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Zeitschrift für Naturforschung A

New expressions for the Aharonov–Bohm phase and consequences for the fundamentals of quantum mechanics

Alexander L. Kholmetskii, Tolga Yarman, O. Missevitch

Published Jan 8, 2024• 1 citation
Publisher

Abstract

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.

DOI
10.1515/zna-2023-0297

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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.

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

Alexander L. Kholmetskii, O. Missevitch, Tolga Yarman

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.

International Journal of Modern Physics A•Mar 30, 2026•0 citations
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.
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.
We analyse the physical meaning of quantum phase effects for point-like charges and electric (magnetic) dipoles in an electromagnetic (EM) field. At present, there are known eight effects of such a kind: four of them (the magnetic and electric Aharonov – Bohm phases for electrons, the Aharonov – Casher phase for a moving magnetic dipole and the He – McKellar – Wilkens phase for a moving electric dipole) had been disclosed in 20th century, while four new quantum phases had recently been found by our team (A. L. Kholmetskii, O. V. Missevitch, T. Yarman). In our analysis of physical meaning of these phases, we adopt that a quantum phase for a dipole represents a superposition of quantum phases for each charge, composing the dipole. In this way, we demonstrate the failure of the Schrödinger equation for a charged particle in an EM field to describe new quantum phase effects, when the standard definition of the momentum operator is used. We further show that a consistent description of quantum phase effects for moving particles is achieved under appropriate re-definition of this operator, where the canonical momentum of particle in EM field is replaced by the interactional EM field momentum. Some 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 AB (Aharonov-Bohm) effect is a pivotal quantum mechanical phenomenon that illustrates the fundamental role of the electromagnetic vector potential A in determining the phase of a charged particle's wave function, even in regions where the magnetic field B is zero.This effect demonstrates that quantum particles are influenced not only by the fields directly present but also by the potentials associated with those fields.In the AB effect, an electron beam is split into two paths, with one path encircling a solenoid and the other bypassing it.Despite the absence of a magnetic field in the regions traversed by the beams, the vector potential A associated with the magnetic flux  through the solenoid induces a phase shift in the electron's wave function.This phase shift, quantified by qc     , manifests as a change in the interference pattern observed in the detection screen.The phenomenon underscores the principle of gauge invariance in QED (quantum electrodynamics), where physical observables remain invariant under local gauge transformations of the vector and scalar potentials.This reinforces the notion that the vector potential A has a profound impact on quantum systems, beyond its classical role.This article outlines the AB effect, including its theoretical framework, experimental observations, and implications.The focus on the role of the vector potential in quantum mechanics provides a comprehensive understanding of this important phenomenon.
Marletto and Vedral [Phys. Rev. Lett. 125, 040401 (2020)] propose that the Aharonov-Bohm (AB) phase is locally mediated by entanglement between a charged particle and the quantized electromagnetic field, asserting gauge independence for non-closed paths. In this Comment, we critically analyze their model and demonstrate that the AB phase arises from the interaction with the vector potential A\mathbf{A}A, not from entanglement, which is a byproduct of the quantum electrodynamics (QED) framework. We show that their field-based energy formulation, intended to reflect local electromagnetic interactions, is mathematically flawed due to an incorrect prefactor and yields +qv⋅As +q \mathbf{v} \cdot \mathbf{A}_{\mathbf{s}} +qv⋅As​ in the Coulomb gauge, conflicting with QED's −qv⋅As -q \mathbf{v} \cdot \mathbf{A}_{\mathbf{s}} −qv⋅As​. This equivalence to qv⋅As q \mathbf{v} \cdot \mathbf{A}_{\mathbf{s}} qv⋅As​ holds only approximately in the Coulomb gauge under static conditions, failing for time-dependent fields and other gauges, undermining their claim of a gauge-independent local mechanism. Furthermore, we confirm that the AB phase is gauge-dependent for non-closed paths, contradicting their assertion. Our analysis reaffirms the conventional explanation in the semi-classical picture, where the AB phase is driven by the vector potential A\mathbf{A}A, with entanglement playing no causal role in its generation.
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.