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arXiv · 2509.24028

The gauge freedom in the Aharonov-Casher theorem: The problem in two and one dimensions

Lucas Sourrouille
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Abstract

In this note, we investigate the role of gauge freedom in the Aharonov-Casher theorem in one and two dimensions. In particular, we analyze the asymptotic behavior of the gauge freedom. We show that in two dimensional space the gauge field is uniquely determined, whereas in one dimensional space there exists a gauge freedom that enables an infinite family of equivalent gauges. This freedom is determined by a constant k. However, the number $k$ must be restricted in order to guarantee the existence of a normalizable zero mode. This restriction is determined by the inequaly $|k| < \frac{1}{2} \int dx B(x)$, where $B(x)$ is a scalar field localized in a finite region of space. We show that this condition is equivalent to the gauge field taking opposite-sign values at $+\infty$ and $-\infty$. Finally, we illustrate these results with an explicit example.

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Lucas Sourrouille. 2025-09-28. The gauge freedom in the Aharonov-Casher theorem: The problem in two and one dimensions. https://arxiv.org/abs/2509.24028

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