The Physical Seam | phase without force
The Loop Remembers
Enclose magnetic flux without crossing its field, move the electron fringes by phase alone, then scramble every local gauge value and watch the loop holonomy stay fixed. Superconducting half-steps expose the even-odd result measured by Tonomura.
Two electron amplitudes pass on opposite sides of a region they cannot enter. In the ideal magnetic Aharonov-Bohm arrangement, the magnetic field vanishes along both accessible paths. Yet their relative phase knows the flux enclosed by the loop those paths make together. There is no local magnetic force on either path in this model. There is a gauge-invariant phase around the hole.
Layer one | move the fringesA field kept inside, a phase found outside
Drag the flux. The pale paths never cross the cyan field region. The two small arrows are the path amplitudes when they meet. Their angle is calculated from the enclosed flux, and the screen below is rebuilt from their sum.
θ = qΦ/ℏ = −2π Φ/(h/e)
I(x) = 2 + 2 cos(kx + θ)
phase ready
Positive flux with the chosen loop orientation gives a negative electron phase.
- relative phase
- −π rad electron sign, chosen orientation
- centre intensity
- 0.000000 arbitrary units, equal amplitudes
- loop holonomy
- −1.000000 + 0.000000i exp(iθ), gauge invariant
Half an electron period: bright and dark exchange places.
At zero flux, the centre is bright. At h/(2e), the relative phase is π modulo a full turn and the centre is dark. At h/e, the entire pattern returns. Reversing the flux reverses the lateral shift. The field drawing stays confined because that is the idealized premise being tested, not a numerical consequence of this one-dimensional interference equation.
Layer two | the flagship depth layerScramble everything local
“The vector potential is gauge dependent, so moving a slider attached to it proves nothing. And if the electron period is h/e, Tonomura's h/(2e) steps look inconsistent.”
Good. Do not trust any local arrow. The ring below has 128 sites and one complex hopping phase on each link. Press scramble gauge: every site phase and every local link arrow changes. Move the branch cut and the same total phase is stored on a different link. The shipped engine then checks the transformed Hamiltonian link by link.
αⱼ → αⱼ + χⱼ − χⱼ₊₁
H′ = UHU†
W = ∏ⱼ exp(iαⱼ) = exp(iθ)
representation zero | invariant ready
Enabled only in the ideal flux-quantized shield mode. This samples Φ = n h/(2e).
- Wilson loop W
- −1.000000 + 0.000000i product of all link phases
- H′ versus UHU†
- 0.000e+0 largest directed-link residual
- spectrum change
- 0.000e+0 largest sorted-energy difference
Normal shield: the continuous flux control from layer one sets the holonomy.
What changed: the local phases χⱼ, every displayed site color, the link phases, and the chosen location of the branch cut. Those are representation choices.
What did not: the Wilson loop, all 128 ring energies as a set, the fringe pattern, and the even-odd class. Those depend on the closed loop.
The half-step is not the electron's period
In the ideal thick-shield limit used by this model, the superconducting enclosure admits steps of h/(2e) because its mobile paired carriers have charge magnitude 2e. A single electron assigns each such step a phase of π modulo a full turn. Therefore n even → 0, n odd → π. Two superconducting steps make one electron period h/e. Switch the shield above, then drag n. The apparent factor-of-two conflict becomes a parity sampler.
The experiment behind that switch was not a bare cartoon. The expanded 1986 report describes a toroidal 20 nm Permalloy film, niobium layers of 200 nm and 300 nm, and a 100 nm copper layer that blocked the electron wave from entering the magnet. At 5 K, below the niobium transition, the measured phase was the even-odd result. At 15 K the shield was normal. The reported reversible transition was 9 ± 1 K, compared with the paper's quoted niobium critical temperature of 9.2 K.
The check | every displayed quantity has an owner
The browser has just recomputed the physical constants, fringe samples, lattice spectrum, Wilson product, and a deterministic gauge audit. The audit uses 100 distinct gauge scrambles. It checks 12800 transformed link phases and 25600 directed Hamiltonian entries. The spectrum panel evaluates 51328 energies.
| quantity | live value | status and origin |
|---|---|---|
| exact SI Planck constant h | loading | definition, exact |
| exact SI elementary charge e | loading | definition, exact |
| electron flux period h/e | loading | division performed live |
| superconducting step h/(2e) | loading | division performed live |
| ratio of those scales | loading | division performed live |
| fringe sample count | loading | free numerical truncation |
| spectral grid | loading | free numerical truncation |
| gauge audit maximum residual | loading | floating-point roundoff |
Free choices, all of them
- Charge convention: the electron has q = −e. Positive flux and the displayed counterclockwise loop orientation therefore give negative phase. Reverse either convention and the sign reverses, while every intensity and parity statement survives.
- Interference model: both path amplitudes have unit magnitude, the screen uses 8 spatial cycles, and the intensity has no envelope, decoherence, detector noise, or path imbalance. Intensity units are arbitrary.
- Rendering truncation: the screen is sampled at 1024 points. Canvas pixels are presentation only and depend on the device. The reported centre intensity is evaluated from the equation, not read back from pixels.
- Lattice truncation: 128 sites, nearest-neighbour hopping only, and t = 1 as an arbitrary energy unit. The flux plot uses 401 flux positions. These choices change visual resolution, not the Wilson-loop identity.
- Gauge audit: 100 deterministic pseudorandom transformations start from seed loading. Determinism makes the offline replay exact. It is a numerical stress test, not a probability claim.
- Branch cut: its location is entirely free. Concentrating the unscrumbled phase on one link makes that freedom visible. The scramble then redistributes it over every link.
Uncertainties and limits, all of them
- The SI values of h and e used here are exact definitions. Derived decimal display is rounded. JavaScript uses binary double precision, and the live gauge residual reports the resulting roundoff.
- The fabrication thicknesses above are nominal values reported in the opened paper. The report does not attach uncertainties to those four values in the passage used here, so this page does not invent any.
- The transition carries the paper's stated uncertainty of 1 K. The measurement temperatures and quoted critical temperature are transcriptions, not values inferred by this simulation.
- “Completely shielded” is the experimenters' operational conclusion. This page has not recovered a quantitative upper bound on residual field leakage from the opened sources, so it does not claim mathematical zero.
- In a general superconducting ring it is the fluxoid that is quantized. The live switch uses the ideal flux-quantized, thick-shield limit appropriate to the argument in the cited experiment. It does not model penetration depth, screening current, or fluxoid corrections.
- The accessible region is treated as multiply connected and field-free. A loop around the toroid has no spanning surface lying wholly in that accessible region, so setting B = 0 on the paths does not force the holonomy to vanish.
The independent Node verifier contains more than forty assertions, executes this page's own ABEngine, and deliberately mutates constants and logic to prove the checks can fail. Run node research/aharonov-bohm/verify-aharonov-bohm.mjs.
What is settled, and what is not
The magnetic phase law tested here is not an open empirical question. Ehrenberg and Siday published the enclosed-flux electron-optics result in 1949. Aharonov and Bohm gave their independent, broader formulation in 1959. Tonomura and colleagues' shielded measurements in 1986 closed the specific beam-penetration and ordinary field-leakage loopholes to the operational reach of their apparatus, with the temperature reversal and flux parity as internal checks.
As of 2026, the live dispute is interpretive: whether the most illuminating ontology is built from potentials, gauge-invariant loop variables, or a larger account that includes the sources. These descriptions do not predict different fringes here, and this experiment does not select a unique ontology. The nearest honest open edge is therefore not “does the phase exist?” It is how quantum gauge theories should express locality and subsystem structure without promoting a gauge-dependent A into an observable.
Sources actually opened for this build
- The Refractive Index in Electron Optics and the Principles of Dynamics, W. Ehrenberg and R. E. Siday, Proceedings of the Physical Society, Section B 62, 8 to 21 (1949), DOI 10.1088/0370-1301/62/1/303. A scan was opened because the publisher blocked automated access.
- Significance of Electromagnetic Potentials in the Quantum Theory, Y. Aharonov and D. Bohm, Physical Review 115, 485 (1959), DOI 10.1103/PhysRev.115.485.
- Evidence for Aharonov-Bohm Effect with Magnetic Field Completely Shielded from Electron Wave, A. Tonomura and colleagues, Physical Review Letters 56, 792 (1986), DOI 10.1103/PhysRevLett.56.792.
- Experimental Confirmation of Aharonov-Bohm Effect Using a Toroidal Magnetic Field Confined by a Superconductor, N. Osakabe and colleagues, Physical Review A 34, 815 to 822 (1986), DOI 10.1103/PhysRevA.34.815.
- The International System of Units, Bureau International des Poids et Mesures, SI Brochure, edition 9, version 4.01 (2026), DOI 10.59161/AUEZ1291.