Aharonov-Bohm phase

The Aharonov-Bohm phase is the quantum-mechanical phase Φ_AB = (q/ℏ) ∮ A·dℓ acquired by a charged particle traversing a closed spatial loop in a region where the electromagnetic vector potential A is non-zero. By Stokes's theorem the loop integral equals the magnetic flux Φ_B enclosed by the loop, so Φ_AB = (q/ℏ) Φ_B. The phase is gauge-invariant in the sense that adding a gradient ∂_μΛ to A_μ changes the open-path integral but leaves the closed-loop integral unchanged (the gradient integrates to zero around any closed loop in a simply-connected region of A).

Because phases are physically observable only modulo 2π, the AB phase is periodic in the magnetic flux quantum Φ_0 = h/|q|. For an electron (q = e) this is h/e ≈ 4.14 × 10^{−15} Wb; for a Cooper pair (q = 2e) the relevant quantum is half this, the superconducting flux quantum. As the enclosed flux increases from 0 to Φ_0, the AB phase advances from 0 to 2π and the interference pattern in a two-slit setup shifts laterally by exactly one fringe — at Φ_B = Φ_0/2 the interference fringes flip from constructive to destructive at the original maxima, and at Φ_B = Φ_0 the pattern returns to its original position. The 1959 Aharonov-Bohm prediction and the 1986 Tonomura electron-holography experiment that closed every classical loophole made this the cleanest experimental demonstration that the four-potential A^μ — not just the field tensor F^{μν} — is the fundamental observable in quantum electrodynamics.