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arXiv:2402.13883

Isospin precession in non-Abelian Aharonov-Bohm scattering

Zhang, Peng-Ming, P. A. Horváthy

Published Feb 21, 2024• 1 citation

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.

DOI: 10.48550/arxiv.2402.13883

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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

SU(2)

N=2k

acts trivially but for

SU(2)

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.

The scattering of a nucleon beam around a cylinder containing a non-Abelian flux is studied. We confirm all the previsions of Wu and Yang. We consider the generalization to the gauge group

SU(N)

, and derive a classification scheme. Isospin precession is recovered also at the classical limit.

We consider a particle in a coherent superposition of states with different electric charges moving in the vicinity of a magnetic flux. Formally, it should acquire a (gauge-dependent) Aharonov–Bohm (AB) relative phase between the charge states, even for an incomplete loop. If measurable, such a geometric, rather than topological, AB phase would seem to break gauge invariance. Wick, Wightman and Wigner argued that since (global) charge-dependent phase transformations are physically unobservable, charge state superpositions are unphysical (‘charge superselection rule’). This would resolve the apparent paradox in a trivial way. However, Aharonov and Susskind disputed this superselection rule: they distinguished between such global charge-dependent transformations and transformations of the relative inter-charge phases of two particles, and showed that the latter could in principle be observable! Finally, the paradox again disappears once we consider the ‘calibration’ of the phase measured by the Aharonov–Susskind phase detectors, as well as the phase of the particle at its initial point. It turns out that such a detector can only distinguish between the relative phases of two paths if their (oriented) difference forms a loop around the flux.

The Aharonov-Bohm (AB) effect highlights the fundamental role of electromagnetic potentials in quantum mechanics, manifesting as a phase shift for a charged particle in field-free regions. While well-established for static magnetic fluxes, the effect's behavior under time-varying fluxes remains an open and debated question. Employing the WKB method, we derive the AB phase shift for a time-dependent magnetic vector potential, demonstrating that for circular paths in the quasistatic regime, it is proportional to the time-averaged enclosed magnetic flux,

Δφ_{\rm AB} = \frac{1}{T} \int_0^T e Φ(t) \, dt

, with the total phase shift, including kinetic contributions, equaling

e Φ(0)

. For non-circular paths, the phase shift depends on both the flux history and path geometry, revealing the effect's hybrid nature involving gauge potentials and induced electric fields. We verify the consistency of our gauge choice with Maxwell's equations and discuss the implications for local versus nonlocal interpretations of the AB effect. We also generalize the results to scenarios with nonzero external magnetic fields, where the enclosed flux is through the actual electron paths, and for circular paths of radius

R

, the AB phase shift is also proportional to the time average of the enclosed flux

Φ_{\rm enc}(R,t)

, with the total phase shift depending only on the initial enclosed flux

e Φ_{\rm enc}(R,0)

; for general non-circular paths, the external magnetic field affects trajectories and phase accumulation through the Lorentz force, leading to additional path dependence. These findings clarify the role of gauge-dependent potentials and induced fields in the generalized AB effect, offering new theoretical insights and potential applications in quantum technologies.