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arXiv:quant-ph/0402203 · Physical Review Letters

The Effect of a Magnetic Flux Line in Quantum Theory

Yakir Aharonov, T. Kaufherr

Published Feb 20, 2004• 26 citations
arXivHTMLPDF

Abstract

The nonlocal exchange of the conserved, gauge invariant quantity ${e}^{\frac{i}{\ensuremath{\hbar}}({p}_{k}\ensuremath{-}\frac{e}{c}{A}_{k}){L}^{k}}$, ${L}^{k}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, $k=1,2$ between the charged particle and the magnetic flux line (in the $k=3$ direction) is responsible for the Aharonov-Bohm effect. This exchange occurs at a definite time, before the wave packets are brought together to interfere, and can be verified experimentally.

DOI
10.1103/physrevlett.92.070404

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The nonlocal exchange of the conserved, gauge invariant quantity ei\ensuremathℏ(pk\ensuremath−ecAk)Lk{e}^{\frac{i}{\ensuremath{\hbar}}({p}_{k}\ensuremath{-}\frac{e}{c}{A}_{k}){L}^{k}}e\ensuremathℏi​(pk​\ensuremath−ce​Ak​)Lk, Lk=const{L}^{k}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}Lk=const, k=1,2k=1,2k=1,2 between the charged particle and the magnetic flux line (in the k=3k=3k=3 direction) is responsible for the Aharonov-Bohm effect. This exchange occurs at a definite time, before the wave packets are brought together to interfere, and can be verified experimentally.
The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region inaccessible to the particle. Classically, such a nonlocal effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle and the electromagnetic vector potential A\mathbf{A}A, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics A\mathbf{A}A is in some sense more ``fundamental'' than B\mathbf{B}B in spite of the former being gauge dependent and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB effect can be fully accounted for by considering only the gauge-invariant B\mathbf{B}B field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge-invariant formulation is that we must give up locality---the AB phase for the particle will arise from the change in the action for the B\mathbf{B}B field in the region inaccessible to the particle.
The Aharonov-Bohm effect is a physical phenomenon in which the quantum state of a charged particle acquires a phase shift that is directly proportional to the magnetic flux, ΦΦΦ, due to a (classical) magnetic field, B{\mathbf B}B, which is confined in a spatial region from which the magnetic field cannot escape. Even though the charged particle is not allowed to interact with the magnetic field, it accumulates a phase shift that affects the interference pattern produced. Not surprisingly, this apparent nonlocality is puzzling and counter intuitive. In this work, we provide an explanation that explains the physics underlying this apparent nonlocality. We find that the role of the confined magnetic field is to impart a puncture in the configuration space, R2\mathbb{R}^2R2, of the charge. Therefore, the quantum state corresponding to the charged quantum particle acquires the phase shift due to its response to the modified topology of the configuration space, R2−{0}\mathbb{R}^2-\{0\}R2−{0}, corresponding to the charge.
The Aharonov-Bohm (AB) effect highlights the fundamental role of electromagnetic potentials in quantum mechanics, manifesting as a phase shift for a charged particle in field-free regions. While well-established for static magnetic fluxes, the effect's behavior under time-varying fluxes remains an open and debated question. Employing the WKB method, we derive the AB phase shift for a time-dependent magnetic vector potential, demonstrating that for circular paths in the quasistatic regime, it is proportional to the time-averaged enclosed magnetic flux, ΔφAB=1T∫0TeΦ(t) dtΔφ_{\rm AB} = \frac{1}{T} \int_0^T e Φ(t) \, dtΔφAB​=T1​∫0T​eΦ(t)dt, with the total phase shift, including kinetic contributions, equaling eΦ(0)e Φ(0)eΦ(0). For non-circular paths, the phase shift depends on both the flux history and path geometry, revealing the effect's hybrid nature involving gauge potentials and induced electric fields. We verify the consistency of our gauge choice with Maxwell's equations and discuss the implications for local versus nonlocal interpretations of the AB effect. We also generalize the results to scenarios with nonzero external magnetic fields, where the enclosed flux is through the actual electron paths, and for circular paths of radius RRR, the AB phase shift is also proportional to the time average of the enclosed flux Φenc(R,t)Φ_{\rm enc}(R,t)Φenc​(R,t), with the total phase shift depending only on the initial enclosed flux eΦenc(R,0)e Φ_{\rm enc}(R,0)eΦenc​(R,0); for general non-circular paths, the external magnetic field affects trajectories and phase accumulation through the Lorentz force, leading to additional path dependence. These findings clarify the role of gauge-dependent potentials and induced fields in the generalized AB effect, offering new theoretical insights and potential applications in quantum technologies.
We present a rigorous no-go theorem that excludes \emph{all} gauge-invariant dynamical explanations of the generalized, time-dependent Aharonov--Bohm (AB) effect. In this generalization, the AB phase is determined by the time-averaged magnetic flux enclosed by the particle's path, ΔϕAB=eℏ⋅1T∫0TΦ(t) dt\Delta \phi_{\mathrm{AB}} = \frac{e}{\hbar} \cdot \frac{1}{T} \int_0^T \Phi(t) \, dtΔϕAB​=ℏe​⋅T1​∫0T​Φ(t)dt. Using a carefully designed protocol in which a constant flux Φ\PhiΦ persists only for a finite interval Δt\Delta tΔt after wavepacket splitting and is switched to zero before detection, we demonstrate that during the phase-generating period all gauge-invariant observables of the electron remain identical to those in a zero-flux experiment. Relativistic causality further prohibits backreaction from the source. Consequently, no gauge-invariant quantity---whether local or nonlocal, field-based or source-mediated---can account for the observed phase, which depends solely on flux history. We also critically examine Healey's holonomy interpretation and show that it, too, fails to explain the time-dependent AB effect, as its static formalism lacks the dynamical structure needed to capture temporal averaging. This result establishes the inescapable physical reality of gauge potentials in quantum theory and clarifies the foundational status of the AB effect as a genuine manifestation of potentials rather than fields.
The Aharonov-Bohm effect is a fundamental topological phenomenon with a wide range of applications. It consists of a charge encircling a region with a magnetic flux in a superposition of wave packets having their relative phase affected by the flux. In this work we analyze this effect using an entropic measure known as realism, originally introduced as a quantifier of a system's degree of reality and mathematically related to notions of global and local quantum coherence. More precisely, we look for observables that lead to gauge-invariant realism associated with the charge before it completes its loop. We find that the realism of these operators has a sudden change when the line connecting the center of both wave packets crosses the solenoid. Moreover, we consider the case of a quantized magnetic-field source, pointing out similarities and differences between the two cases. Finally, we discuss some consequences of these results.