We consider a particle in a coherent superposition of states with different electric charges moving in the vicinity of a magnetic flux. Formally, it should acquire a (gauge-dependent) Aharonov–Bohm (AB) relative phase between the charge states, even for an incomplete loop. If measurable, such a geometric, rather than topological, AB phase would seem to break gauge invariance. Wick, Wightman and Wigner argued that since (global) charge-dependent phase transformations are physically unobservable, charge state superpositions are unphysical (‘charge superselection rule’). This would resolve the apparent paradox in a trivial way. However, Aharonov and Susskind disputed this superselection rule: they distinguished between such global charge-dependent transformations and transformations of the relative inter-charge phases of two particles, and showed that the latter could in principle be observable! Finally, the paradox again disappears once we consider the ‘calibration’ of the phase measured by the Aharonov–Susskind phase detectors, as well as the phase of the particle at its initial point. It turns out that such a detector can only distinguish between the relative phases of two paths if their (oriented) difference forms a loop around the flux.
We consider a particle in a coherent superposition of states with different electric charges moving in the vicinity of a magnetic flux. Formally, it should acquire a (gauge-dependent) Aharonov–Bohm (AB) relative phase between the charge states, even for an incomplete loop. If measurable, such a geometric, rather than topological, AB phase would seem to break gauge invariance. Wick, Wightman and Wigner argued that since (global) charge-dependent phase transformations are physically unobservable, charge state superpositions are unphysical (‘charge superselection rule’). This would resolve the apparent paradox in a trivial way. However, Aharonov and Susskind disputed this superselection rule: they distinguished between such global charge-dependent transformations and transformations of the relative inter-charge phases of two particles, and showed that the latter could in principle be observable! Finally, the paradox again disappears once we consider the ‘calibration’ of the phase measured by the Aharonov–Susskind phase detectors, as well as the phase of the particle at its initial point. It turns out that such a detector can only distinguish between the relative phases of two paths if their (oriented) difference forms a loop around the flux.
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 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, 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 is in some sense more ``fundamental'' than 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 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 field in the region inaccessible to the particle.
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.
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.
Contrary to long-standing textbook doctrine, we demonstrate that the Aharonov-Bohm (AB) phase for a closed interferometric loop — the relative phase governing interference — becomes gauge dependent when backreaction of the test charge on the electromagnetic source is included in a fully quantum treatment. The standard Stokes-theorem argument fails because different interfering paths become entangled with distinct field configurations, permitting branch-dependent gauge transformations. We prove that the total phase decomposes into (1) a gauge-invariant core tied to the fixed background flux, which fully accounts for the observed interference shift, and (2) a purely gauge-dependent backreaction term that can be eliminated by an appropriate gauge choice and carries no observable consequences. This decomposition follows from a general framework for observables in gauge theories, which we establish from a foundational axiom that purely gauge-dependent quantities have no observational consequences. From this axiom, we derive theorems showing that any gauge-dependent quantity contains an extractable gauge-invariant core that constitutes its measurable content, while the remaining gauge-dependent component is physically irrelevant. This framework resolves the apparent paradox between formal gauge dependence and robust experimental observations, and offers a conceptually sharper understanding of observables in quantum electrodynamics, non-abelian gauge theories, general relativity, and other gauge-based theories.
Recent works showed that the Aharonov-Bohm (AB) phase difference for a quantum charged particle can be written in terms of electric and magnetic fluxes in a space-time surface whose boundaries are the possible particle worldlines in the interferometer, relative to the possible paths. After presenting this result in a more detailed way, reinforcing its topological nature, we study the magnetic and electric versions of the AB effect in different inertial reference frames. We find a particular reference frame for a magnetic AB effect where the magnetic flux has a null contribution for the AB phase difference, which is entirely due to an electric flux. Also, we find a particular reference frame for an electric AB effect where the electric flux has a null contribution for the AB phase difference, which is entirely due to a magnetic flux. In this sense, the nomenclatures magnetic AB effect and electric AB effect lose their meaning. We have electromagnetic AB effects.