Electrons travel through a double-slit. Behind the slits sits an idealized solenoid carrying magnetic flux Φ — but its field B is perfectly confined inside. The electrons travel where B = 0 everywhere along their paths. Yet the interference pattern shifts as you turn the flux up: the vector potential A is non-zero outside, and quantum phases pick it up along the way. Bohm and Aharonov, 1959; first experimentally confirmed by Tonomura (1986).
when the field doesn't matter but the potential does
Classical electromagnetism says: charged particles feel forces from fields, not potentials. The vector potential A is a mathematical convenience; only B = ∇ × A and E = −∇φ − ∂A/∂t have physical meaning. Different gauge choices give different A but the same B, so A "can't be real."
Quantum mechanics says otherwise. The Schrödinger equation for a charged particle is:
iℏ ∂ψ/∂t = (1/(2m))(−iℏ∇ − qA)² ψ + qφ ψ
A appears directly in the equation. A wavefunction going from a to b picks up a phase:
ψ → ψ · exp(iq/ℏ · ∫_a^b A · dl)
For the double-slit-plus-solenoid setup, even though B = 0 along both electron paths, the line integral of A around the closed loop (path through slit 1, around, back through slit 2) equals the enclosed flux Φ. The phase difference between the two paths is:
Δφ = (q/ℏ) · ∮ A · dl = (q/ℏ) · Φ
With each flux quantum Φ₀ = h/q ≈ 4.14·10⁻¹⁵ Wb, the interference pattern shifts by one full fringe. The electrons never go near the field, yet they "know" about it. The vector potential A — not just the field — is physically real. Aharonov-Bohm is the first experimentally confirmed gauge-theoretic phenomenon, and the prototype for everything in modern gauge theory: Wilson loops, Berry phases, topological insulators.