Electrons feel electromagnetic potentials even where fields are zero—proving potentials are physically real
Classical electromagnetism says that forces on charged particles come from electric and magnetic fields (E and B). The potentials (φ and A) are just mathematical conveniences—useful for calculations but not "real" in themselves. Change the potential by adding a constant? No physical effect.
Consider a long solenoid carrying electric current. Inside, there's a uniform magnetic field B. Outside, the field is exactly zero—the solenoid's design confines the field perfectly. Classical physics says nothing outside should "know" about the magnetic field inside.
But the vector potential A, which satisfies B = ∇ × A, extends outside the solenoid even where B = 0. It circles around the solenoid, and its line integral around any closed path equals the enclosed magnetic flux.
Split an electron beam into two paths that go around opposite sides of the shielded solenoid, then recombine them on a detector screen. The electrons never enter the region with non-zero B—they stay entirely in the field-free zone outside.
Classical physics predicts: no phase difference between paths (no force acted on either beam). The interference pattern should be independent of the enclosed flux.
Quantum mechanics predicts: the phase difference depends on the enclosed flux Φ. The interference pattern shifts when you change the current in the solenoid—even though the electrons never encounter any magnetic field!
The effect was definitively confirmed by Akira Tonomura and colleagues in 1986. Using electron holography and a superconducting shield to guarantee zero magnetic field leakage, they observed the predicted interference shifts. The results matched quantum mechanics exactly.
Non-locality: The electron's behavior depends on the enclosed flux— information from a region it never visits. This is a form of quantum non-locality, though it doesn't allow faster-than-light signaling.
Gauge invariance: Classical gauge freedom (ability to add gradients to A without changing physics) becomes more subtle in quantum mechanics. While gauge invariance holds for observables, the potential itself carries physical content.
Topological effects: The AB effect depends on the topology of the electron paths—whether they enclose flux or not. This connects to modern ideas in topological quantum matter and gauge theories.
Similar effects exist for electric potentials (Aharonov-Bohm electric effect) and even gravitational potentials (gravitational AB effect). The principle generalizes: in quantum mechanics, potentials—not just fields—are the fundamental physical quantities.