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International Journal of Modern Physics A

Different ways to the Aharonov–Bohm effect and elucidation of its physical meaning

Alexander L. Kholmetskii, O. Missevitch, Tolga Yarman

Published Mar 30, 2026• 0 citations
Publisher

Abstract

We compare different ways to the Aharonov–Bohm (A–B) phase for an electrically charged point-like particle in an electromagnetic field and disclose three mathematically equivalent representations for the A–B phase, which explicitly demonstrate its Lorentz invariance and gauge invariance. Based on these results, we obtain a linear dependence of the wave vector of a moving charged particle on the vector composition of mechanical and electromagnetic momenta, which sheds new light on the physical origin of the A–B effect.

DOI
10.1142/s0217751x26500776

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We compare different ways to the Aharonov–Bohm (A–B) phase for an electrically charged point-like particle in an electromagnetic field and disclose three mathematically equivalent representations for the A–B phase, which explicitly demonstrate its Lorentz invariance and gauge invariance. Based on these results, we obtain a linear dependence of the wave vector of a moving charged particle on the vector composition of mechanical and electromagnetic momenta, which sheds new light on the physical origin of the A–B effect.
We analyse the physical meaning of the Aharonov–Bohm (AB) phase based on its representation through electromagnetic (EM) potentials as a sum of four components, which, in addition to the known electric and magnetic phase components, contains two more terms recently disclosed by our team in the analysis of quantum phase effects for dipoles and charges, and which we named the complementary electric AB phase and the complementary magnetic AB phase. Using the complete expression for the AB phase, we reveal that the phase component, explicitly depending on time, is determined by the interactional electric energy, while the phase component, explicitly depending on the velocity of charge, is determined by the interactional EM momentum for an isolated system ‘source of EM field and charge’. These findings shed new light on the origin of the AB phase and, in particular, allow us to generalize the de Broglie relationship and the Heisenberg uncertainty relations for a charged particle in an EM field.
Abstract We focus on new expressions for the Aharonov–Bohm (AB) phase proposed recently by C. Marletto and V. Vedral (Phys. Rev. Lett. 125, 040401 (2020)) and by our team (A. L. Kholmetskii et al., Ann. Phys., 434, 168627 (2021)). We emphasize the mismatch between the standard expression for the electric component of the AB phase and the expression by Marletto and Vedral for this component, which, as we show, has fundamental roots and indicates the need to use different expressions for the Hamiltonian of the radiative component and bound component of the electromagnetic (EM) field. We further show that the corrected expression for the AB phase of Marletto and Vedral and our expression for the AB phase do not coincide with its standard representation through the potentials of EM field (Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485) and contain additional terms, which straightforwardly allow to re-express the AB phase via electric and magnetic fields instead of scalar and vector potentials. The recognition of the field-dependent origin of the AB effect also requires to re-express the Hamiltonian of a charged particle through the EM field, which we show can be done in the only possible way. We discuss the physical meaning of the AB effect with the primary role of the EM field and also discuss some general quantum mechanical implications.
Abstract We analyze the physical meaning of quantum phases for moving electric/magnetic dipoles through a superposition of quantum phases for point-like charges of the dipole, and in this way we disclose two new quantum phases for moving charges, next to the well-known magnetic and electric Aharonov-Bohm phases. We find that a consistent description of the full set of quantum phase effects for charged particles requires to modify the standard definition of the momentum operator via the canonical momentum of a particle towards a more inclusive definition, where this operator is associated with the vector sum of mechanical and interactional electromagnetic (EM) momenta for a charged particle in the EM field. Some principal implications of this result are discussed.
Abstract We present an updated analysis of the total expression for the Aharonov-Bohm (AB) phase of a charged particle in an electromagnetic field, which we previously obtained through the superposition principle for quantum phases of charges and dipoles ( Sci. Rep. , 8 (2018) 11937), and here we re-derive it directly in the framework of the general approach, when the source, the electromagnetic field and the charged particle are quantized. The disclosure of the full set of quantum phase effects for a moving charged particle allows an important update of the wave-particle duality concept by generalizing the de Broglie relationship, where the wave vector associated with the particle is proportional to the vector sum of the mechanical and interactional electromagnetic momenta.
The gauge invariance of the Aharonov-Bohm (AB) effect with a quantum treatment for the electromagnetic field is demonstrated. We provide an exact solution for the electromagnetic ground energy due to the interaction of the quantum electromagnetic field with the classical charges and currents that act as sources of the potentials in a classical description, in the Lorenz gauge. Then, we use first-order perturbation theory to compute an extra change on the electromagnetic ground energy due to the presence of a quantum charged particle with known wave function in the system. This energy in general depends on the quantum particle path in an interferometer, which results in an AB phase difference between the paths. The gauge invariance of this AB phase difference is then shown for the magnetic, electric, and the recently proposed electrodynamic versions of the AB effect. However, the AB phase difference could depend on the gauge for nonclosed paths, which reinforces the view that it only can be measured in closed paths.
This article describes the Aharonov-Bohm (AB) effect of electron waves travelling in free space and its application to the observation of gauge fields (vector potentials). The AB effect is inconceivable in classical physics since an observable effect is produced on electrons passing through field-free spaces. Electrons can be affected only by Lorentz forces due to electromagnetic fields. The situation is different in quantum mechanics, where electrons show wave properties: the concept of force is no longer relevant, so electric field E and magnetic field B, defined as forces acting on a unit charge, take on secondary meanings. “Phase shifts” come into play, and the primary physical entities become neither E nor B but electrostatic potential V and vector potential A. These potentials interact with electron waves and shift their phases.