Chapter 02
The Donut.
A magnetic field hidden inside a tiny donut. Electrons that never touch the field still feel it. Classical physics says this is impossible. Quantum mechanics says it's a loop talking to itself.
Set up.
A toroidal magnet — a donut — confines all of its magnetic field B inside the torus. Outside the donut, in the entire region where electrons travel, B = 0. Classically, no force, no effect.
Send electrons through a two-slit setup with the donut between the slits and the screen. Vary the flux Φ_B trapped inside the donut. The interference fringes shift.
The fringe pattern shifts as you change Φ_B even though both electron paths run through B = 0. Local field reasoning predicts no effect; loop holonomy predicts exactly the shift you see.
What just happened?
The two electron paths form a closed loop around the donut. Quantum mechanically, each path picks up a phase that depends on the electromagnetic potential A, not just the field B. Around the loop:
The local value of A is gauge-dependent — you can change it freely by a gauge transformation. The loop integral is not. It's gauge invariant. It's physical. It's what shifts the fringes.
OPH translation.
Each electron path is a patch. Both patches report "B = 0 here." But when we glue the two patches together at the screen, the loop remembers the enclosed flux. The fringe shift is the observer-facing record of that holonomy.
Local field-free does not mean globally effect-free.
Same shape as three-body.
| Local story | Loop story | |
|---|---|---|
| Aharonov–Bohm | No B touches the electron. | Loop carries enclosed flux. |
| Three-body | Each pair-force is simple. | Triangle carries action holonomy. |
The same OPH machinery handles both. The bottom line: physics is what survives gluing.