The most popular interpretation of the Aharonov–Bohm (AB) effect is that the electromagnetic potential locally affects the complex phase of a charged particle’s wave function in the magnetic field free region. However, since the vector potential is a gauge-variant quantity, multiple researchers suspect that it is just a convenient tool for calculating the force field. This motivates them to explain the AB effect without using the vector potential, which inevitably leads to some sort of non-locality. This frustrating situation is shortly summarized by the statement by Aharonov et al. that the AB effect may be due to a local gauge potential or due to non-local gauge-invariant fields. In the present paper, we shall give several convincing arguments which support the viewpoint that the vector potential is not just a convenient mathematical tool with little physical entity. Despite its gauge arbitrariness, the vector potential certainly contains a gauge-invariant piece, which solely explains the observed AB phase shift. Importantly, this component has a property such that it is basically unique and cannot be eliminated by any regular gauge transformations. To complete the discussion, we also discuss the role of remaining gauge arbitrariness still contained in the entire vector potential.The most popular interpretation of the Aharonov–Bohm (AB) effect is that the electromagnetic potential locally affects the complex phase of a charged particle’s wave function in the magnetic field free region. However, since the vector potential is a gauge-variant quantity, multiple researchers suspect that it is just a convenient tool for calculating the force field. This motivates them to explain the AB effect without using the vector potential, which inevitably leads to some sort of non-locality. This frustrating situation is shortly summarized by the statement by Aharonov et al. that the AB effect may be due to a local gauge potential or due to non-local gauge-invariant fields. In the present paper, we shall give several convincing arguments which support the viewpoint that the vector potential is not just a convenient mathematical tool with little physical entity. Despite its gauge arbitrariness, the vector potential certainly contains a gauge-invariant piece, which solely explains the observed AB phase shift. Importantly, this component has a property such that it is basically unique and cannot be eliminated by any regular gauge transformations. To complete the discussion, we also discuss the role of remaining gauge arbitrariness still contained in the entire vector potential.

arXiv:2506.07018SymmetryJun 12, 20250 citations

Revisiting Modular Momentum in the Aharonov-Bohm Effect: A Resolution of Aharonov’s Paradox

Gao, Shan

The Aharonov-Bohm (AB) effect demonstrates that electromagnetic potentials can influence quantum systems in field-free regions, challenging classical intuitions. In 2004, Aharonov and Kaufherr proposed that the AB phase shift arises from a nonlocal exchange of modular momentum, a gauge-invariant quantity, occurring at a definite time before wave packet interference. This framework, however, introduces a paradox: the expectation value of the modular momentum operator, $\langle \hat{M}_y \rangle$ , changes abruptly when the electron’s wave packets cross the solenoid, yet local, gauge-invariant properties of the electron, such as its velocity distribution, remain continuous. We resolve this paradox by showing that $\hat{M}_y=\exp(i \frac{m v_y L}{\hbar})$ and the velocity operator $\hat{v}_y$ have distinct spectral measures, and the abrupt change in $\langle \hat{M}_y \rangle$ reflects a global topological change in the electromagnetic environment, not a dynamical alteration in the electron’s state. Through a corrected, gauge-invariant formulation, we demonstrate that $\langle \hat{M}_y \rangle$ encodes the AB phase but does not explain the AB effect’s mechanism. This resolution reaffirms the conventional explanation of the AB effect while clarifying the nonlocal nature and experimental limitations of modular momentum.The Aharonov-Bohm (AB) effect demonstrates that electromagnetic potentials can influence quantum systems in field-free regions, challenging classical intuitions. In 2004, Aharonov and Kaufherr proposed that the AB phase shift arises from a nonlocal exchange of modular momentum, a gauge-invariant quantity, occurring at a definite time before wave packet interference. This framework, however, introduces a paradox: the expectation value of the modular momentum operator, $\langle \hat{M}_y \rangle$, changes abruptly when the electron’s wave packets cross the solenoid, yet local, gauge-invariant properties of the electron, such as its velocity distribution, remain continuous. We resolve this paradox by showing that $\hat{M}_y=\exp(i \frac{m v_y L}{\hbar})$ and the velocity operator $\hat{v}_y$ have distinct spectral measures, and the abrupt change in $\langle \hat{M}_y \rangle$ reflects a global topological change in the electromagnetic environment, not a dynamical alteration in the electron’s state. Through a corrected, gauge-invariant formulation, we demonstrate that $\langle \hat{M}_y \rangle$ encodes the AB phase but does not explain the AB effect’s mechanism. This resolution reaffirms the conventional explanation of the AB effect while clarifying the nonlocal nature and experimental limitations of modular momentum.

Sep 10, 20250 citations

Gauge Fields, Electromagnetism and the Bohm-Aharonov Effect

Chen Ning Yang

WORLD SCIENTIFIC eBooksApr 30, 20136 citations

\langle \hat{M}_y \rangle

, changes abruptly when the electron’s wave packets cross the solenoid, yet local, gauge-invariant properties of the electron, such as its velocity distribution, remain continuous. We resolve this paradox by showing that

\hat{M}_y=\exp(i \frac{m v_y L}{\hbar})

and the velocity operator

\hat{v}_y

have distinct spectral measures, and the abrupt change in

\langle \hat{M}_y \rangle

reflects a global topological change in the electromagnetic environment, not a dynamical alteration in the electron’s state. Through a corrected, gauge-invariant formulation, we demonstrate that

\langle \hat{M}_y \rangle

encodes the AB phase but does not explain the AB effect’s mechanism. This resolution reaffirms the conventional explanation of the AB effect while clarifying the nonlocal nature and experimental limitations of modular momentum.