Gao, Shan (2025) Revisiting Modular Momentum in the Aharonov-Bohm Effect: A Resolution of Aharonov’s Paradox. [Preprint]

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Abstract

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.

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Item Type:	Preprint
Creators:	Creators Email ORCID Gao, Shan sgao7319@uni.sydney.edu.au
Keywords:	Aharonov-Bohm Effect; AB phase; nonlocality; modular momentum; velocity distribution; spectral measure
Subjects:	Specific Sciences > Physics > Quantum Mechanics Specific Sciences > Physics > Symmetries/Invariances
Depositing User:	Prof. Shan Gao
Date Deposited:	10 Sep 2025 14:26
Last Modified:	10 Sep 2025 14:26
Item ID:	26613
Subjects:	Specific Sciences > Physics > Quantum Mechanics Specific Sciences > Physics > Symmetries/Invariances
Date:	10 September 2025
URI:	https://philsci-archive.pitt.edu/id/eprint/26613

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