2025/09/28 by Lucas Sourrouille, Sourrouille, Lucas
Mathematics · #FOS: Physical sciences #Quantum Physics (quant-ph) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2509.24028
openalex publication_date 2025/09/28 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
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| < (1)/(2) ∫ 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 +∞ and -∞. Finally, we illustrate these results with an explicit example.