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The non-Abelian Aharonov-Bohm effect

P. A. Horvathy

TL;DR

This work studies the non-Abelian Aharonov–Bohm effect for Yang–Mills flux with vanishing field outside a cylinder, modeling a fixed background for a non-relativistic, spinless nucleon. By employing a diagonal gauge, the $SU(N)$ vacuum splits into $N$ independent BA vacua, allowing the Schrödinger equation to reduce to $N$ Abelian BA problems and the S-matrix to factorize accordingly. It derives a representation-theoretic classification of vacua, demonstrates isospin precession and a square-root relation between non-Abelian and Abelian phases, and shows how different YM vacua can yield the same scattering for non-faithful representations. The classical limit via Wong’s equations aligns with the quantum results, establishing a consistent bridge between gauge structure, representation theory, and observable phases. The results generalize the non-Abelian BA framework to arbitrary SU($N$) representations and provide insight into observability and equivalence of YM vacua in scattering processes.

Abstract

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.

The non-Abelian Aharonov-Bohm effect

TL;DR

This work studies the non-Abelian Aharonov–Bohm effect for Yang–Mills flux with vanishing field outside a cylinder, modeling a fixed background for a non-relativistic, spinless nucleon. By employing a diagonal gauge, the vacuum splits into independent BA vacua, allowing the Schrödinger equation to reduce to Abelian BA problems and the S-matrix to factorize accordingly. It derives a representation-theoretic classification of vacua, demonstrates isospin precession and a square-root relation between non-Abelian and Abelian phases, and shows how different YM vacua can yield the same scattering for non-faithful representations. The classical limit via Wong’s equations aligns with the quantum results, establishing a consistent bridge between gauge structure, representation theory, and observable phases. The results generalize the non-Abelian BA framework to arbitrary SU() representations and provide insight into observability and equivalence of YM vacua in scattering processes.

Abstract

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 , and derive a classification scheme. Isospin precession is recovered also at the classical limit.
Paper Structure (8 sections, 7 theorems, 49 equations, 1 table)

This paper contains 8 sections, 7 theorems, 49 equations, 1 table.

Table of Contents

  1. Introduction
  2. Yang-Mills vacua in the generalized Bohm-Aharonov experiment
  3. Nucleon scattering around a Y-M flux
  4. Generalization to $SU(N)$
  5. Classical limit
  6. Discussion
  7. Acknowledgement
  8. Appendix

Key Result

Theorem 1

Two Y-M vacua $A^{(1)}$ and $A^{(2)}$ are gauge-equivalent if there exists a fixed element $h \in S U(N)$ such that

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 2