Aharonov–Bohm Effect

An experiment that proves phase transport, not force fields

Claim: The Aharonov–Bohm effect is pure transport phase accumulation.
No force, no field interaction, no nonlocal influence.

Experimental Facts

Path A — Transport Geometry (Loop Non-Closure)

Primitive: Phase accumulated under transport around a loop

Two transport corridors enclose a flux tube.

Although no force acts locally, the transport loop does not close.

$$ \Delta \phi = \oint \mathbf{A} \cdot d\mathbf{l} $$

The line integral measures phase residue from loop transport.

The magnetic field never appears — only the transport history does.

Path B — Phase Accounting (Gauge-Invariant Statistics)

Primitive: Phase differences determine detection statistics

The probability distribution depends only on phase difference:

$$ I \propto \cos^2\!\left( \frac{\Delta \phi}{2} \right) $$

Gauge transformations shift phase uniformly and cancel.

Only the enclosed transport residue survives.

Agreement Condition

$$ \Delta \phi_{\text{geometry}} = \Delta \phi_{\text{statistics}} $$

Both paths predict the same fringe shift.

What This Experiment Actually Proves

Why Gauge Fields Exist at All

Gauge potentials encode transport constraints.

They are not physical substances — they track phase non-closure.

Connection to π and Transport Invariants

The phase shift depends on enclosed area in phase space.

Once again, invariants arise from loop transport.