There recently appear some interesting attempts to explain the AB-effect through the interaction between the charged particle and the solenoid current mediated by the exchange of a virtual photon. A vital assumption of this approach is that AB-phase shift is proportional to the change of the interaction energy between the charged particle and solenoid along the path of the moving charge. Accordingly, they insist that the AB-phase change along a path does not depend on the gauge choice so that the AB-phase shift for a non-closed path is in principle measurable. We however notice the existence of two fairly different discussions on the interaction energy between the solenoid and a charge particle, the one is due to Boyer and the other is due to Saldanha and others. In the present paper, based on a self-contained quantum mechanical treatment of the combined system of a solenoid, a charged particle, and the quantized electromagnetic fields, we show that both interaction energies of Boyer and of Saldanha are in fact gauge invariant at least for non-singular gauge transformations but they are destined to cancel each other. Our analysis rather shows that the origin of the AB-phase can be traced back to other part of our effective Hamiltonian. Furthermore, based on the path-integral formalism with our effective Lagrangian, we explicitly demonstrate that the AB-phase shift for a non-closed path is not a gauge-variant quantity, which means that it would not correspond to direct experimental observables.
There recently appear some interesting attempts to explain the AB-effect through the interaction between the charged particle and the solenoid current mediated by the exchange of a virtual photon. A vital assumption of this approach is that AB-phase shift is proportional to the change of the interaction energy between the charged particle and solenoid along the path of the moving charge. Accordingly, they insist that the AB-phase change along a path does not depend on the gauge choice so that the AB-phase shift for a non-closed path is in principle measurable. We however notice the existence of two fairly different discussions on the interaction energy between the solenoid and a charge particle, the one is due to Boyer and the other is due to Saldanha and others. In the present paper, based on a self-contained quantum mechanical treatment of the combined system of a solenoid, a charged particle, and the quantized electromagnetic fields, we show that both interaction energies of Boyer and of Saldanha are in fact gauge invariant at least for non-singular gauge transformations but they are destined to cancel each other. Our analysis rather shows that the origin of the AB-phase can be traced back to other part of our effective Hamiltonian. Furthermore, based on the path-integral formalism with our effective Lagrangian, we explicitly demonstrate that the AB-phase shift for a non-closed path is not a gauge-variant quantity, which means that it would not correspond to direct experimental observables.
The Aharonov-Bohm (AB) phase is usually associated with a line integral of the electromagnetic vector potential generated by an external current source, such as a solenoid. According to this interpretation, the AB phase of a nonclosed path cannot be observed, as the integral depends on the gauge choice of the vector potential. Recent attempts to explain the AB effect through the interaction between a charged particle and an external current, mediated by the exchange of quantum photons, have assumed that the AB phase shift is proportional to the change in interaction energy between the charged particle and the external current source. As a result, these attempts argue that the AB phase change along a path does not depend on the gauge choice, and that the AB phase shift for a nonclosed path is in principle measurable. In this paper, we critically examine this claim and demonstrate that the phase obtained through this approach is actually gauge dependent and not an observable for a nonclosed path. We also provide a brief critical discussion of the proposed experiment for observing the AB phase shift of a nonclosed path.
In the Aharonov-Bohm (AB) effect, a superposed charge acquires a detectable phase by enclosing an infinite solenoid, in a region where the solenoid's electric and magnetic fields are zero. Its generation seems therefore explainable only by the local action of gauge-dependent potentials, not of gauge-independent fields. This was recently challenged by Vaidman, who explained the phase by the solenoid's current interacting with the electron's field (at the solenoid). Still, his model has a residual nonlocality: it does not explain how the phase, generated at the solenoid, is detectable on the charge. In this Letter, we solve this nonlocality explicitly by quantizing the field. We show that the AB phase is mediated locally by the entanglement between the charge and the photons, like all electromagnetic phases. We also predict a gauge-invariant value for the phase difference at each point along the charge's path. We propose a realistic experiment to measure this phase difference locally, by partial quantum state tomography on the charge, without closing the interference loop.
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.
The standard semiclassical treatment of the Aharonov-Bohm (AB) effect, which predicts no observable phase shift for open electron paths in field-free regions due to gauge dependence, is shown to be incomplete. By fully quantizing both the probe electron and the electromagnetic source within quantum electrodynamics (QED), we derive an effective action that decomposes the AB phase into two equal contributions: a gauge-dependent probe phase from the source's potential acting on the electron, and a gauge-invariant source phase arising from the electron's quantum backreaction on the source via virtual-photon exchange. For open paths, where the probe phase vanishes under gauge choice, the source phase persists as a measurable part of the closed-loop value, isolated through a polarization--magnetization decomposition that separates its invariant bulk from a gauge-dependent boundary term. This uncovers a leading-order QED correction to nonrelativistic quantum mechanics, resolves longstanding debates on the AB phase's origin and locality, and refines gauge principles by revealing hidden gauge invariants in formally gauge-dependent quantities. While challenging to test directly, a conceptual scheme involving phase transfer to auxiliary qubits illustrates its potential observability, emphasizing its novelty beyond conventional closed-loop paradigms.
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, 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 in the Coulomb gauge, conflicting with QED's −qv⋅As. This equivalence to 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, with entanglement playing no causal role in its generation.