The Vector Potential: the field behind the fields

Level 0 · The picture

Everyone has seen a magnet pick up a paperclip. Most people have heard that a magnet is surrounded by a field — an invisible push-and-pull that fills the space around it. Fewer people know that physicists found something underneath the field: a quieter, deeper quantity that the field is made from. Its name is the vector potential, and physicists write it with a single bold letter, A.

Here is the whole course in one sentence. The magnetic field tells you where the push is; the vector potential tells you how the space itself is "flowing", and electrons can feel that flow even where there is no push at all.

Now the surprising part. Take a coil of wire and run a current through it. In the textbook idealisation — an infinitely long solenoid, so that end effects vanish — the magnetic field is uniform inside the coil and exactly zero outside it. A real, finite coil leaks a little field outside; the ideal case is the one where the argument is clean, and it is the case drawn in the diagram above. But the flow map — the vector potential — circulates around the outside, in wide rings, and it does not vanish where B does. Keep hold of what those arrows are: an assignment of a vector to each point, not water and not a wind. Classical physics says an electron flying past the outside of the coil should feel nothing, because there is no field there. Quantum mechanics says otherwise: the electron's wave picks up a twist from the circulating A, and that twist shows up in an interference pattern. This was predicted in 1959 by Yakir Aharonov and David Bohm, first seen in 1960, and confirmed beyond doubt by Akira Tonomura's team in 1986 with a magnet wrapped in superconductor so that not one line of field could leak out. The electrons still noticed.

The interferometer with the field switched off

Split an electron beam, send the two halves either side of a magnetic flux that is completely confined and completely shielded, and bring them back together. On the paths themselves the magnetic field is exactly zero — the shaded corridor below is field-free the whole way. The vector potential there is not zero, and the fringes move. Dial the enclosed flux and watch the pattern slide sideways by exactly one fringe for every flux quantum. Then try the second control. It changes the gauge — the potential itself, at every point on both paths, as the arrows show — and the fringes do not move at all. Both controls are live; only one of them changes anything a detector can see.

Two controls: one moves the fringes, one cannot

Enclosed flux, in flux quanta h/e

0.00

Advance by one whole flux quantumReturn to zero enclosed flux

Gauge choice λ — the freedom in the potential itself

0.00

Return to the textbook gauge

• Upper path
• Lower path
• Confined flux
• Vector potential on the paths

What is happening: The enclosed flux is a whole number of quanta, so the phase difference is a whole number of turns and the pattern sits exactly where it started. This is the periodicity that makes the effect a measurement rather than a drift.

Magnetic field on the electron paths

zero, at every setting

Vector potential on the upper path, in this gauge

0.00 nWb/m

Vector potential on the lower path, in this gauge

0.00 nWb/m

Phase the gauge adds to each half

0.00 radians — the same for both halves, so it cancels

Phase difference between the two halves

0.00 radians

Fringe shift

0.00 fringe widths

What no choice of gauge can change

the field on the paths, the phase difference, the fringe shift

The precision rule this instrument exists to protect: the vector potential controls phase. No energy is stored in it, and none needs to be — the phase difference is set by the flux the two paths enclose between them, which is exactly the quantity you are dialling. And the folklore reading — that the effect proves the potential is physically real at a point — is exactly what the second control is here to foreclose: the honest observable is the loop integral, never the potential at a point.

Nothing happening is the result here, not a broken control. The potential is not unique: for any smooth function χ you may replace A by A + ∇χ, which changes the potential at every point and changes the magnetic field nowhere, because the curl of a gradient is zero. This instrument performs that substitution for real and then integrates the full potential, gauge term included, around the closed loop the two halves of the beam make between them. Each half does pick up a phase from the gauge term — exactly 2λ radians in this family — and both halves pick up the same one, so it cancels out of their difference. What survives is the loop integral, which is the enclosed flux, and that is the observable.

Tonomura's 1986 experiment answered the obvious objection by construction: the flux sat inside a toroidal magnet wrapped in a superconducting shield, so stray field was not merely small on the electron wave but excluded from it. The shift was still there, and it was still one fringe per flux quantum.

Scope: a drawn two-path interferometer with a perfectly confined flux, which is the geometry of the published experiment. Beam energy, path length and magnet size are not modelled, and no absolute intensity is asserted. Φ₀ = h/e ≈ 4.136e-15 Wb.

The source sheet: Tonomura et al. 1986, Evidence for Aharonov–Bohm effect with magnetic field completely shielded from electron wave

That is why Charles Chase — a Lockheed Skunk Works veteran who now runs a laboratory called the UnLab — said in his August 2026 interview that the vector potential "is actually more fundamental than the fields", that it "can exist with no fields being present", and that when it is there "it changes the phase of things with no energy exchange". He was describing the tool his team uses to try to put electrons and atoms in step with one another, the way a laser puts light in step. We will get there by Level 5.

Ways to think about it

• The field is the weather; the vector potential is the wind map the weather is drawn from.
• The vector potential is momentum per unit charge: it is how much momentum a charge "borrows" from the electromagnetic environment just by being there.
• Where the field is zero but A is not, nothing pushes you — but your clock runs differently depending on which way round you go.

Level 1 · Foundations

Michael Faraday, the greatest experimentalist of the nineteenth century, could not do the mathematics of fields, so he thought in pictures. In 1852 he described a state of the space around a magnet that he called the electrotonic state — a kind of stored, ready-to-act condition that showed itself only when something changed. James Clerk Maxwell, who turned Faraday's pictures into equations, gave that state a symbol and a name: the electromagnetic momentum. Today we call it the vector potential. So the idea is not a modern exotic add-on; it was there at the birth of the subject.

Three facts to carry forward

1. A is a vector field: an arrow at every point in space, with a size and a direction, measured in units of momentum per charge (volt-seconds per metre).
2. The magnetic field B is the swirl of A. Where A circulates, B points through the centre of the circulation.
3. The electric field has two sources: a slope in the scalar potential, and a change in time of the vector potential. When A changes, an electric field appears — that is electromagnetic induction, and it is what turns generators.

Level 2 · The rules

Now we can write the rules down. Two short equations connect the potentials to the fields.

Fields from potentials(1)

$$\mathbf{B} = \nabla \times \mathbf{A}, \qquad \mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}$$

Because B is only the swirl of A, you can change A without changing B, as long as the change has no swirl of its own. Adding the gradient of any smooth function to A leaves B untouched. This freedom is called gauge freedom, and it is the source of a century of arguments about whether A is "real". Level 3 settles the argument: the answer is that closed loops of A are real, even if the value at a single point is a matter of choice.

Gauge freedom(2)

$$\mathbf{A}' = \mathbf{A} + \nabla \chi, \qquad \phi' = \phi - \frac{\partial\chi}{\partial t}$$

The coil, done properly. A long solenoid of radius R carrying current has a uniform field B inside and zero outside. Outside, at distance r from the axis, the vector potential circulates around the axis with size

The whirlpool outside a solenoid(3)

$$A_\theta(r) = \frac{B\,R^2}{2r} \quad (r > R)$$

That last sentence is the key to everything above Level 2. The circulation of A around a closed loop equals the magnetic flux Φ through the loop. It does not matter that the field is zero on the loop; what matters is what the loop encloses.

Stokes' theorem for A(4)

$$\oint_{\text{loop}} \mathbf{A}\cdot d\mathbf{l} = \Phi_{\text{enclosed}}$$

Ways to think about it

• A single arrow of A is like a single reading on a meter whose zero you are free to set. A loop integral of A is like the difference between two readings: the arbitrary zero cancels, and what is left is physical.
• Gauge freedom is not a flaw. It is a symmetry, and in Level 4 it turns out to be the symmetry that generates electromagnetism.

The loop, not the local value. An interferometer never reads A at a point; two paths make a loop, and the loop sees only the flux it encloses. Move the loop yourself: put it right beside the coil, where the arrows are longest, and the phase is zero; put it around the coil, far out where the arrows are short, and the phase is the whole flux.

The loop, not the local value

An interferometer does not read the vector potential at a point. It compares two paths, and two paths make a loop. Drag the loop, or use the sliders, and watch which number the phase follows.

The loop and the coil

Loop centre, across

2.6 coil radii

Loop centre, up and down

0.0 coil radii

Loop radius

1.2 coil radii

Flux inside the coil

1.0 flux quanta

Put the loop around the coilPut the loop right beside the coilStart again

You can also drag the loop in the drawing with a pointer.

What the interferometer sees: The loop does not contain the coil. The vector potential along it is not zero, yet it adds up to nothing round the loop, so the phase is zero.

Flux enclosed by the loop:

0.00 flux quanta

Vector potential added up round the loop:

0.00 flux quanta

Phase difference:

0.00 fringes

Largest vector potential anywhere on the loop:

0.114 flux quanta per coil radius

The point: The local value of the vector potential can be large while the phase is zero, and small while the phase is a whole fringe. What is physical is the total round the closed loop, and that total is the enclosed flux.

Scope: an ideal long coil with uniform flux inside, drawn in one convenient gauge. Changing the gauge changes the arrows and leaves every reading below them unchanged.

The source sheet: the measured Aharonov–Bohm effect

Level 3 · Undergraduate

Momentum, done honestly. In classical mechanics with a magnetic field, the momentum that appears in the equations is not mv. It is the canonical momentum:

Canonical momentum(5)

$$\mathbf{p} = m\mathbf{v} + q\mathbf{A}$$

The Hamiltonian that produces the correct equations of motion is built from that combination, and it is the same combination that appears in the Schrödinger equation — which is why quantum mechanics cannot avoid A even when it could, classically, get away with B alone:

Minimal coupling(6)

$$\hat H = \frac{(\hat{\mathbf p} - q\mathbf{A})^2}{2m} + q\phi$$

The Aharonov–Bohm phase. Send an electron wave along two paths that pass on either side of a shielded solenoid and recombine. Along each path the wave's phase advances by (q/ħ) times the line integral of A. The difference between the two paths is a loop integral, and by equation (4) that is the enclosed flux:

The Aharonov–Bohm phase(7)

$$\Delta\varphi = \frac{q}{\hbar}\oint \mathbf{A}\cdot d\mathbf{l} = \frac{q\,\Phi}{\hbar}$$

The Aharonov–Bohm effect is measured, replicated, and used every day in electron holography. That the vector potential acts on quantum phase where fields vanish is established physics, not interpretation.

A worked example, with the signs kept straight. Two different quanta live in this subject and confusing them is the classic slip. For single-electron interference the natural period is h/e ≈ 4.14 × 10⁻¹⁵ weber: an electron carries q = −e, so a flux of h/e gives Δφ = −2π, a full cycle, and the fringes return to where they started. Half that, h/(2e) ≈ 2.07 × 10⁻¹⁵ weber, gives Δφ = −π and shifts the pattern by exactly half a fringe: bright becomes dark. That same number h/(2e) is also the superconducting flux quantum Φ₀, but for the opposite reason — there the carriers are pairs with q = −2e — and the two facts should never be run together. Now the illustration. A solenoid one micrometre across carrying a field of one millitesla encloses B·πr² ≈ 7.85 × 10⁻¹⁶ Wb, which is 0.19 of the electron period h/e: a shift of about a fifth of a fringe. This is an illustrative calculation, not Tonomura's apparatus, but a shift of that size is comfortably measurable in an electron microscope, which is how measurements of this kind are made.

Ways to think about it

• Phase is a clock hand on each electron. A turns the hand without touching the electron's speed or energy: "a change of phase with no energy exchange", as Chase put it.
• Gauge freedom is harmless because the phase of a single path is unmeasurable, but the difference between two paths is a loop, and loops see only flux.

Level 4 · Graduate

The vector potential is a connection. In modern language, the wavefunction of a charged particle is not a function with a fixed phase; it is a section of a bundle in which the phase reference can be rotated independently at every point in space. A rule is needed for comparing phases at neighbouring points. That rule is the connection, and the connection is precisely qA/ħ. The field strength is the curvature of the connection — the failure of parallel transport around a small loop to return the phase to its starting value. Equation (7) is then the statement that the Aharonov–Bohm phase is a holonomy: carry the phase around a closed loop and it comes back rotated by the enclosed curvature.

Local U(1) gauge symmetry(8)

$$\psi(\mathbf{x}) \to e^{\,i q \chi(\mathbf{x})/\hbar}\,\psi(\mathbf{x}), \qquad \mathbf{A} \to \mathbf{A} + \nabla \chi$$

Berry's generalisation. In 1984 Michael Berry showed that any quantum system carried slowly around a closed loop in its parameter space acquires a phase that depends only on the geometry of the loop. The Aharonov–Bohm phase is the special case where the parameter is position and the connection is the electromagnetic one. The same mathematics governs the polarisation of light in a coiled fibre, the anomalous Hall effect, and the topological classification of materials.

Superconductors: where A becomes visible to the naked eye. In a superconductor all the electron pairs share one macroscopic phase θ. The supercurrent is set by the gauge-invariant combination of the phase gradient and A:

The London equation(9)

$$\mathbf{J}_s = \frac{n_s q^2}{m}\left(\frac{\hbar}{q}\nabla\theta - \mathbf{A}\right)$$

Two consequences follow immediately, and the first is worth writing exactly. The shared phase must return to itself around any closed path, so what is quantised is not the bare flux but the fluxoid: the flux plus the line integral of the supercurrent term around the path.

Fluxoid quantisation(9a)

$$\Phi + \oint_C \frac{m}{n_s q^2}\mathbf{J}_s\cdot d\mathbf{l} = n\frac{h}{2e}, \qquad n\in\mathbb{Z}$$

Second, if a thin barrier separates two superconductors, the phase difference across it drives a current with no voltage at all — the Josephson effect, which has its own course on this site. Chase's interview moves from the vector potential to Josephson junctions for exactly this reason: they are the devices in which quantum phase is an engineering variable.

A caution that belongs beside all of this: pairing does not suspend the Pauli exclusion principle. The electrons in a Cooper pair are still fermions with spin ½, and it is the pair — a composite with integer spin — that can share a single quantum state. Cooling never repeals exclusion; it changes what the relevant particle is.

Flux quantisation, the Meissner effect and the Josephson effects are measured to extraordinary precision; the volt is now defined through them.

Ways to think about it

• A is not a force field; it is the rulebook for comparing phases between neighbouring points. Curvature of the rulebook is what you feel as a magnetic field.
• Gauge invariance is not "A is unphysical". It is that the observables are the gauge-invariant combinations — loop integrals of A, and the bracket in equation 9 — and superconductors turn the first of those into integers you can count.

Level 5 · Research frontier

The phase-controlled matter beam. Here is what Charles Chase and his former Lockheed colleague Dr. Moe Arman are trying to build, in Chase's own words from the interview: "we're trying to put particles like electrons or atoms in phase together, like a laser". A laser works because photons are bosons — they are happy to share a state. Electrons and atoms with odd numbers of constituents are fermions, and fermions "cannot occupy the same state like a photon can". Bose–Einstein condensates get round this by cooling to almost absolute zero. The UnLab's idea is different: use the vector potential to steer the phase of each particle's wave, with no energy exchange, so that particles "that normally don't want to be in phase get in phase". Chase names the Aharonov–Bohm effect as the mechanism and calls it, without hedging, "the secret sauce".

What would such a beam be for? Chase gives two answers. The first is power: "think of a beam that is a million times more powerful than a laser". The second, which he says now excites him more, is chemistry: "all molecules are waves… by controlling the phase of those waves we can make them combine in different ways that we currently can't", including direct atomic assembly with a calculated resolution of about 0.2 nanometres. Both applications rest on the same physics you learned in Level 3: phase is a controllable variable, and A is the control.

The matter-wave beam is a proposal with patents, a physical mechanism, and a stated target. The proposed test is the first published measurement of induced coherence in a fermion beam. That is the experiment to watch.

Conditioned fields and the non-Abelian idea. In 1997 H. David Froning and Terence Barrett proposed that a beam whose polarisation is deliberately structured — "conditioned" — could carry a field symmetry richer than ordinary electromagnetism's U(1): the SU(2) symmetry of the weak interaction, with its own vector potentials that do not commute. Froning's experiment proposal appears in NASA's Breakthrough Propulsion Physics workshop proceedings. In gauge-theory language the claim is that a suitably shaped light beam could carry a non-Abelian connection, and therefore couple to matter in ways — including, they argued, to inertia — that ordinary radiation cannot. The vector potential is the natural language for this because, as Level 4 showed, it is the connection. This is the thread the channel followed in July 2026 as the ancestor of the Pais effect, and it is where the vector potential meets the rest of this compendium.

Non-Abelian structure in engineered beams is a well-posed proposal with a named experiment and no measurement yet.

What to watch

1. The UnLab's first published coherence measurement on an electron or atom beam.
2. Any group reporting a conditioned-beam inertia test on a suspended mass, with the field configuration documented.
3. Josephson-junction arrays as phase-coherent emitters — the subject of this site's companion course and of the "gaser" proposal in the research log.

Teaching aids

Three paper exercises

1. Assign, do not transport. On squared paper, draw a circle for the coil and mark twelve points around it. At each point draw an arrow tangent to its own circle, with length proportional to 1/r using equations 3 and 8 as the reference for what the arrows mean. Now ask what is moving. Nothing is: each arrow is a value assigned to a point, and no matter crosses the page. Students who can say why this is not a picture of a fluid have understood the Aharonov–Bohm setup before meeting it.
2. Close the loop on paper. Draw two different paths from a start point to an end point, one either side of the coil, and shade the region they enclose. Read equations 4 and 5 and have students argue why the two paths give different phases while a single path's phase is a matter of convention. The enclosed area is the whole answer, and the argument works entirely with a pencil.
3. A common offset changes nothing. Give every student a personal offset — add 40 minutes to your watch — and ask them to time the same song and compare. The durations agree because a common, constant offset cancels in every difference. Say clearly where the analogy stops: gauge freedom lets the offset differ from point to point and from moment to moment, and equation 2 shows what the scalar potential has to do in return.

Self-check (answers below)

1. Outside an ideal infinite solenoid the magnetic field is zero. Is the vector potential zero there too?
2. Two electron paths enclose a flux of h/2e. By how much do the interference fringes shift, and what is the sign of the phase for an electron?
3. A physicist adds the gradient of a time-dependent function χ to A. What must happen to the scalar potential, and what stays the same?
4. What exactly is quantised around a superconducting ring, and when is it the flux itself?
5. In one sentence: why does a laser need bosons, and does cooling a metal switch the Pauli exclusion principle off?

Answers. (1) No — it circulates around the coil, falling off as 1/r. (2) Half a fringe, bright becoming dark; with q = −e the phase is Δφ = −π. (3) The scalar potential must shift by −∂χ/∂t at the same time; the fields, the loop integrals of A and every measurable quantity stay the same, while the value of A at each point and the wavefunction's phase convention change together. (4) The fluxoid — the flux plus the circulation of the supercurrent — in units of h/2e; it reduces to the flux alone on a path deep inside a thick ring where the supercurrent has died away. (5) Photons can share a state, so they fall into step easily; cooling switches nothing off, and pairing works because a Cooper pair of two spin-½ fermions is a composite that behaves as a boson.

One-page summary for the wall

• A is a flow map; B is its swirl; E is the slope of φ plus the change of A in time.
• The value of A at a point is a convention; its integral around a loop is the enclosed flux, and that is physics. A gauge change moves φ by −∂χ/∂t at the same time.
• Quantum phase feels A directly, with no force and no energy exchange (Aharonov–Bohm, 1959; measured 1960 and 1986).
• Gauge symmetry — the freedom to reset phase locally — is what generates electromagnetism.
• Superconductors quantise the fluxoid in units of h/2e — the pairs carry q = −2e, and Pauli exclusion still holds for the electrons inside them — and Josephson junctions make phase an engineering variable.
• The frontier: steer phase with A to bring fermions into step (the UnLab), and shape beams to carry richer gauge structure (Froning–Barrett).