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.

A 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)

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.

Scramble 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)

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.

Sources actually opened for this build

1. 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.
2. Significance of Electromagnetic Potentials in the Quantum Theory, Y. Aharonov and D. Bohm, Physical Review 115, 485 (1959), DOI 10.1103/PhysRev.115.485.
3. 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.
4. 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.
5. The International System of Units, Bureau International des Poids et Mesures, SI Brochure, edition 9, version 4.01 (2026), DOI 10.59161/AUEZ1291.