The physical interpretation of the Aharonov–Bohm effect: action of the vector potential versus local accessible fields

In the Aharonov–Bohm (AB) effect, charged-particle interference fringes shift by an amount depending on the magnetic flux confined inside an impenetrable cylinder, even though the particles never enter a region of nonzero magnetic field. Since Aharonov and Bohm's 1959 prediction, a foundational controversy has persisted over what physically causes the fringe shifts: the electromagnetic vector potential acting where no field is accessible, or infinitesimal accessible fields (induced electric fields during flux switching, leakage from finite solenoids, penetration into the cylinder). Berry, Chambers, Large, Upstill and Walmsley (1980) reviewed the limiting arguments showing that the wavefunction tends to the AB solution regardless, judged the question 'metaphysical and devoid of observational implication', and left it unsettled. The question has since re-emerged as a sharp local-versus-nonlocal dilemma in a continuing debate (Vaidman 2012; Aharonov, Cohen and Rohrlich 2015, 2016; and analyses through 2025) with mutually inconsistent proposed resolutions and no accepted decisive experiment. The problem is to settle the dynamical status of the AB phase — local gauge-invariant interaction, primary role of the potential, or genuine nonlocality — and to determine whether any operational criterion can decide it.

Background

The Aharonov–Bohm effect (Aharonov and Bohm 1959) concerns the scattering of a beam of particles of charge $q$ by an impenetrable cylinder of radius $R$ (a solenoid) containing a confined magnetic flux $\Phi$. Classically, no force acts on the particles because they never enter the magnetic field. Quantum mechanically, the wavefunction picks up a phase difference proportional to the flux parameter $\alpha = q\Phi/h$ (with $h$ Planck's constant) between paths passing on opposite sides of the cylinder, so the interference fringe pattern shifts periodically as $\Phi$ varies, although all observable quantities depend only on $\alpha$ modulo an integer. In classical electrodynamics the electromagnetic vector potential $\mathbf{A}$ (defined by $\mathbf{B} = \nabla \times \mathbf{A}$) is a mathematical convenience: it is gauge-dependent, since the transformation $\mathbf{A} \to \mathbf{A} + \nabla\Lambda$ leaves the fields unchanged. The AB effect was the first clear indication that in quantum mechanics potentials have observable significance, and electron interference experiments confirmed the fringe shifts (Chambers 1960; Tonomura and co-workers 1986, using a toroidal ferromagnet with the flux completely shielded from the electron wave, which removed leakage and penetration fields as practical explanations of the observed shifts). A review of the experiments and of common misconceptions is given by Batelaan and Tonomura (2009).

The source of this question is a study by Berry, Chambers, Large, Upstill and Walmsley (1980) of the wavefronts, i.e. the surfaces of constant phase, of the AB wavefunction $\psi_{\mathrm{AB}}$. They showed that $\psi_{\mathrm{AB}}$ possesses a wavefront dislocation on the flux line — a phase singularity where a definite number of wave crests terminate, that number being the integer nearest to $\alpha$ — and they realized the same topological structure in a classical analogue: surface water waves crossing an irrotational 'bathtub' vortex, whose circulating but vorticity-free flow plays the role of the vector potential, with photographed dislocation patterns matching the theory. In the water-wave analogue the phase shift has an unproblematic local cause: the moving medium physically carries the waves. This motivated the paper's closing discussion of what, in the quantum case, actually causes the fringe shifts.

That discussion (section 6) reviews three theoretical arguments that the effect can always be explained by electromagnetic fields accessible to the particles, however small: an argument that switching the flux on slowly lets the accessible induced electric field outside the cylinder account for the evolution of the wavefunction to $\psi_{\mathrm{AB}}$; the analysis of Casati and Guarneri (1979), who took the cylinder to be slightly penetrable and showed that for any finite barrier height the shifts are explainable by the magnetic field now accessible to the particles; and the result of Roy (1980), who took a solenoid of finite length and showed that a gauge can always be chosen in which the vector potential is expressed entirely in terms of the weak magnetic field leaking from the solenoid ends into the accessible region. In every case, however, the limiting process always yields $\psi_{\mathrm{AB}}$ itself, so a purely field-based explanation must accept that infinitesimal causes produce finite effects, whereas taking the vector potential (in the gauge-invariant, quantum-mechanically observable deviation of $\alpha$ from an integer) as a primary local causative agent avoids that discordance. The authors concluded that 'the question of the reality of the AB effect' is metaphysical and devoid of observational implication, but the controversy did not end: it re-emerged in modern form as a dilemma — the AB phase is due either to a local but gauge-dependent potential, or to nonlocal gauge-invariant fields, but not to local gauge-invariant fields (Stewart 2016; Aharonov, Cohen and Rohrlich 2015) — and it has been debated in physical and philosophical literature ever since (Healey 1997).

Problem Statement

Determine the physical interpretation of the Aharonov–Bohm effect. As posed in the source discussion: are the predicted and observed fringe shifts really effects of the vector potential acting on the particles in regions where no electromagnetic field is accessible to them, or can they be explained entirely in terms of accessible electromagnetic fields — such as the induced electric field when the confined flux is switched on, however slowly; the magnetic field of a cylinder that is penetrable at any finite barrier height; or the weak leakage field of a solenoid of arbitrarily great but finite length? Any explanation of the second kind must confront the fact that the wave always tends to the Aharonov–Bohm wavefunction $\psi_{\mathrm{AB}}$ no matter how the limiting process is performed, so that infinitesimal causes would produce finite effects. In the modern restatement, the task is to decide which of the following accounts of the AB phase is correct: a dynamical explanation purely in terms of local interactions with gauge-invariant accessible quantities (fields, or field variables of a quantized source entangled with the particle); an account in which the vector potential (or a gauge-invariant component of it) is a primary local causative agent; or an account in which the effect is a genuinely nonlocal influence of gauge-invariant fields. It must further be determined whether any operational criterion or experiment can decide among these accounts — in particular whether the nonlocal or the local description can be discriminated in principle, or whether, as the source authors maintained, the question is devoid of observational implication. Whether any operational criterion or experiment can discriminate among these accounts is part of the question, and an argument that the accounts are operationally indistinguishable — as the source authors concluded — is itself a substantive answer. The scope is the Aharonov–Bohm phase in standard quantum mechanics and quantum electrodynamics, in the full generality in which the source and the subsequent debate state it, without restriction to particular geometries, flux values, or source models.

References

1. M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, and J. C. Walmsley (1980), 'Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue', European Journal of Physics 1, 154-162. DOI: 10.1088/0143-0807/1/3/008, https://doi.org/10.1088/0143-0807/1/3/008
2. Y. Aharonov and D. Bohm (1959), 'Significance of Electromagnetic Potentials in the Quantum Theory', Physical Review 115, 485. DOI: 10.1103/PhysRev.115.485, https://doi.org/10.1103/PhysRev.115.485
3. R. G. Chambers (1960), 'Shift of an Electron Interference Pattern by Enclosed Magnetic Flux', Physical Review Letters 5, 3. DOI: 10.1103/PhysRevLett.5.3, https://doi.org/10.1103/PhysRevLett.5.3
4. Giulio Casati and Italo Guarneri (1979), 'Aharonov-Bohm Effect from the "Hydrodynamical" Viewpoint', Physical Review Letters 42, 1579. DOI: 10.1103/PhysRevLett.42.1579, https://doi.org/10.1103/PhysRevLett.42.1579
5. S. M. Roy (1980), 'Condition for Nonexistence of Aharonov-Bohm Effect', Physical Review Letters 44, 111. DOI: 10.1103/PhysRevLett.44.111, https://doi.org/10.1103/PhysRevLett.44.111
6. Akira Tonomura, Nobuyuki Osakabe, Tsuyoshi Matsuda, Takeshi Kawasaki, Junji Endo, Shinichiro Yano, and Hiroji Yamada (1986), 'Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave', Physical Review Letters 56, 792. DOI: 10.1103/PhysRevLett.56.792, https://doi.org/10.1103/PhysRevLett.56.792
7. Herman Batelaan and Akira Tonomura (2009), 'The Aharonov–Bohm effects: Variations on a subtle theme', Physics Today 62 (9), 38. DOI: 10.1063/1.3226854, https://doi.org/10.1063/1.3226854
8. Lev Vaidman (2012), 'Role of potentials in the Aharonov-Bohm effect', Physical Review A 86, 040101. DOI: 10.1103/PhysRevA.86.040101, https://doi.org/10.1103/PhysRevA.86.040101 (preprint arXiv:1110.6169, https://arxiv.org/abs/1110.6169).
9. Yakir Aharonov, Eliahu Cohen, and Daniel Rohrlich (2015), 'Comment on "Role of potentials in the Aharonov-Bohm effect"', Physical Review A 92, 026101. DOI: 10.1103/PhysRevA.92.026101, https://doi.org/10.1103/PhysRevA.92.026101 (preprint arXiv:1604.05748, https://arxiv.org/abs/1604.05748).
10. Lev Vaidman (2015), 'Reply to "Comment on "Role of potentials in the Aharonov-Bohm effect"', Physical Review A 92, 026102. DOI: 10.1103/PhysRevA.92.026102, https://doi.org/10.1103/PhysRevA.92.026102
11. Yakir Aharonov, Eliahu Cohen, and Daniel Rohrlich (2016), 'Nonlocality of the Aharonov-Bohm effect', Physical Review A 93, 042110. DOI: 10.1103/PhysRevA.93.042110, https://doi.org/10.1103/PhysRevA.93.042110 (preprint arXiv:1502.05716, https://arxiv.org/abs/1502.05716).
12. A. M. Stewart (2016), 'On the paper "Role of potentials in the Aharonov-Bohm effect"', arXiv:1605.05470, https://arxiv.org/abs/1605.05470
13. M. V. Berry (2017), 'Gauge-invariant Aharonov–Bohm streamlines', Journal of Physics A: Mathematical and Theoretical 50, 43LT01. DOI: 10.1088/1751-8121/aa8b2d, https://doi.org/10.1088/1751-8121/aa8b2d
14. Kolahal Bhattacharya (2021), 'Demystifying the nonlocality problem in Aharonov–Bohm effect', Physica Scripta 96, 084011. DOI: 10.1088/1402-4896/ac0189, https://doi.org/10.1088/1402-4896/ac0189 (preprint arXiv:2108.07701, https://arxiv.org/abs/2108.07701).
15. Alexander L. Kholmetskii, Oleg V. Missevitch, and Tolga Yarman (2024), 'Role of electromagnetic energy and momentum in the Aharonov–Bohm effect', Proceedings of the Royal Society A 480, 20230286. DOI: 10.1098/rspa.2023.0286, https://doi.org/10.1098/rspa.2023.0286
16. Masashi Wakamatsu (2025), 'Unveiling the Role of Vector Potential in the Aharonov–Bohm Effect', Symmetry 17, 935. DOI: 10.3390/sym17060935, https://doi.org/10.3390/sym17060935
17. Henrique Gomes (2025), 'The Aharonov-Bohm effect: reality and folklore', arXiv:2509.19407, https://arxiv.org/abs/2509.19407
18. Richard Healey (1997), 'Nonlocality and the Aharonov-Bohm Effect', Philosophy of Science 64, 18-41. DOI: 10.1086/392534, https://doi.org/10.1086/392534