cdot) = D\psi \quad \text{where} \quad \psi = W - A(X) \qquad (III.11)$$
for some Lie-algebra valued fields W or $\psi$, respectively [75–78]. Then the adapted form of the Noether theorem allows one to derive conservation laws [66, 74, 78].

As a first illustration, we consider the angular momentum in the field of a Wu-Yang monopole, (III.6), for which the general formulæ yield the conserved total angular momentum [37, 79, 80],

$$ \vec{J} = \vec{x} \times \vec{\pi} - q\hat{\vec{r}} \qquad (III.12) $$

where $q$ is the conserved charge (III.8). This expression is analogous to the one in the field of a Dirac monopole [36, 66–68, 74, 78–82]. Eliminating $\vec{\pi}$ in favor of $m\vec{x} = \vec{p} = \vec{\pi} + \vec{A}$, (III.12) $\vec{x} \times \vec{A}$ and $q\hat{\vec{r}}$ combine into the isospin vector, and the angular momentum is separated into the sum of orbital and internal terms,

$$ \vec{J} = \vec{L} - \vec{Q}, \quad \text{where} \quad \vec{L} = \mathbf{r} \times \vec{p}, \qquad (III.13) $$

highlighting the “spin from isospin” phenomenon [74, 78, 81–83]. Similar formulæ will show up again in sec.VI C.

Now we spell out the general theory for internal symmetries and charges in the non-Abelian Aharonov-Bohm context. We first consider a general compact and connected gauge group $G$ before returning to $G = SU(2)$.

The symmetry condition (III.11) assumes that the (infinitesimal) transformation is implemented on the fields by the Lie derivative by the vector field $X$ on the spacetime. But a gauge transformation acts on space-time trivially, $X = 0$; Then how does it act on fields ? And if it has been implemented, is it a symmetry ? The answer, which generalizes what we said in sec.III A for the electric charge, proceeds in three steps [27, 28] :

1. Question : Which elements of the gauge group can be implemented on the gauge field, i.e., for which the action of a $g \in G$ on the gauge potential can be defined ?

2. Question : Let $K$ be an implementable subgroup of the gauge group $G$ and $k \in K$. Does it act as a symmetry ?

3. Question : What is the conserved quantity associated with a vector $\kappa$ of the Lie algebra $\mathfrak{R}$ of the internal symmetry group $K$ ?

Coming to details,
• A subgroup K of the gauge group G is said implementable if there exists, in each point x of spacetime, an automorphism of K into of the gauge group G,

$$ \Omega_{\{\cdot\}}(x) : K \to G \quad \text{such that} \quad \Omega_{k_1 k_2}(x) = \Omega_{k_1}(x) \Omega_{k_2}(x). \qquad (\text{III.14}) $$

Such a mapping can be made to act on the gauge potential according to <fnref n="7" />,

$$ k \cdot A_{\mu}(x) = \Omega_{k}(x) A_{\mu}(\Omega_{k}(x))^{-1} - \partial_{\mu} \Omega_{k}(x) (\Omega_{k}(x))^{-1}. \qquad (\text{III.15}) $$

If the underlying space has non-trivial topology, a topological obstruction against implementing certain elements of the gauge group can arise, and K will be a proper subgroup of G. This is what happens for non-Abelian monopoles [23–28]. For the Non-Abelian Aharonov-Bohm Effect no such obstruction does arise, though [27, 28], therefore the general theory will be skipped; the interested reader is advised to check the literature. However, a (sub)group K can have different inequivalent implementations, – and it is what happens for the Non-Abelian Aharonov-Bohm Effect as it will be shown in sec. VI.

Returning to the general theory, let us assume that $K \subset G$ is an implementable subgroup. The infinitesimal action of a vector $\kappa \in \mathfrak{K}$, the Lie algebra of K, is then a $\mathfrak{G}$-valued Higgs-type field $\hat{\Psi}_{\kappa}(x)$ normalized as $|\hat{\Psi}_{\kappa}| = 1$ which acts on the gauge potential by covariant derivative [27, 28, 74–78],

$$ \hat{\Psi}_{\kappa} \cdot A_{\mu} = D_{\mu} \hat{\Psi}_{\kappa}. \qquad (\text{III.16}) $$

Then the answers to the questions above are:

• $\kappa$ is an infinitesimal symmetry when $\omega_{\kappa}$ is covariantly constant

$$ D_{\mu} \hat{\Psi}_{\kappa} = 0. \qquad (\text{III.17}) $$

• The Noether theorem applied to an internal symmetry then says that

$$ q_{\kappa} = -2 \operatorname{tr}(Q \cdot \hat{\Psi}_{\kappa}) = Q^{a} \hat{\Psi}_{\kappa}^{a} \qquad (\text{III.18}) $$

is ordinarily conserved, $\dot{q}_{\kappa} = 0$ generalizing (III.5).

<footnote n="7">⁷ For comparison: a gauge transformation by $k = k(x)$ would act instead as $A_{\mu} \to k^{-1}A_{\mu}k + k^{-1}\partial_{\mu}k$. In bundle language, the gauge group acts from the right, while the implementation (III.15) acts from the left [28].</footnote>
For aficionados of fiber bundles and differential geometry [18, 19, 27, 28], we first recall two definitions:

(i) the centralizer of a subgroup $K \subset G$ is defined as all $g$ elements of $G$ which commute with all $k \in K$, i.e., whose adjoint action leaves all $k \in K$ invariant,

$$Z_G(K) = \{g^{-1}kg = k \mid \text{for all } k \in K\}. \qquad (\text{III.19})$$

The centralizer of $K = G$ itself is thus the center, $Z(G)$, of $G$.

(ii) Using fiber bundle language [18, 19], the horizontal lift, obtained by calculating the integral (I.3) of the YM potential along a loop, ends in the same fiber it started from, and can therefore be obtained by (right) action of a group element. The holonomy group consists of all possible values of the Wu-Yang factor (I.3).

In the non-Abelian Aharonov-Bohm context, the holonomy group consists of integer powers of the Wu-Yang factor (I.3), calculated for an arbitrary loop which generates $\pi_1 \approx \mathbb{Z}$ of the punctured plane. Then we quote PROP. 4.3 of [28] :

**Theorem III.1.** A subgroup $K$ of the gauge group $G$ is an internal symmetry group for the YM field $A = A_\mu dx^\mu$ if and only if the centralizer of $K$ in $G$, $H = Z_G(K)$, contains the holonomy group of $A$. The bundle then reduces to one with structure group $H = Z_G(K)$ obtained by solving the internal symmetry equation (III.17). The entire $G$ is itself an internal symmetry if and only if the holonomy group belongs to the center, $Z(G)$.

In sec.VI we show that, in the non-Abelian Aharonov-Bohm context, the symmetry condition (III.17) may have several inequivalent solutions.

## IV. ISOSPIN PRECESSION

As emphasised in the introduction, there is no classical Aharonov-Bohm effect in the Abelian theory : charged particles move freely in the curl-free magnetic potential. The effect is purely quantum mechanical and is related to the phase freedom [1–3, 5–7, 69].

What about the non-Abelian case ? This is the question to which this paper is devoted.

The isospin couples to the non-Abelian field [55, 56] and the results obtained in an analogous but different context, namely for diatomic molecules [35, 36] hint at isospin precession, recovered also at the (pseudo-) classical level [37].
In this paper we find (pseudo-) classical isospin precession also in the non-Abelian Aharonov-Bohm context.

Focusing our attention at the gauge group $G = SU(2)$ we consider a non-Abelian flux confined to a thin cylinder upon which particles with isospin are scattered, as illustrated schematically in FIG.1 and, in more detail, in FIG.2.

We shall need a first but important result. Unlike as for monopoles [74, 78–82], the orbital angular momentum alone,

$$ \vec{L} = \vec{x} \times \dot{\vec{x}}, \qquad (IV.1) $$

(for unit mass) is conserved, for all values of the non-Abelian flux, in the non-Abelian Aharonov-Bohm context. This is seen directly by using (III.1a) with $F \equiv 0$: $\dot{\vec{L}} = \dot{\vec{x}} \times \dot{\vec{x}} + \vec{x} \times \ddot{\vec{x}} = 0$. The sign of $\vec{L}$ will be used to distinguish the half-planes in the two sides of the flux, as illustrated schematically in FIG.2. Further aspects of the angular momentum will be studied in sec. VIC.

<figure><img src="image_3.png" /><figcaption>FIG. 2: Scattering of a nucleon on a non-Abelian flux $\vec{\Phi}$, whose direction is perpendicular to the proton/neutron axis. The isospin is rotated by an angle $\pm \Delta\vartheta$ a with the sign depending on the side the particles passes, i.e., the sign, $L > 0$ or $L < 0$, 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 (II.9).</figcaption></figure>

The classical dynamics is given by the Kerner-Wong equations (III.1). Outside the cylin-
der the Yang-Mills field is flat, $F = DA = 0$, therefore the generalized Lorentz force in (III.1a) vanishes and the space-time motion is free, just like in the Abelian case.

As to the isospin, we first remark that, in the free case, $a = 0$ in the diagonal gauge (II.5), therefore the flux $\vec{\Phi}$ vanishes and the isospin eqn. (III.1b) implies that,

$$Q(t) = Q_0 = \text{const.} \qquad (\text{IV.2})$$

However switching to another gauge (III.2) by an arbitrary gauge transformation $g(x^\mu)$,

$$Q \rightarrow \tilde{Q}(t) = g^{-1}(x^{\mu}(t)) \tilde{Q}_{0} g(x^{\mu}(t)), \qquad (\text{IV.3})$$

the trajectory will look substantially different — but this is a mere gauge artifact.

Do we find genuine, non-gauge-artifact classical motions ?

When the non-Abelian flux $\Phi$ and the incoming isospin $\vec{Q}_{in}$ are parallel in internal isospace (considered by Wu and Yang) the answer is negative by (III.1b). However in the second case proposed in [9], i.e., when $\vec{Q}_{in}$ and the non-Abelian flux $\vec{\Phi} = (\Phi^a)$ are not parallel in isospace, then genuine internal motion can be found, as we show now.

To explain how this comes about, let us assume for simplicity that $\vec{Q}_{in}$ is perpendicular to the flux in isospace, $Q_{in}^3 = 0$, as depicted in FIG. 2.

First of all, we fix our terminology: consistently with the Yang-Mills argument [20] quoted in the introduction we shall call, in the diagonal gauge, the particle incoming from far on the right,

$$\left\{ \begin{array}{ll} \text{proton if } Q_{in}^p = \frac{1}{2i}\sigma_1 = \frac{1}{2i}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} & (\text{IV.4}) \\ \text{neutron if } Q_{in}^n = \frac{1}{2i}\sigma_2 = \frac{1}{2}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} & \end{array} \right. $$

which both have unit length and are orthogonal to each other w.r.t. the scalar product (III.18),

$$-2 \operatorname{tr}(Q_{in}^2) = 1 \quad \text{and} \quad \operatorname{tr}(Q_{in}^n \cdot Q_{in}^p) = 0. \qquad (\text{IV.5})$$

This fixes the choice at one point – and, as Yang and Mills say [20], at all other points by gauge covariance.

Now we study the motion of the isospin. Recall first that (III.3) implies that $|Q| = \text{const}$, therefore $Q$ moves on a circle. Eliminating $t$ in favor of the polar angle $\vartheta$, $\dot{\vartheta}$ drops out from the parallel transport equation (III.1b), leaving us with $\frac{dQ}{d\vartheta} + [A_\vartheta, Q] = 0$. It follows that
$Q^3 = \text{const.}$ We put $Q^3 = 0$ for simplicity. Let us assume that the incoming particle is a proton, $Q_{in} = Q_{in}^p$. First consider $L > 0$. Then

$$Q_{+}^{p}(\varphi) = \cos(\varphi\mathfrak{a}) \frac{\sigma_1}{2i} - \sin(\varphi\mathfrak{a}) \frac{\sigma_2}{2i} \quad \text{i.e.,} \quad Q_{+}^{p}(\varphi) = \frac{1}{2i} \begin{pmatrix} 0 & e^{+i\varphi\mathfrak{a}} \\ e^{-i\varphi\mathfrak{a}} & 0 \end{pmatrix}, \qquad (\text{IV.6})$$

where we shifted to $\varphi = \vartheta - \pi$, $0 \le \varphi \le \pi$. The initial position, far on the left, has $\varphi_{in} \approx 0$. A similar formula works for $L < 0$ with $\varphi = \pi - \vartheta$ which changes just a sign,

$$Q_{-}^{p}(\varphi) = \cos(\varphi\mathfrak{a}) \frac{\sigma_1}{2i} + \sin(\varphi\mathfrak{a}) \frac{\sigma_2}{2i}, \quad \text{i.e.,} \quad Q_{-}^{p}(\varphi) = \frac{1}{2i} \begin{pmatrix} 0 & e^{-i\varphi\mathfrak{a}} \\ e^{+i\varphi\mathfrak{a}} & 0 \end{pmatrix}. \qquad (\text{IV.7})$$

The motion of the isospin can be viewed as a rotation of the isospin around the 3rd axis,

$$\vec{Q}_{out}^{p} = \begin{cases} R_{+\varphi\mathfrak{a}} \vec{Q}_{in}^{p} & \text{for } L \ge 0 \\ R_{-\varphi\mathfrak{a}} \vec{Q}_{in}^{p} & \text{for } L \le 0 \end{cases}, \qquad (\text{IV.8})$$

as suggested by Kerner [55], – but the rotation is in opposite directions as shown in FIGs.2-3.

The intuitive picture of the opposite shifts suffered on the two sides is : sailing downwind resp. upwind shifts the phase by the relative phase difference.

Notice however that for $\varphi > 0$ the upper and lower values of the isospin in (IV.6) and (IV.7), respectively, do not match unless $\mathfrak{a}$ is an integer,

$$\mathfrak{a} = 2k \quad \text{or} \quad \mathfrak{a} = 1 + 2k. \qquad (\text{IV.9})$$

Similarly for an incoming neutron, $Q_{in}^n = \frac{\sigma_2}{2i}$, we get,

$$Q_{\pm}^{n}(\varphi) = \cos(\varphi\mathfrak{a}) \frac{\sigma_2}{2i} \mp \sin(\varphi\mathfrak{a}) \frac{\sigma_1}{2i} \quad \text{i.e.,} \quad Q_{\pm}^{n}(\varphi) = \frac{1}{2} \begin{pmatrix} 0 & -e^{\pm i\varphi\mathfrak{a}} \\ e^{\mp i\varphi\mathfrak{a}} & 0 \end{pmatrix}. \qquad (\text{IV.10})$$

The formulae (IV.6)-(IV.7) and (IV.10) will be recovered in sec.IV, # (VII.11) using a different framework.

## V. ISOSPIN SCATTERING

We are be particularly interested by the value of the isospin far on the right, at $\varphi_{out} = \pi$, shown in FIGs.2-3-4. In other words, we want to study the Non-Abelian Aharonov-Bohm
<figure><img src="image_4.png" /><figcaption>FIG. 3: The “isospin clockwork” for non-Abelian flux $\Phi_a = -i\pi a\sigma_3$, ($a \neq 0$). The incoming proton (at $L_{\pm}$) is aligned with $\vec{p} = \sigma_1/2i$ (at “6 o’clock”), $Q_{in}^P = \vec{p}$. For $L > 0$ the isospin rotates clockwise, but for $L < 0$ it rotates counter-clockwise. For odd integer flux ($a = 1$, 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 $a = \frac{1}{2}$ (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 $a$.</figcaption></figure>

Effect from the viewpoint of scattering theory [22]. In terms of the polar angle, $\Delta\vartheta = \pm\pi$,

$$Q_{out}^{p} = \frac{1}{2i} \begin{pmatrix} 0 & e^{\pm i\pi a} \\ e^{\mp i\pi a} & 0 \end{pmatrix}, \quad Q_{out}^{n} = \frac{1}{2} \begin{pmatrix} 0 & -e^{\pm i\pi a} \\ e^{\mp i\pi a} & 0 \end{pmatrix}, \qquad (V.1)$$

with the sign depending on which side of the flux line had the particle passed, correlated in turn with the sign of the vertical component, $L \equiv L_z$, of the conserved orbital the angular momentum $\vec{L} = \mathbf{x} \times \dot{\mathbf{x}}$ in (IV.1). The rotations in (V.1) [or more generally in (IV.6)-(IV.7)] on the two sides do not match for $L \to 0$ in general, though. However in the non-trivial exceptional case $a = 1 + 2k$, for example, the outgoing result becomes side-independent,

$$Q_{out}^{p} = -\frac{1}{2i}\sigma_{1} = -Q_{in}^{p} \qquad (V.2)$$
<figure><img src="image_5.png" /><figcaption>FIG. 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.3. The incoming neutron (in purple and green) is along $\hat{z}$ (at “3 o’clock”), $Q_{in}^n = \sigma_2/2i$ on both sides of the flux. For $L > 0$ the isospin rotates clockwise for both $Q^p$ and $Q^n$ and for $L < 0$ 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 $S = -I$ in (V.4). The centers of the “shifting clocks” are at $\varphi = 0, \pi/2, \pi$.</figcaption></figure>

consistently also with $R_{\pm\pi} = -I$ in (IV.8) and shown in FIG.4. More generally, the isospin is reversed in both sectors. The rôle of the minus sign here will be explained in sec.III A.

Eqns. (IV.6)-(IV.7) with $\Delta\theta = \pi$ allow us to express the result in terms of the Wu-Yang factor (II.6) in matrix resp. in vector form [22],

$$Q_{out}^{p} = (\mathfrak{F})^{-1/2} Q_{in}^{p} (\mathfrak{F})^{1/2} \quad \text{i.e.,} \quad \vec{Q}_{out}^{p} = \mathfrak{F} \vec{Q}_{in}^{p}. \qquad (V.3)$$

These formulae, consistent with (II.3), confirm that the isospin is rotated by $\mathcal{F}(\varphi)$ cf. (II.9), as shown in FIGs.3 and 4. They are gauge-covariant when the incoming direction also transforms covariantly, $Q_{out}^p = \mathfrak{F} Q_{in}^p \rightarrow (g^{-1}\mathfrak{F}g)(g^{-1}Q_{in}g) = g^{-1}(\mathfrak{F}Q_{in})g = g^{-1}Q_{out}g$. For both exceptional values $a = 2k$ and $a = 1 + 2k$ the phase difference accumulated on the two sides is $\pm 2\pi$, implying that the non-Abelian fluxes differ on by an even integer and
thus yield identical Wu-Yang factors,

$$ \mathcal{F} = \begin{cases} \mathbb{I} & \text{when } \mathfrak{a} = 2k \\ -\mathbb{I} & \text{when } \mathfrak{a} = 1 + 2k \end{cases}. \qquad (V.4) $$

Therefore $\vec{Q}_{out} = \pm \vec{Q}_{in}$ on both half-planes, (V.2), as shown in FIGs. 3 and 5.

The non-integrable phase factor for the half-loop (II.9) is the square root of the Wu-Yang factor (II.6),

$$ \mathcal{F}(\pm\pi) = (\mathcal{F})^{\pm 1/2} = \begin{pmatrix} e^{\pm i\pi(\frac{1}{2}\mathfrak{a}+k)} & 0 \\ 0 & e^{\mp i\pi(\frac{1}{2}\mathfrak{a}+k)} \end{pmatrix}, \qquad (V.5) $$

where $k$ is an integer $^8$. This expression is explained as follows. The Wu-Yang factor is obtained by integrating along a generating loop of $\pi_1$ put together from two half-loops, starting from $\vartheta = -\pi$, going first in the $L > 0$ half-plane to the turning point at $\vartheta = 0$ and then completing the loop by returning to $\vartheta = \pi$ in the $L < 0$ half-plane. Each half-loop contributes equally. Thus $\mathcal{F} = \mathcal{F}(+\pi)\mathcal{F}^{-1}(-\pi) = \text{diag}(e^{i\pi\mathfrak{a}}, e^{-i\pi\mathfrak{a}})$.

Hence we focus our attention at the odd case. Identifying the external and internal coordinates by mimicking monopole theory [52, 70–72], we get a sort of “Aharonov-Bohm hedgehog”. Then the trajectory of the isospin is that of a point on a rolling circle whose center moves on the spatial trajectory, — i.e., a cycloid segment, depicted in FIG.5. In detail,

• For a proton

$$ Q_1^p = |\vec{Q}_0|(\varphi - \sin\varphi), \quad Q_2^p = -|\vec{Q}_0|\cos\varphi, \quad 0 \le \varphi = \vartheta - \pi \le \pi \quad \text{L > 0} \qquad (V.6a) $$

$$ Q_1^p = |\vec{Q}_0|(\varphi + \sin\varphi), \quad Q_2^p = -|\vec{Q}_0|\cos\varphi, \quad 0 \le \varphi = \pi - \vartheta \le \pi \quad \text{L < 0} \qquad (V.6b) $$

• For a neutron

$$ Q_1^n = |\vec{Q}_0|(\varphi + \cos\varphi), \quad Q_2^n = -|\vec{Q}_0|\sin\varphi, \quad 0 \le \varphi = \vartheta - \pi \le \pi \quad \text{L > 0} \qquad (V.7a) $$

$$ Q_1^n = |\vec{Q}_0|(\varphi + \cos\varphi), \quad Q_1^n = |\vec{Q}_0|\sin\varphi, \quad 0 \le \varphi = \pi - \vartheta \le \pi \quad \text{L < 0} \qquad (V.7b) $$

<footnote n="8">⁸ $\mathcal{F}$ in (V.5) acts on the su(2) matrix $Q$ in the adjoint representation. In vector terms, $Q \rightsquigarrow \vec{Q}$, the arguments of the exponential are doubled, cf. (V.3).</footnote>
<figure><img src="image_6.png" /><figcaption>FIG. 5: Isospin motion for $a = 1$ for both protons (V.6) and neutrons (V.7), in both respective half-planes. The trajectories follow cycloid segments, which all satisfy $\vec{Q}_{out} = -\vec{Q}_{in}$.</figcaption></figure>

S-matrix [22].

Eqn (V.3) allows us to consider a pseudo-classical S-matrix defined, separately, on the two half-planes distinguished by the sign of the orbital angular momentum $L$

$$S_{\pm} = \mathcal{F}(\pm\pi) = \begin{pmatrix} e^{\pm i\pi(\frac{1}{2}a+k)} & 0 \\ 0 & e^{\mp i\pi(\frac{1}{2}a+k)} \end{pmatrix} \qquad (V.8)$$

cf. (V.5) and (IV.8). The $\pm$ expressions match for $a = 0$ or $a = 1 \text{ mod. even integers.}$ In conclusion,

**Theorem V.1.** When the incoming isospin $\vec{Q}_{in}$ and the non-Abelian flux $\vec{\Phi}$ are *not* parallel, the isospin vector $\vec{Q}$ *precesses* in opposite directions, depending on which side the particle goes and distinguished by the sign of the orbital angular momentum $L$. The Wu-Yang factor
F (I.3) acts as “classical S-matrix” by rotating the isospin,

$$Q_{out} = S Q_{in} = (\mathfrak{F})^{-1/2} Q_{in} (\mathfrak{F})^{1/2} \qquad (V.9)$$

i.e.,

$$\vec{Q}_{out} = S \vec{Q}_{in} \quad \text{where} \quad S = R_{\pm \pi a}, \qquad (V.10)$$

illustrated in FIGs. 3 and 4. In the odd case i.e., for non-Abelian flux

$$\Phi^{\text{odd}} = (1 + 2k) \Phi_0, \qquad \Phi_0 = \pi \sigma_3 / i, \qquad (V.11)$$

the S-matrix is minus the identity on both sides, cf. (V.4) and FIG.4.

For non-integer values of a the S-matrix will be side-dependent. For a = 1/2 (shown in FIG.3 in magenta and in cyan), for example,

$$Q_{out}^{p}(a = \frac{1}{2}) = \begin{cases} -\frac{\sigma_2}{2i} & \text{for } L > 0 \\ +\frac{\sigma_2}{2i} & \text{for } L < 0 \end{cases} \qquad (V.12)$$

so that the incoming the isospin is rotated by $\mp\pi/2$, depending on the side.

These properties are related to charge conservation, studied in the next section.

## VI. INTERNAL CHARGES

Does the minus signs in (V.2) imply that a proton would have become an “antiproton”?! We argue that no! The reason is gauge covariance. Our clue is that following Yang-Mills [20] the proton/neutron identity of a particle is defined by its charge derived by the Noether theorem, and not by the apparent value of its isospin, which is gauge-dependent. The charge should be defined by setting up a direction field, as we now explain by applying the general theory to the Non-Abelian Aharonov-Bohm Effect outlined in sec. III A.

### A. Implementability and internal symmetry

Implementability. Unlike as for monopoles [24–28], no topological obstruction arises against implementing gauge transformations in the non-Abelian Aharonov-Bohm case. Leaving the mathematical subtleties to the literature, we merely cite the statement proved in [27–29]:
**Theorem VI.1.** In the Non-Abelian Aharonov-Bohm Effect the entire gauge group SU(2) is implementable, namely as in (III.15). However we have two inequivalent implementations, namely a trivial one and a twisted one, defined, in the diagonal gauge, by

$$ \Omega_k^{(1)}(x) = k = (k_{ij}) \quad \text{and} \quad \Omega_k^{(2)}(x) = \begin{pmatrix} k_{11} & e^{i\varphi} k_{12} \\ e^{-i\varphi} \bar{k}_{12} & \bar{k}_{11} \end{pmatrix}, \qquad (\text{VI.1}) $$

where $\det k = |k_{11}|^2 + |k_{12}|^2 = 1$. Infinitesimally,

$$ \hat{\Psi}_{\kappa}^{(1)}(\varphi) = \frac{1}{2i} \kappa = \frac{1}{2i} (\kappa_{ab}) \quad \text{and} \quad \hat{\Psi}_{\kappa}^{(2)}(\varphi) = \frac{1}{2i} \begin{pmatrix} \kappa_{11} & e^{i\varphi} \kappa_{12} \\ e^{-i\varphi} \bar{\kappa}_{12} & -\kappa_{11} \end{pmatrix}, \qquad (\text{VI.2}) $$

respectively, where we normalized to $|\hat{\Psi}| = 1$.

Recalling that $\varphi = \pm(\vartheta - \pi)$, the second implementation can also be written in terms of the polar coordinate $0 \le \vartheta \le 2\pi$ as,

$$ \Omega_k^{(2)}(\vartheta) = \begin{pmatrix} k_{11} & -e^{\pm i\vartheta} k_{12} \\ -e^{\mp i\vartheta} \bar{k}_{12} & -k_{11} \end{pmatrix}, \quad \hat{\Psi}_{\kappa}^{(2)}(\vartheta) = \frac{1}{2i} \begin{pmatrix} \kappa_{11} & -e^{\pm i\vartheta} \kappa_{12} \\ -e^{\mp i\vartheta} \bar{\kappa}_{12} & -\kappa_{11} \end{pmatrix}, \qquad (\text{VI.3}) $$

respectively, with the sign depending that of the orbital angular momentum, L. Reassuringly, for $\kappa = \sigma_1$ we recover an incoming proton,

$$ \hat{\Psi}^{(1,2)}(\varphi = 0) = \hat{\Psi}^{(1,2)}(\vartheta = \pi) = Q_{in}^{p}. \qquad (\text{VI.4}) $$

The 2nd implementation, by $\Omega_h^{(2)}$, could actually be brought also to position-independent form by gauge transformations, defined separately in the two contractible domains,

$$ \left\{ \begin{array}{ll} g_{-} = \operatorname{diag}(e^{i\vartheta/2}, e^{-i\vartheta/2}) & \text{in } V_{-} = 0 \le \vartheta \le \pi \\ g_{+} = \operatorname{diag}(e^{-i\vartheta/2}, e^{i\vartheta/2}) & \text{in } V_{+} = \pi \le \vartheta \le 2\pi \end{array} \right. \qquad (\text{VI.5}) $$

$$ \Omega_k^{(2)}(\vartheta) \rightarrow g_{\pm} \Omega_k^{(2)}(\vartheta) g_{\pm}^{-1} = \begin{pmatrix} k_{11} & -k_{12} \\ -\bar{k}_{12} & \bar{k}_{11} \end{pmatrix}. \qquad (\text{VI.6}) $$

However the gauge transformations do not match on the negative x axes, $g_+(\vartheta = \pi) = -i\sigma_3$, $g_-(\vartheta = \pi) = i\sigma_3$. “Dressing” $\Omega_k^{(2)}$ to a constant introduces a non-trivial transition function, $-\mathbb{I}^9$. In what follows, we consider the infinitesimal version.

<footnote n="9">⁹ The inequivalent implementations correspond to the center of the gauge group [28]. The trivial one corresponds to $\mathbb{I}$ and the twisted one in (VI.1) corresponds to $-\mathbb{I}$. For $G = SO(3)$ one has only $\mathbb{I}$.</footnote>
Internal symmetries.

Which of the implementations (VI.2) are symmetries ? We recall that (as said in sec.III A) : $\kappa = (\kappa_{ab}) \in \mathfrak{G}$ generates an internal symmetry when $\hat{\Psi}_{\kappa}$ is covariantly constant, $D_{\mu}\hat{\Psi}_{\kappa} = 0$, cf. (III.17). Then putting the covariant derivates to zero and taking into account that $\mathfrak{a}$ and $\mathfrak{a} - 2k$ with $k$ an arbitrary integer are equivalent, (II.7)-(II.8),

$$D_{\mu} \hat{\Psi}_{\kappa}^{(1)} = \frac{\mathfrak{a} - 2k}{i} \begin{pmatrix} 0 & \kappa_{12} \\ -\bar{\kappa}_{12} & 0 \end{pmatrix}, \qquad (\text{VI.7a})$$

$$D_{\mu} \hat{\Psi}_{\kappa}^{(2)} = \frac{\mathfrak{a} - (1 + 2k)}{i} \begin{pmatrix} 0 & e^{i\varphi} \kappa_{12} \\ -e^{-i\varphi} \bar{\kappa}_{12} & 0 \end{pmatrix}. \qquad (\text{VI.7b})$$

Thus the covariant derivatives vanish and thus yield an internal symmetry only for

$$\text{either } \mathfrak{a} = 2k \text{ or } \mathfrak{a} = 1 + 2k \text{ (k an integer),} \qquad (\text{VI.8})$$

respectively. In both cases, the enclosed magnetic flux is quantized for both series,

$$\Phi_k^{\text{even}} = (2k) \Phi_0, \quad \Phi_0 = (-i\pi \sigma_3), \quad k = 0, \pm 1, \pm 2, \dots, \qquad (\text{VI.9a})$$

$$\Phi_k^{\text{odd}} = (1 + 2k) \Phi_0, \quad \Phi_0 = (-i\pi \sigma_3), \quad k = 0, \pm 1, \pm 2, \dots, \qquad (\text{VI.9b})$$

Comparison with (I.2) shows that the Abelian series splits into two series, one even and one odd. The Wu-Yang factor takes its values in the center $Z(G) = \{\mathbb{I}, -\mathbb{I}\}$ of the gauge group SU(2) $^{10}$, as seen in (V.4). We have, of course also

$$\hat{\Psi}_{\sigma_3} = \frac{\sigma_3}{2i}. \qquad (\text{VI.10})$$

**Theorem VI.2.** For $\mathfrak{a} = 0$ or for $\mathfrak{a} = 1$ (mod even numbers) i.e., for Wu-Yang factor $\mathfrak{F} = \pm \mathbb{I}$, each $\kappa \in \text{su}(2)$ generates, via the first resp. the second implementation in (VI.2) an internal symmetry and thus generate 3 conserved internal charges, (III.18),

$$\begin{align} q_b^{(1)} &= Q^b & \text{for } \mathfrak{a} = 2k \tag{VI.11} \\ q_b^{(2)} &= -2\text{tr} (Q \cdot \hat{\Psi}_{\sigma_b}^{(2)}) & \text{for } \mathfrak{a} = 1 + 2k \nonumber \end{align}$$

<footnote>$^{10}$ For SO(3) = SU(2)/Z$_2$ the center would be just {I} and the twisted implementation would not be symmetry, leaving us only with the trivial one.</footnote>
where $b = 1, 2, 3$ are internal indices. For non-integer values of $a$ in the vector potential i.e., for $\mathfrak{F} \neq \pm \mathbb{I}$, there are no internal symmetries and thus no conserved internal charges.

Thus the conserved charges are $q_b^{(1)} = Q^b$ for the trivial implementation $a = 2k$ and $q_b^{(2)}$ in the twisted case with vector potential $a = 1 + 2k$. Then for the initial choices (IV.4) we recover the proton, for $b = 1$ and the neutron for $b = 2$ with charges $q_{proton} = 1$ and $q_{neutron} = 0$, respectively, consistently with (IV.5). The conservation of the charges (VI.11) follows also from covariant-constancy, $\dot{q} = -2\text{tr}(D_t Q \cdot \hat{\Psi} + Q \cdot D_t \hat{\Psi}) = 0$.

The full gauge group $G = \text{SU}(2)$ is a symmetry in both cases [27, 28]. However in the even case $a = 2k$ the field (II.5) is a pure gauge

$$A_{\mu} = g^{-1} \partial_{\mu} g \quad \text{for} \quad g(x) = e^{ik\vartheta} \sigma_3 = \text{diag}(e^{ik\vartheta}, e^{-ik\vartheta}). \qquad (\text{VI.12})$$

In the non-trivial odd case, $a = 1 + 2k$, when the flux is quantized in odd units as in (VI.9b). For this value of $a$ the field is not a gauge transform of the vacuum: although the it could seemingly be removed by gauge transforming by $g(\vartheta) = e^{i\vartheta\sigma_3/2}$ However this function is not well-defined on the plane, because $g(\vartheta = 0) = \mathbb{I}$ but $g(\vartheta = 2\pi) = e^{i\pi\sigma_3} = -\mathbb{I} \neq \mathbb{I}^{11}$.

## B. Conserved charges in the non-Abelian Aharonov-Bohm effect

Henceforth we focus our attention at $a = 1 + 2k$ an odd integer, we fix the implementation to be that of the initial proton direction, (IV.4), $Q_{in}^p = \frac{1}{2i}\sigma_1 = \frac{1}{2i}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. Then parallel transport implies that $^{12}$,

$$\hat{\Psi}_{\sigma_1}(\varphi) = Q_{\pm}^{p}(\varphi) = \cos \varphi \frac{\sigma_1}{2i} \mp \sin \varphi \frac{\sigma_2}{2i} = \frac{1}{2i} \begin{pmatrix} 0 & e^{\pm i\varphi} \\ e^{\mp i\varphi} & 0 \end{pmatrix}. \qquad (\text{VI.13})$$

The charge is the projection onto the direction field, (III.18). Spelling out for the proton trajectories (IV.6)-(IV.7) we get indeed, for all $\varphi$

$$q_{proton} = -2\text{tr}(Q_{\pm}^{p}(\varphi))^2 = 1, \qquad (\text{VI.14})$$

as expected: $\sigma_1 \in \text{su}(2)$ generates an internal symmetry and thus the charge is conserved: the incoming proton remains a proton during the scattering. For the S-matrix for example,

<footnote>$^{11}$ Remember that for $a = 1 + 2k$ the S-matrix is minus the identity, $S = -\mathbb{I}$, as noted in (V.4).</footnote>

<footnote>$^{12}$ The two signs correspond to the two half-planes: the sign should change when we switch from one to the other. Remember that $\varphi = \pm(\vartheta - \pi)$, depending on the sign of the orbital angular momentum, $L$.</footnote>
the minus sign in (V.4) combines with the minus sign of the direction field and thus yield conserved charge equal to 1, interpreted as the electric charge of the proton.

For $\sigma_2$ and $\sigma_3$ we have instead, by (IV.10),

$$ \hat{\Psi}_{\sigma_2}(\varphi) = \frac{1}{2} \begin{pmatrix} 0 & -e^{\pm i\varphi} \\ e^{\mp i\varphi} & 0 \end{pmatrix} \quad \Rightarrow \quad q_{\text{neutron}} = 0, \qquad (\text{VI.15a}) $$

$$ \hat{\Psi}_{\sigma_3}(\varphi) = \frac{1}{2i} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \quad \Rightarrow \quad q_{\text{vert}} = Q^3(0). \qquad (\text{VI.15b}) $$

The 3rd charge here corresponds to $\sigma_3$ perpendicular to the plane of the motion. For our choice it vanishes. These formulæ are consistent with (IV.5) and confirm that when the flux is quantized in odd units, $\Phi = -(1+2k)\pi(\sigma_3/i)$ with $k$ an integer, cf. (VI.9b), then we have three conserved charges. However if $a$ is not an integer i.e., for flux $\vec{\Phi} \neq N\pi\sigma_3$ the charge is not conserved. For $a = 1/2$, for example,

$$ \left(Q_{\frac{1}{2}}^{p}\right)_{\pm}(\varphi) = \frac{1}{2i} \begin{pmatrix} 0 & e^{\pm i\varphi/2} \\ e^{\mp i\varphi/2} & 0 \end{pmatrix} \qquad (\text{VI.16}) $$

and the charge is not conserved,

$$ \Rightarrow q_{\frac{1}{2}}(\varphi) = -2\text{tr}(\hat{\Psi}_{\sigma_1} \cdot Q_{\frac{1}{2}}) = \cos \frac{\varphi}{2} \neq \text{const.} \qquad (\text{VI.17}) $$

Thus the incoming proton loses its initial charge $q_{in} = 1$ and goes out with zero charge, $q_{out} = 0$, i.e., as a neutron,

$$ Q_{in} = Q^p \rightarrow Q_{out} = Q^n, \qquad (\text{VI.18}) $$

as argued by Wu and Yang [9].

Intuitively, the “electromagnetic direction field” $\hat{\Psi}$ is in fact that of a proton for $a = 1+2k$ (drawn in red and in blue in FIG.3), and the isospin for $a = \frac{1}{2}$ (drawn in magenta and cyan) turn at different angular velocity, therefore the projection of the latter onto the former one is not a constant, (VI.17). Similarly, an incoming neutron would come out as a proton, as shown in FIG.3.

Generalizing our previous investigations, we now consider motion with the same external initial conditions as before but allowing a non-zero vertical component for the isospin,

$$ Q_{in} = \left( \sin \alpha \frac{\sigma_1}{2i} + \cos \alpha \frac{\sigma_3}{2i} \right), \qquad (\text{VI.19}) $$
where $0 \le \alpha \le \pi$ is the angle of the initial isospin with the $\sigma_3$-oriented flux, shown in FIG.6. The spatial trajectory remains unchanged, whereas the isospin equation is solved by

$$Q_{\pm}(\varphi) = \frac{1}{2i} \left\{ \sin \alpha \begin{pmatrix} 0 & e^{\pm i\varphi a} \\ e^{\mp i\varphi a} & 0 \end{pmatrix} + \cos \alpha \sigma_3 \right\}. \qquad (\text{VI.20})$$

Then the projection onto the flux' direction is

$$\vec{Q} \cdot \vec{\Phi} = Q^3(0) = \cos \alpha = \text{const}, \qquad (\text{VI.21})$$

which implies that the isospin precesses on a circular section at constant height of a cones with opening angle $\alpha$ w.r.t. the $\sigma_3$-oriented flux $\vec{\Phi}$, whereas theirs apexes travel along the free spatial trajectories as for $\alpha = \pi/2$, see FIG.6. Maintaining our charge-based proton/neutron definition with direction field $\hat{\Psi}_{\sigma_1}$,

$$\tilde{q} = \sin \alpha \qquad (\text{VI.22})$$

which is conserved, is however neither a proton nor a neutron except for $\alpha = 0$ (when it is a neutron) or $\alpha = \pi/2$ (when it is a proton) $^{13}$. Similarly, the charge generated by $\hat{\Psi}_{\sigma_3} = \sigma_3$,

$$-2\operatorname{tr}(\tilde{Q}_{\pm}(\varphi) \cdot \hat{\Psi}_{\sigma_1}) = \cos \alpha, \qquad (\text{VI.23})$$

although a constant of the motion, is 1 only for $\alpha = 0$ but is zero $\alpha = \pi/2$ (VI.15b).

However as emphasised by Yang and Mills, we are allowed to change our proton/neutron definition and call instead proton an incoming particle with $\tilde{\Psi}(\varphi = 0) = Q_{in}$ in (VI.19) and promote (VI.20) with $\mathfrak{a} = 1$ to a “moving reference frame”

$$\tilde{\Psi}(\varphi) = \frac{1}{2i} \left\{ \sin \alpha \begin{pmatrix} 0 & e^{\pm i\varphi} \\ e^{\mp i\varphi} & 0 \end{pmatrix} + \cos \alpha \sigma_3 \right\}, \quad 0 \le \varphi \le \pi, \qquad (\text{VI.24})$$

Then projecting onto the latter,

$$\tilde{q} = -2\operatorname{tr}(\tilde{\Psi} \cdot \tilde{Q}) = \frac{1}{2}\operatorname{tr} \left\{ \sin^2 \alpha \begin{pmatrix} e^{i\varphi(\mathfrak{a}-1)} & 0 \\ 0 & e^{-i\varphi(\mathfrak{a}-1)} \end{pmatrix} + \cos^2 \alpha \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \right\}, \qquad (\text{VI.25})$$

would again be a satisfactory charge definition in that it would be (i) gauge-covariant and (ii) for $\mathfrak{a} = 1$ equal to 1 for all $\varphi$ by Pythagoras' theorem. The redefined charge (VI.25) would again be conserved and remain a “proton” with respect the new direction field $\tilde{\Psi}$.

<footnote n="13">13 The situation is analogous to that observed for classical spin, which becomes a half-integer only under quantization [69, 84].</footnote>
<figure><img src="image_7.png" /><figcaption>FIG. 6: The isospin vector $\vec{Q}$ whose initial direction has an angle $\alpha$ with that of the non-Abelian flux $\vec{\Phi}$ rotates along a horizontal circle on the cone of opening angle $\alpha$ around the flux' direction. The sense whose the rotation depends on the side the. The apex of the cone is carried along the free motion in space. The charge defined as the projection of $\vec{Q}$ onto the comoving "electric direction" $\hat{\Psi}$ is conserved when the flux is quantized in odd units (VI.9b).</figcaption></figure>

The electric-type internal charges defined by projecting onto a covariantly constant moving reference frame, (III.18), are conserved.

Another curious feature of the inherent freedom advocated by Yang and Mills [20] to choose what we call a proton [9] is the following. Let us assume that $a = 1$; then we have 3 internal symmetries. Selecting again the covariantly constant direction field $\hat{\Psi}_{\sigma_1}$ (VI.13) above provides us, in the twisted case, with 3 conserved charges as seen above. However, the components of a constant isospin trajectory,

$$ \vec{Q}(\varphi) = \vec{Q}_0 = \text{const}, \qquad (\text{VI.26}) $$

which would provide us with trivially conserved charges in the even case $a = 2k$ will not be conserved in the twisted case $a = 1$ in general. Using (VI.11) we find indeed,

$$ \left\{ \begin{array}{ll} Q^1(\varphi) = \frac{\sigma_1}{2i} & q^1 = \cos \varphi \neq \text{const} \\ Q^2(\varphi) = \frac{\sigma_2}{2i} & q^2 = \sin \varphi \neq \text{const} \\ Q^3(\varphi) = \frac{\sigma_3}{2i} & q^3 = 0 = \text{const} \end{array} \right. \qquad (\text{VI.27}) $$
### C. Spin from isospin

To get an insight, we start with the trivial case $a = 2k$ i.e., $\mathfrak{F} = \mathbb{I}$. The particle is free and the total angular momentum,

$$ \vec{J} = \vec{L} - \vec{Q} \quad \text{where} \quad \vec{L} = \mathbf{r} \times \vec{p}, \qquad (\text{VI.28}) $$

is conserved. The two terms here are in fact separately conserved: $\vec{L}$, the orbital one is conserved, because the spatial motion is free; the internal one, $\vec{Q}$, is conserved because in the diagonal gauge the commutator drops out from the isospin equation (III.1b) leaving us with $d\vec{Q}/dt = 0$. For $a = 2k$ the gauge potential can be gauged away (as noted above) and the equation is gauge-covariant.

This scenario realizes Souriau’s “décomposition barycentrique” [84] chap. III. p.163 <fnref n="14" />, which says that when the “space of motions” (his replacement for the phase space) has at least 8 dimensions which is our case here because of the isospin, then, for an isolated system with pairwise internal interactions, the rotations of the center of mass and the rotations around the latter act separately and both contribute to the angular momentum. Thus the SO(3) symmetry is in fact enlarged to $SO(3)_{ext} \times SO(3)_{int}$.

In the non-Abelian Aharonov-Bohm setup the flux is enclosed into a thin cylinder that arguably reduces the external symmetry to an axial $SO(2)_{ext}$. However this mechanical constraint is unrelated to the internal structure (isospin) of our particles, leaving us with the enlarged rotational symmetry

$$ \mathrm{SO}(2)_{\mathrm{ext}} \times \mathrm{SO}(3)_{\mathrm{int}}, \qquad (\mathrm{VI.29}) $$

generated by the orbital momentum $\vec{L}$ and the isospin 3-vector $\vec{Q}$, respectively, as in (VI.28).

Souriau illustrates his “théorème de décomposition barycentrique” by a non-relativistic particle with spin [84]. However, expressed in Souriau’s language, the internal symplectic form for the isospin is in fact identical to that for his classical non-relativistic spin [84], namely the surface form of the two-sphere of radius $|Q_0|$, $\mathcal{O} \sim \mathbb{S}^2 \subset \mathrm{su}(2)$. The (separately) conserved quantities are the orbital angular momentum $\vec{L}$ and the 3 internal charges which are in fact the 3 conserved components $Q^i$ of the isospin in eqn. (VI.11).

<footnote n="14">¹⁴ Anticipated by Jacobi in his Königsberg lectures [85].</footnote>
What happens in the non-trivial case ? Our general theory implies that when a is not an integer then no covariantly constant “direction fields” and thus no internal symmetries do exist, implying that the total angular momentum (VI.28) can not be conserved, leaving us with the mere orbital expression.

We are still left with the other exceptional case, $a = 1 + 2k$. However recall that the orbital angular momentum, $\vec{L}$, is conserved for all $a \neq 2k$ as we have seen in Sec.III B. But this implies that the total angular momentum, (VI.28) can not be conserved whenever the isospin is not a constant of the motion (except for a particular initial condition). We conclude that the internal structure breaks the full rotational symmetry (VI.28) to mere orbital one: for $a = 1 + 2k$, $\sigma_3$ generates, as before, internal rotations around the flux' direction which yields the internal angular momentum,

$$L_{int} = Q^3 \frac{\sigma_3}{2i} \qquad (VI.30)$$

which is conserved since $\hat{\Psi}_{\sigma_3} = \sigma_3$ and $Q^3 = Q^3(0)$. Thus the non-trivial flux breaks the symmetry (VI.29) down to a double axial one,

$$SO(2)_{ext} \times SO(2)_{int}, \qquad (VI.31)$$

generated by independent external and internal rotations around the (respective) 3rd directions. For $\alpha = 0$ the cone degenerates to a vertical vector of unit length which is conserved; for $\alpha = \pi/2$ we recover our previous result in FIGs.3-4-5.

The total angular momentum (VI.28) is conserved in the vacuum case $a = 2k$ ($\mathfrak{F} = \mathbb{I}$) but broken to (VI.31) for $a = 2k + 1$ ($\mathfrak{F} = -\mathbb{I}$), leaving us with the 3rd component only,

$$J_3 = L_3 - Q^3. \qquad (VI.32)$$

The spin from isospin term (although only along the 3rd direction) but the two other components in (VI.28) are not more conserved — the isospin precesses.

## VII. CONSERVED QUANTITIES BY VAN HOLTEN'S ALGORITHM

To deepen our insight, we revisit the internal symmetry question from a rather different point of view, namely by using van Holten's approach [79, 80]. His algorithm amounts to developing the conserved quantity we are looking for into powers $\Pi_i = \dot{x}_i$ (we took $m = 1$),

$$I = C + C_i \Pi_i + \frac{1}{2} C_{ij} \Pi_i \Pi_j + \dots \qquad (VII.1)$$
whose conservation, {$I, H$} = 0 requires

$$
\begin{array}{ll}
\text{order 1 :} & D_i C = Q^a F_{ij}^a C_j \\
\text{order 2 :} & D_i C_j + D_j C_i = Q^a (F_{ik}^a C_{kj} + F_{jk}^a C_{ki}), \\
\dots & \text{(VII.2)}
\end{array}
$$

which stops when we have a Killing tensor. Applied to the Non-Abelian Aharonov-Bohm
Effect, $F_{ij}^a = 0$, this simplifies to

$$
\begin{array}{ll}
\text{order 1 :} & D_i C = 0, \\
\text{order 2 :} & D_i C_j + D_j C_i = 0, \\
\dots & \text{(VII.3)}
\end{array}
$$

For simplicity we restrict our attention at planar real-space motion, i.e., assume that the initial conditions imply that the real-space motion is in the $x-y$ plane with the $z$-direction identified with $\sigma_3$ in isospace, $\hat{z} \sim \sigma_3$. Planar rotations, for example, are generated by the Killing vector $\hat{z} \times \vec{x}$. Then the algorithm yields $C = 0$, $C_i = -\epsilon_{ij}x^j$ [79, 80] and we recover the 3rd component of the *conserved orbital angular momentum* (IV.1), $L \equiv L_3 = \epsilon_{ij}x^i\dot{x}^j$, as expected.

Now we rederive our previous results about internal charges by the van Holten method.
Searching for the first-order in $\vec{Q}$ coefficient in the form

$$
C(\vec{x}, Q) = C^a(\vec{x}) Q^a \tag{VII.4}
$$

and using the parallel transport eqn. $D_i Q^a = 0$, we get

$$
D_i C^a = \partial_i C^a - \epsilon^{abc} A_i^b C^c = 0. \qquad (\text{VII.5})
$$

Inserting here the non-Abelian Aharonov-Bohm gauge potential $A_i^a = -\frac{B^a}{r^2} \epsilon_{ij} x^j$ with ($B^a$) = (0, 0, a), these equations become

$$
\left\{
\begin{array}{l}
\partial_x C^1 - a C^2 \frac{y}{r^2} = 0, \quad \partial_x C^2 + a C^1 \frac{y}{r^2} = 0, \quad \partial_x C^3 = 0, \\
\\
\partial_y C^1 + a C^2 \frac{x}{r^2} = 0, \quad \partial_y C^2 - a C^1 \frac{x}{r^2} = 0, \quad \partial_y C^3 = 0,
\end{array}
\right.
\qquad (\text{VII.6})
$$

which admits the nontrivial solution

$$
C^1(x, y) = \alpha \sin \mu, \quad C^2(x, y) = \alpha \cos \mu, \quad \text{where} \quad \mu = \mu(x, y) = a \arctan \frac{x}{y} + \phi_0, \quad \text{(VII.7)}
$$

where $\phi_0$ is an arbitrary constant. This should be completed with $C^3 = \beta = \text{const}$ which
follows from having directed the vector potential into the 3rd direction.
For $\alpha = 0$, the isospin has only a $\sigma_3$ component and (VI.30) is recovered. We put hence $\alpha = 1, \beta = 0$ and focus our attention at the two remaining components. Then the algorithm yields,

$$I = \sin \mu Q^1 + \cos \mu Q^2, \qquad (VII.8)$$

This eqn. is plainly valid for $y \neq 0$, i.e., in the upper and lower half-planes separated by the x-axis because of $y$ in the denominator for $\mu$. However letting $y \to 0$ from the two sides yields, for fixed $x > 0$, $x/y \to \pm\infty$, depending on the sign of $y$. Then the asymptotic behavior of the arctan implies,

$$\mu \to \begin{cases} \mu_+ = a\pi/2 + \phi_0 & \text{when } y > 0 \\ \mu_- = -a\pi/2 + \phi_0 & \text{when } y < 0 \end{cases} \quad \Rightarrow \quad \mu_- = \mu_+ - a\pi. \qquad (VII.9)$$

Thus the trigonometric functions do not match in general, $I_+ \neq I_-$: there is no well-defined conserved quantity on the entire plane, consistently with the breaking of the internal symmetries found earlier.

However remember that there might be a transition function (in fact a constant element) between the two half-planes which should belong to the center of the gauge group [27, 28],

$$I_+ = h I_-, \quad \text{where} \quad h \in Z_G = \{\mathbb{I}, -\mathbb{I}\}. \qquad (VII.10)$$

For $h = \mathbb{I}$ this requires $\sin \frac{a\pi}{2} = 0$ so that $a$ is an even integer, $a = 2k$. For $h = -\mathbb{I}$ we get instead $\cos \frac{a\pi}{2} = 0$ implying that $a$ is an odd integer, $a = 1 + 2k$. In both cases we get from by using (VII.4) with (VII.7) (VII.8), which is a well-defined conserved quantity for both of exceptional values of $a$.

Conversely, remembering that the length of $\vec{Q}$ is a constant, $|\vec{Q}|^2 = I^2$, allows us to recover the trajectory using the conserved quantity $I$ as<fnref n="15" />

$$Q^1(x, y) = I \sin \mu(x, y) \quad \text{and} \quad Q^2(x, y) = I \cos \mu(x, y) \qquad (VII.11)$$

consistently with (IV.6)-(IV.7) and (IV.10). Inserting (VII.11) into (VII.8) identity as it should. The conserved quantity (VII.8) is indeed the conserved charge $q = Q^a \Psi^a$ in (III.5) or more generally, in (VI.11) with (VII.11) for $\hat{\Psi}_{\sigma_1}$ chosen as in (VI.13).

<footnote n="15">¹⁵ More generally, $|\vec{Q}|^2 = I^2 + |Q^3|^2$.</footnote>
In particular for $a = 2k$ (even) $\mu \to \pm k\frac{\pi}{2} + \phi_0 = \text{const}$, implying that $\vec{Q}$ is a constant,

$$Q^1 = (-1)^k I \sin \phi_0 = \text{const} \quad \text{and} \quad Q^2 = (-1)^k I \cos \phi_0 = \text{const}. \qquad (\text{VII.12})$$

For $a = 1 + 2k$ (odd) we have $\mu = \mu(x, y) \neq \text{const}$, (VII.8), however the limits from the two sides, $\mu_{\pm} \to \pm\frac{\pi}{2}(1 + 2k) + \phi_0$, do match when $y \to 0$, (VII.9).

These results confirm those found in sec. V. as we show for $a = 1 + 2k$ with $y > 0$. For the “in” (resp. “out”) states, choosing $\phi_0 = (1 + k)\pi$ (VII.11) implies,

$$Q_{in}^1 = 1, \quad Q_{in}^2 = 0 \quad \text{resp.} \quad Q_{out}^1 = -1, \quad Q_{out}^2 = 0, \qquad (\text{VII.13})$$

consistently with (V.2). The other values shown in FIG. 4 are recovered also by appropriate choices of $\phi_0$.

Returning to the angular momentum, we get, consistently with our previous results,

**Theorem VII.1.** When the Wu-Yang factor does not belong to the center, $\mathfrak{F} \neq \pm I$, then only the 3rd component of the orbital angular momentum, $L = L_3$, is conserved. However when the flux is quantized in either of the two series in (VI.9), then $L_3$ and $Q^3$ in (IV.1) and (VI.30), are both separately conserved and the symmetry is doubled, (VI.31), consistently with Souriau's décomposition barycentrique [84].

## VIII. CONCLUSION AND OUTLOOK

This project, suggested by T. T. Wu to PAH over 40 years ago [22, 29, 86–89] <fnref n="16" />, studies the Non-Abelian Aharonov-Bohm Effect proposed by Wu and Yang in their celebrated paper on the non-integrable phase factor [9]. While the original proposal is field-theoretical/quantum mechanical, here we argue that a “shadow” of it, namely isospin precession<fnref n="17" />, can be recovered also at the pseudo-classical level, with the isospin represented by a vector $\vec{Q}$ of the Lie algebra of the gauge group <fnref n="18" />. This is in sharp contrast with the Abelian (electromagnetic) case : Aharonov-Bohm effect is purely quantum mechanical [1].

<footnote n="16">¹⁶ H. Bacry, musing over a piece of paper in Marseille in the eighties, more than once whispered : “On écrit le même article durant toute sa vie” (One writes the same paper during all his life).</footnote>

<footnote n="17">¹⁷ Isospin precession was found also for diatomic molecules [35, 36]; see the recent study [37].</footnote>

<footnote n="18">¹⁸ Subtleties of the concept of classical isospin were discussed in [90–92].</footnote>
Comparing the Abelian and the non-Abelian case, we argue that “the electric charge is a *constant scalar* (in fact the same constant for all motions), but the non-Abelian isospin *vector* has a proper dynamics, – even if it is rather poor and, being parallel transported, is determined by the space-time motion. Thus the difference comes from the (partial) observability of the phase.

Physical interpretation in terms of protons and neutrons can be obtained by following the charge-based arguments of Yang and Mills [20]. The charge is the projection of the isospin vector onto a chosen direction field $\hat{\Psi}(x^\mu)$, (III.18). Then a proton is a particle with charge 1 and a neutron has zero charge. However, the charge is conserved only when $\hat{\Psi}(x^\mu)$ is covariantly constant (III.17), $D_\mu \hat{\Psi} = 0$, i.e., when $\hat{\Psi}$ corresponds to an internal symmetry [27, 28].

For non-Abelian monopoles this happens outside their core [52, 71, 72], allowing to reduce the dynamics of a test particle to an Abelian problem [93, 94].

In the non-Abelian Aharonov-Bohm case, though, a covariantly constant $\hat{\Psi}$ exists only when the non-Abelian *flux is quantized* in either of the two distinct series (VI.9), then we get 3 conserved internal charges [27, 28]; for other flux values we do not get any. The flux quantization we find here is reminiscent of the one in superconducting rings [95-97].

This surprising behaviour hints at the *global structure* of the gauge group: it comes from the center of SU(2) being composed of $I$ and $-I$; for SO(3) = SU(2)/Z₂ we have only $I$. Thus answering the question : — “Is the non-Abelian flux quantized ? And if it is, then in which units ?” would depend on what the “true” gauge group is.

The most dramatic prediction of Wu and Yang [9] is that an incoming proton might be turned into an outgoing neutron under specific circumstances. We confirm their statement when the flux is not quantized : for half-integer flux $a = \frac{1}{2}$, for example, then $D_\mu \hat{\Psi} \neq 0$ and an *incoming proton could indeed turn into an outgoing neutron*.

However the initial choice can be changed yielding different casts, illustrating the arbitrariness of identifying protons and neutrons, as emphasised by Yang and Mills [20].

Such an eventuality raises the fundamental question : Is such a transition indeed possible or not ? What would happen with the electric charge ? What could be the physical mechanism for it. We confess not to be able to answer. It might well happen that the test particle framework we follow in this paper (or its quantum mechanical antecedents [22, 86-88]) are just not sufficient to answer. Another idea could be to enlarge the commutative
part of the gauge group from SU(2) to Weinberg-Salam U(2) (locally (U(2) × U(1))).

The direction field $\hat{\Psi}(x^\mu)$ behaves, mathematically, as a *non-dynamical Higgs field* in the adjoint representation – which however plays only a passive role, as a sort of “measuring apparatus”. For a Wu-Yang monopole [70], the archi-type of non-Abelian monopoles, for example, it corresponds to their radial “hedgehog” direction (III.7). For a ’t Hooft-Polyakov monopole instead, the direction field is promoted to a dynamical field which allows for finite-energy soliton solutions of the coupled Yang-Mills-Higgs equations [52, 71, 72]. The direction field chosen tacitly is the normalized physical Higgs field, $\hat{\Psi} = \Psi/|\Psi|$. Outside the monopole core the SU(2) gauge symmetry is spontaneously broken to a residual U(1) implying that the Higgs field is covariantly constant there.

The non-Abelian flux line we studied in this paper corresponds, intuitively, to a pure YM vortex whose “outside region” is the entire plane with the origin removed. Form the point of view of a field-theoretical YMH Lagrangian, the kinetic term $D_\mu\Psi D^\mu\Psi$ is switched off, reducing $\Psi$ to a non-dynamical quantity.

The contrast between pure YM and YMH becomes even more salient in Chern-Simons theory [98, 99], whose non-Abelian version [100, 101] is strongly reminiscent of non-Abelian Aharonov-Bohm – except for that the Higgs field is, once again, dynamical. However the analytic solutions found in [100, 101] are as remote as possible from the non-Abelian Aharonov-Bohm in that they are *self-dual*, $\frac{1}{2}\epsilon_{ijk}F^{jk} = D_i\Psi$. instead of our cast with $F^{jk} \propto \delta(x)$.

The angular momentum in the Non-Abelian Aharonov-Bohm Effect exhibits the “spin from isospin” phenomenon [66, 74, 81–83]. The orbital angular momentum is conserved for all fluxes. However for the two particular values of the flux in (VI.9), we have additional terms generated by internal symmetry which is *independent* of space-time rotations. The total angular momentum, *J* in (VI.28) splits into two separately conserved pieces, one external (orbital) and one internal (isospin), realizing Souriau’s doubled rotational symmetry idea [84]. When the flux quantized in odd units, eqn (VI.9b), the remnant of the internal symmetry is detected by the precessions of the isospin on horizontal circles, shown in FIG.6.

Further insight can be gained by looking at the problem from a different point of view, advocated by van Holten [79, 80], see sec.VII.

The flux quantization we found above has some analogy with the Dirac quantization of a monopole: choosing the Dirac string along (say) the negative *z* axis, in its neighborhood the
vector potential will behave approximately as an Aharonov-Bohm solenoid, which remains unobservable when the flux is quantized [52, 102, 103].

Thus the non-Abelian Aharonov-Bohm setup is indeed the planar analog of the non-Abelian monopole [52, 104, 105]. Our investigations confirm the *credo* of Wu and Yang [9] :

> ... what constitutes an intrinsic and complete description of electromagnetism?
> 
> [...] An examination of the Bohm-Aharonov experiment indicates that in fact
> only the phase factor (I.1) and not the phase is physically meaningful.

Quantum aspects were studied in [22, 27–29, 86–89]. See [37–41] for a (very incomplete) selection. It is remarkable that part of the Non-Abelian Aharonov-Bohm Effect can be captured by the (pseudo-)classical isospin of Kerner and Wong [55, 56], confirming that the isospin is neither a truly classical nor a truly quantum concept: it is at half ways between the two, justifying our terminology.

Yang et al [47, 106] attribute the non-Abelian Aharonov-Bohm effect to the breaking of time-reversal symmetry. However parallel transport implies that real time is replaced, in our framework, by the angle $\varphi$. Thus the effect comes from “navigating” on either sides of the flux. However we found that when the flux is quantized, (VI.9) then the time reversal symmetry disappears: side-dependent phase shifts yield identical contributions on both sides, as highlighted by doubling the rotational symmetry, as in (VI.28)-(VI.31).

Intensive ongoing work on effective/artificial gauge fields [35–51] provide promising testing ground of the conceptual suggestions of Wu and Yang, allowing us to hope that the Non-Abelian Aharonov-Bohm Effect may, on one day, not be considered anymore as ... “dry water” (using a celebrated expression put forward by J. von Neumann).

## Acknowledgments

This paper is dedicated to the memory of “Tai” (Tai-Tsun Wu : 1933-2024)). Correspondance with Janos Balog, Gary Gibbons, Csaba Sükösd and A. Zeilinger is gratefully acknowledged. PMZ was partially supported by the National Natural Science Foundation of China (Grant No. 11975320).
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