Stefano Filippi

Category: Quantum & Gauge

  • What the Electrons Respond To

    A conducting sphere is charged, and in the region outside it the electromagnetic field is exactly zero — no electric field, no magnetic field, no Poynting flux — for as long as the charging lasts. A receiver placed in that region nevertheless produces a measurable voltage, arriving at the speed of light.

    The first half of that has been established elsewhere, along with the considerable qualifications it requires. This article is about the second half: what the receiver is responding to, given that there is nothing there to push its electrons around.

    The route that leads nowhere

    The obvious place to start is the classical equation of motion. A conductor responds to fields; write the force on a conduction electron and see what it gives.

    It gives zero. The Lorentz force is the charge times the electric field plus the velocity cross the magnetic field, and both fields vanish. The equation is satisfied identically, for an electron of any velocity, at every point and every instant.

    That is not a difficulty to be worked around. It is correct, and it is the whole content of the classical level: no force, no classical dynamics. An identity of the form nothing equals nothing determines nothing, and no amount of care in solving it will produce a mechanism.

    So the classical equation of motion is the wrong instrument. It is not that it gives the wrong answer — it gives no answer, and says so honestly.

    Where the coupling actually lives

    The right instrument is the Hamiltonian, and the reason is worth stating in general terms before looking at this particular case.

    The relation between a force balance and a Hamiltonian is the relation between statics and analytical mechanics. Knowing that the normal force on a block equals its weight tells you nothing about how a constrained system moves; the dynamics come from the Lagrangian or the Hamiltonian, and they contain terms that no force diagram displays. The same asymmetry applies here, and it is the standard structure of mechanics rather than anything peculiar to this problem.

    Written out for a conduction electron in the Lorenz gauge, the minimal-coupling Hamiltonian is:

    H=p^22m−qm𝐀⋅p^+iℏq2mb+q2A22m+qφH = \frac{\hat{p}^2}{2m} – \frac{q}{m}\mathbf{A}\cdot\hat{p} + \frac{i\hbar q}{2m}b + \frac{q^2 A^2}{2m} + q\varphi

    Four terms beyond the free kinetic energy. The last of them is the one that matters, and the crucial point about it is structural: the term involving the scalar potential is not part of the kinetic sector. It does not descend from the momentum operator, it is not what generates the Lorentz force, and it does not vanish when the fields vanish. It sits there as long as the potential is non-zero, whatever the fields are doing.

    What it does, in a conductor, is shift the electrochemical potential:

    μ=EF+qφ(t)\mu = E_F + q\varphi(t)

    uniformly across the receiver. Every conduction electron finds its energy reference displaced, in step, by an amount that tracks the potential in time.

    What happened to the other three terms

    This is where the argument could easily have gone wrong, and where checking it changes the answer.

    Two of the remaining terms — the paramagnetic and the diamagnetic — are exactly the ones a bulk conductor usually picks up, and they involve the vector potential, which is emphatically not zero here. A first estimate, taking the paramagnetic term alone against a single-electron energy scale, suggested they might contribute a per cent or a few per cent of the response.

    That estimate does not survive the correct treatment, and the reason it fails is instructive. In any bulk conduction response, gauge invariance of the conductivity — expressed through the f-sum rule, standard in linear response theory — forces the paramagnetic and diamagnetic terms to combine. They cannot appear separately in an observable. What they combine into is a response that depends only on the physical field:

    jbulk(ω)=σ(ω)E(ω)≡0because 𝐄tot≡0j_{bulk}(\omega) = \sigma(\omega) E(\omega) \equiv 0 \qquad \text{because } \mathbf{E}_{tot} \equiv 0

    and the physical field is identically zero throughout the dynamic regime. So the bulk conduction current is not one to four per cent. It is exactly zero — the same null result as at the classical level, now extended from the Lorentz force to the entire bulk-conduction channel, paramagnetic and diamagnetic terms included.

    The estimate was superseded rather than confirmed, and the correction went in the direction of strengthening the null result rather than weakening it. Recording that matters more than the number: an approximate answer of one to four per cent and an exact answer of zero are different kinds of statement, and the second is the one that stands.

    The fourth term, the non-Hermitian one, is bounded separately and comes out around one part in a million. It is not the mechanism either.

    What the electrons respond to

    That leaves the potential term, and the question of how it produces a current. Here the linear-response machinery has to be applied with some care, because the natural way to set it up gives nothing.

    Standard practice couples the perturbation to the electric field and computes the current response. Do that here, in the long-wavelength limit appropriate to a receiver small compared to the distances involved, and the answer vanishes — consistently with the field being zero, and uselessly. That route is not wrong, it is empty. It cannot be used.

    The physically correct route couples the perturbation to the charge density instead, which is to say to the potential itself rather than to its gradient. The response function obtained that way remains finite in the same limit. And the physical reading of that statement is the point of the whole exercise:

    The electrons are not responding to a spatial gradient of the potential across the receiver. To leading order there is none — the potential is essentially uniform over the device. They are responding to the change in time of their own local electrochemical potential.

    That is an ordinary mechanism. It is classical linear response of the Kubo–Drude kind, the same machinery used for conductivity in metals, with a relaxation time in the tens of femtoseconds — nine orders of magnitude faster than the transients involved. Nothing exotic is being invoked. What is unusual is only the perturbation it is being applied to.

    The agent and the condition

    One distinction deserves to be made explicit, because the two halves of it are easy to conflate and they are not the same statement.

    The agent is the potential itself, not its rate of change. The coupling term in the Hamiltonian contains the potential, and it is present whenever the potential is non-zero, whether or not it is changing. A constant potential still shifts the electrochemical potential and still displaces the equilibrium charge distribution of the conductor accordingly.

    What a constant potential does not do is sustain a current. The displaced distribution, once reached, is stationary — the conductor equilibrates to its new reference and then nothing further happens.

    The rate of change enters only through continuity:

    ∂ρ∂t+∇⋅𝐉=0\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0

    If the charge density at a point is changing, a current must flow. So a non-zero rate of change is what turns a one-off equilibration into a continuing, measurable current — the condition under which a receiver can register the effect, rather than a second mechanism competing with the first.

    Two levels of the same physics: the coupling in the Hamiltonian, and the observability enforced by continuity. Not two rival explanations.

    What the bookkeeping quantities are, and are not

    A programme of this kind accumulates derived quantities — a scalar built from the divergence of the vector potential, an energy density formed from it, a flux, several integrated energies. It is worth being blunt about their status, because the temptation to promote them is real and the discipline against it is what keeps the account honest.

    None of them is independent physical energy, flux, or radiated power.

    The scalar quantity itself is gauge-dependent by construction: it vanishes identically in Coulomb gauge, which is enough to settle the question on its own. The energy density and flux built from it are local diagnostics, and where they are non-trivial the ordinary electromagnetic stress-energy tensor is identically zero — so there is nothing for them to be carrying. Several of the integrated quantities numerically anticipate a physical energy, which is a useful property and not the same as being one. Others turn out to be local re-expressions of circuit power that has already been counted once at the terminals; counting it twice would be an error, not a discovery.

    Exactly one quantity in the family is physical field energy, and it is the settled electrostatic energy after the transient is over.

    What all the others share is a single common status, and it is better stated once than corrected six times: they are gauge-dependent diagnostics of where and when the physical process is active — that process being the potential coupling in the Hamiltonian, and nothing else. A diagnostic that tells you accurately when something is happening is worth having. It is not the thing that is happening.

    Not an Aharonov–Bohm effect

    The comparison is unavoidable and should be made carefully, because the resemblance is real and the identification would be wrong.

    Aharonov and Bohm established that a potential can have observable consequences in a region where the gauge-invariant field vanishes. That principle is what is at work here too, and the debt should be acknowledged.

    But the scalar version of their effect requires the electric field to vanish at the location of the charge, which is achieved with a Faraday-shielded enclosure, and the observable is a quantum phase shift measured by interference. Here the geometry is open and unshielded, a particular field quantity at the receiver is not zero, and the mechanism is classical linear response producing a terminal voltage — no interference, no phase measurement, no shielding.

    Same principle, different mechanism. Not a macroscopic reproduction, and describing it as one would be an overstatement that the details do not support.

    What remains open

    One structural question in this account is not settled. The construction of the vector potential used throughout is argued to be the unique causal one consistent with the source and the gauge condition, but the argument is presented informally rather than as a proof. A formal demonstration combining the causality requirement with the multipole selection rule has not been given.

    Until it is, the correct description is that the construction is well motivated and consistent, not that it is unique. The distinction matters, and marking it is cheaper than defending it later.

    Sources

  • S. Filippi, “The Scalar Longitudinal b-Channel in Lorenz Gauge: Foundations and an Explicit Coulomb-Gauge Verification,” 2026.
  • S. Filippi, “Local, Causal Field Seeding in Lorenz Gauge via Quaternion Formalism: Longitudinal Scalar Wavefront and Experimental Evidence,” Zenodo, 2026. doi:10.5281/zenodo.21032019
  • S. Filippi, “Field Exclusion for a Monopolar Source in Lorenz Gauge,” 2026.
  • R. Kubo, “Statistical-Mechanical Theory of Irreversible Processes. I,” J. Phys. Soc. Jpn., vol. 12, pp. 570-586, 1957.
  • G. D. Mahan, Many-Particle Physics, 3rd ed. New York: Kluwer, 2000.
  • Y. Aharonov and D. Bohm, “Significance of Electromagnetic Potentials in the Quantum Theory,” Phys. Rev., vol. 115, pp. 485-491, 1959.
  • Preprints and technical notes are deposited on Zenodo under ORCID 0009-0007-3660-8498.

  • The Term That Isn’t Hermitian

    Write down the Schrödinger equation for a charged particle in an electromagnetic field. Replace the momentum operator by the canonical one, expand, and collect terms. Four of them are familiar. The fifth is not:

    H^=p^22m−qm𝐀⋅p^+iℏq2mb+q2A22m+qφ,b=∇⋅𝐀\hat{H} = \frac{\hat{p}^2}{2m} – \frac{q}{m}\mathbf{A}\cdot\hat{p} + \frac{i\hbar q}{2m} b + \frac{q^2 A^2}{2m} + q\varphi, \qquad b = \nabla\cdot\mathbf{A}

    where b is the divergence of the vector potential. The factor of i in front is not decoration. It means this term is not Hermitian: take its adjoint and the sign flips.

    That should stop anyone who has been taught quantum mechanics properly. The Hamiltonian is supposed to be Hermitian. Hermiticity is what guarantees real energies and unitary evolution, and unitary evolution is what guarantees that probability is conserved. A non-Hermitian term in the Hamiltonian of an ordinary electron in an ordinary field looks like a claim that something very basic has gone wrong.

    It has not. What follows is why the term is there, what it does, and why the honest conclusion is deflationary — which is the interesting part, not a disappointment.

    Where it comes from

    The term is not inserted. It appears on its own, from the standard minimal-coupling prescription, provided one thing: that the Lorenz gauge is used, so that the divergence of the vector potential is not set to zero.

    In Coulomb gauge that divergence vanishes by definition, and the term disappears with it. This is worth noting immediately, because it locates the term precisely: it is a feature of a particular representation, not of the physics. Nothing observable can depend on which gauge one chose, and nothing here will.

    What the term does, though, is different from what the others do. Every other term in the expansion is Hermitian and generates a phase — the wavefunction rotates in the complex plane and its magnitude is untouched. This one generates a real exponential:

    ψ(t)=ψ0Tb(t),Tb=exp⁡(−qφ(t)2mc2)\psi(t) = \psi_0 \, T_b(t), \qquad T_b = \exp\left(-\frac{q\varphi(t)}{2mc^2}\right)

    which multiplies the amplitude. Not the phase. The probability density is rescaled.

    An open system, not a broken one

    The immediate consequence is that the continuity equation for probability acquires a source:

    ∂ρ∂t+∇⋅𝐉=qmb(t)ρ\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = \frac{q}{m} b(t) \, \rho

    Probability is no longer locally conserved. Written down cold, that is alarming.

    But this is a familiar structure, and it has a name. It is exactly what happens when a quantum system is coupled to an environment and one writes an equation for the system alone. Probability flows out of the description not because it is destroyed but because it has gone somewhere the description does not cover. Non-Hermitian effective Hamiltonians are standard in open-system quantum mechanics for precisely this reason.

    So the question becomes: what is the environment here? And the answer is the scalar potential itself, or more precisely the source that is driving it. The electron is not isolated. It sits in a potential that is changing in time because something elsewhere is moving charge around, and the term in question is the channel through which that dependence enters.

    Read this way, the non-Hermiticity is not a defect. It is the signature of having written down a system that is genuinely open, and it is telling us the size of the coupling to what was left out.

    Why it does not violate anything

    There is a deeper reason not to be alarmed, and it is the point of this article.

    The Schrödinger equation with minimal coupling is not fundamental. It is the non-relativistic limit of the Dirac equation — the first term in a systematic expansion in powers of the particle’s velocity over the speed of light, obtained by the Foldy–Wouthuysen procedure. The full theory it comes from is unitary, with a Hermitian Hamiltonian and conserved probability, with no exceptions.

    A term appearing at first order in that expansion inherits the status of the expansion. It is a statement about the truncated description, not about nature. Non-Hermiticity here is a property of where the series was cut, not a property of the electron.

    This is testable within the formalism rather than by appeal to authority. Carry the expansion to the next order and no second non-Hermitian term appears — the structure does not proliferate, which is what one would expect if it were a genuine feature and not an artefact of truncation.

    The same reasoning disposes of a more dramatic reading. The amplitude factor above involves the ratio of the electrostatic energy to the rest energy of the electron, and it appears to diverge when that ratio approaches one, at around a million volts. It is tempting to see a threshold there — the Klein paradox, mixing of positive and negative energy states, pair production out of the vacuum.

    That reading fails, and it fails on a second condition that is independent of the first. Mixing the two Dirac sectors requires not only a large enough potential but one that varies on the scale of the Compton wavelength and the Compton time: about four hundred femtometres, about a zeptosecond. A macroscopic apparatus — centimetres across, switching in nanoseconds — is away from that condition by ten to twelve orders of magnitude. By the adiabatic theorem, a potential that slow does not mix the sectors however large it becomes.

    So what actually happens if the amplitude condition is met? Nothing exotic. An electron traverses a strong electrostatic potential adiabatically, and the correct description is the exact relativistic energy relation treated in the usual approximation. The apparent divergence is an artefact of stopping the expansion at first order, not a physical instability. Nothing pathological survives.

    How big is it

    Having established that the term is real, calculable, and harmless, the remaining question is whether it matters. It is worth answering with a number rather than an adjective.

    The comparison to make is with ordinary screening. When a potential is applied to a metal, the conduction electrons redistribute — the standard Thomas–Fermi mechanism, in which the perturbation shifts the local Fermi level and charge moves among the states available near it. An electron deep in the Fermi sea cannot participate: there is no empty state for it to move into.

    The amplitude factor works differently, and the difference is instructive. It rescales every occupied state by the same amount regardless of energy, because the term carries no energy label and cannot tell whether a given electron sits at the Fermi edge or far below it. There is no redistribution between states, so there is no Pauli restriction, and the full electron density enters rather than just the density of states at the Fermi level.

    That sounds like an advantage, and in a sense it is: the mechanism reaches electrons that ordinary screening cannot touch. The ratio of the two effects works out to two-thirds of the Fermi energy divided by the rest energy of the electron:

    |δρTbδρTF|=23EFmc2≈9×10−6\left| \frac{\delta\rho_{T_b}}{\delta\rho_{TF}} \right| = \frac{2}{3} \frac{E_F}{mc^2} \approx 9 \times 10^{-6}

    Seven electron-volts against five hundred and eleven thousand. The ratio is about nine parts in a million.

    For the experimental configuration in question, with terminal voltages of a few volts, this puts the contribution of the quantum channel at around thirty microvolts. A standard eight-bit oscilloscope has a single quantisation step several orders of magnitude larger than that on the relevant range. The effect is not merely hard to measure: it is buried beneath the resolution of the instrument one would use to measure everything else in the experiment.

    The observable signal is dominated, entirely, by the ordinary classical mechanism.

    What this is worth

    Stated plainly: a term appears in a standard Hamiltonian, it is not Hermitian, it modifies amplitudes rather than phases, it gives the equations the structure of an open quantum system, it violates nothing, and it is far too small to see.

    That is a negative result, and it deserves to be reported as one rather than dressed up. But it is not an empty one, for three reasons.

    The first is that it delimits. A calculated upper bound on a channel is worth more than an untested suspicion that the channel might matter, and it removes the temptation to attribute an observed signal to something exotic when a classical mechanism is available and adequate.

    The second is methodological. Non-Hermitian effective Hamiltonians turn up in a great many places, and they attract a certain kind of overreading — loss of unitarity, probability leaking out of the universe, energy from nowhere. The discipline that dissolves those readings is always the same: ask what approximation the term lives inside, and check whether the full theory it descends from is unitary. Here it is, and that settles it.

    The third is the one worth keeping. A truncated theory that produces a non-Hermitian term is not misbehaving. It is reporting, in the only language available to it, that something has been left outside the description — and the size of the term is a measure of how much. Read that way, an approximation announcing its own incompleteness is not a failure of the approximation. It is the most useful thing an approximation can do.

    Sources

    • S. Filippi, “Local, Causal Field Seeding in Lorenz Gauge,” 2026.
    • S. Filippi, “Field Exclusion for a Monopolar Source in Lorenz Gauge,” 2026.
    • H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems. Oxford: Oxford University Press, 2002.
    • L. L. Foldy and S. A. Wouthuysen, “On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit,” Phys. Rev., vol. 78, pp. 29-36, 1950.
    • N. W. Ashcroft and N. D. Mermin, Solid State Physics. New York: Holt, Rinehart and Winston, 1976.
    • O. Klein, “Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac,” Z. Phys., vol. 53, pp. 157-165, 1929.
    • Preprints and technical notes are deposited on Zenodo under ORCID 0009-0007-3660-8498.

  • A Phase Without a Field

    For most of the history of electromagnetism, the potentials were understood as a convenience. Useful, often indispensable in practice, but not themselves part of the furniture of the world.

    The reasoning was sound. Maxwell’s theory can be written entirely in terms of the electric and magnetic fields. Those are what push on a charge; those are what an instrument responds to. The potentials φ and A enter as mathematical scaffolding — quantities from which the fields can be derived by differentiation, and which make many calculations far easier than they would otherwise be. And as the previous article discussed, they are not even uniquely defined: infinitely many choices of potential describe the same physical situation. A quantity you can change at will without changing anything measurable does not look like a physical object.

    In 1959 that position became untenable.

    A field that is not there

    Consider a long solenoid — a tightly wound coil carrying a steady current. Inside it there is a magnetic field. Outside it, if the coil is long enough, there is essentially none: the field is confined to the interior.

    The vector potential is a different matter. Outside the solenoid, where B vanishes, A does not. It circulates around the coil, falling off with distance but never reaching zero.

    Classically this is of no consequence whatsoever. A charged particle passing outside the solenoid experiences no magnetic field, therefore no magnetic force, therefore nothing at all. The non-zero A in that region is exactly the sort of thing one expects of a gauge-dependent bookkeeping quantity: present in the mathematics, absent from the physics.

    A classical counterpart of this geometry does exist, and is worth noting in passing. If the solenoid is driven with an alternating current, a conducting loop encircling it — but never entering it — develops a measurable electromotive force, even though the magnetic field along the loop is negligible. Faraday’s law in integral form accounts for this without difficulty, since it depends on the flux threading the circuit rather than on the field at the conductor. Rousseaux and co-workers, who studied the configuration theoretically and experimentally in 2008, named it the Maxwell–Lodge effect after an 1889 paper of Oliver Lodge, and argued that it is best understood in terms of the vector potential.

    Two differences matter. The classical effect requires the current to vary in time; with a steady current there is no electromotive force and nothing to measure. And the field outside a real solenoid is small rather than exactly zero — a gap that would take decades to close for the quantum case.

    What Yakir Aharonov and David Bohm proposed, in a paper published in Physical Review under the title Significance of Electromagnetic Potentials in the Quantum Theory, was that in quantum mechanics this is not true. An electron travelling through a region where the fields are rigorously zero is nonetheless affected by the potential, in a way that produces an observable consequence.

    Why the potentials, and not the fields

    The reason is structural, and it predates the effect by three decades. When electromagnetism is coupled to quantum mechanics, the fields do not appear in the equations at all. The potentials do.

    The coupling is introduced by the minimal substitution: in the Hamiltonian of a particle of charge q, the momentum is replaced by the canonical momentum, and the scalar potential is added to the energy.

    𝐩→𝐩−q𝐀,E→E−qφ\mathbf{p} \rightarrow \mathbf{p} – q\mathbf{A}, \qquad E \rightarrow E – q\varphi

    Written out, this is the operator that governs the evolution of the wavefunction:

    iℏ∂ψ∂t=[12m(−iℏ∇−q𝐀)2+qφ]ψi\hbar\frac{\partial \psi}{\partial t} = \left[\frac{1}{2m}\left(-i\hbar\nabla – q\mathbf{A}\right)^2 + q\varphi\right]\psi

    There is no way to write this in terms of E and B. The Schrödinger equation for a charged particle is a statement about φ and A, and always was — a fact that attracted little comment for as long as everything it predicted could also be obtained from the fields.

    Gauge invariance survives because a change of gauge multiplies the wavefunction by a position-dependent phase, which leaves every probability unchanged. But a phase that cancels in the modulus need not cancel between two interfering paths — and that is the whole of the effect.

    How you would see it

    The prediction concerns interference, so it is worth a sentence on what that means here.

    An electron beam split into two paths and recombined produces a pattern of alternating light and dark bands — fringes — because the electron behaves as a wave, and the two halves of that wave arrive either in step or out of step depending on the difference in what they have encountered along the way. Shift the relative timing of the two halves, and the whole pattern slides sideways. The size of that shift is a measurement.

    Now run one path on each side of the solenoid. Neither electron ever enters a region with a magnetic field. But the two paths, taken together, enclose the solenoid — and the phase difference between them is

    Δθ=qℏ∮𝐀⋅dℓ=qΦℏ\Delta\theta = \frac{q}{\hbar}\oint \mathbf{A}\cdot d\boldsymbol{\ell} = \frac{q\Phi}{\hbar}

    where the integral runs around the closed loop formed by the two paths, and Φ is the magnetic flux threading it. The fringes shift, by an amount fixed by the flux inside a region the electrons never visited.

    The part that is easy to state wrongly

    It is tempting to summarise this as “the vector potential is physically real after all.” That is too quick, and the precise version is more interesting.

    A is still not gauge-invariant. Change gauge and its value at any given point changes, exactly as before. What is gauge-invariant is the integral around a closed loop, because the extra term introduced by a gauge transformation is a gradient, and the integral of a gradient around any closed path vanishes identically:

    ∮∇χ⋅dℓ=0\oint \nabla\chi \cdot d\boldsymbol{\ell} = 0

    So the observable is not A at a point. It is the loop integral — a global quantity, belonging to the path as a whole rather than to any location on it.

    This has a consequence that is easy to miss. Whether the loop integral can be non-zero, in a region where the field vanishes, depends on the shape of the region. The solenoid geometry works because the electron paths enclose something they cannot enter: the accessible region has a hole in it. Given a region without such a hole — one in which every loop can be shrunk continuously to a point — a vanishing field forces the loop integral to vanish too, no matter how large A happens to be locally.

    The requirement that the solenoid be long, which reads at first like an engineering detail, is doing exactly this work. It is what makes the exterior region topologically non-trivial.

    1949

    There is an earlier paper.

    In 1949, Werner Ehrenberg and Raymond Siday published in the Proceedings of the Physical Society a study of the refractive index in electron optics which discusses a substantially equivalent effect — a decade before Aharonov and Bohm, in a journal nobody would have thought to overlook.

    It went essentially unnoticed. The result was framed as a technical point within electron optics rather than as a statement about the status of the potentials, and it reached an audience of instrument specialists rather than of physicists interested in foundations. The effect is now occasionally called Ehrenberg–Siday–Aharonov–Bohm, which is accurate and which almost nobody says.

    The pattern is the same one that swallowed Lorenz’s name, with a different mechanism. There the correct attribution lost to a more famous near-homophone. Here it lost to framing: the same physics, presented as a correction to a specialist calculation instead of as a challenge to a foundational assumption, simply did not register.

    Closing the last gap

    The prediction was contested for years, and the objection was a fair one. Real solenoids are finite. A small amount of magnetic field always leaks into the region the electrons traverse, and a sceptic could maintain that the observed shift was caused by that leakage rather than by the potential in a genuinely field-free region.

    Settling it took until the 1980s, and the work of Akira Tonomura and his collaborators at Hitachi. Electron holography experiments in 1982 established the shift. The decisive experiment came in 1986.

    They built a toroidal magnet roughly six micrometres across — a doughnut, so that the field lines close on themselves inside it with no ends to leak from — and then covered the whole thing in a superconducting shell. A superconductor expels magnetic field from its interior, the Meissner effect, so the shell acted as a barrier that no field line could cross. The magnetic field was not merely small outside the torus. It was excluded, by a mechanism independent of the geometry.

    The observed fringe shift was exactly half a fringe — the value required by the quantisation of magnetic flux in a superconductor, in units of

    Φ0=h2e≈2.07×10−15 Wb\Phi_0 = \frac{h}{2e} \approx 2.07 \times 10^{-15}\ \mathrm{Wb}

    Not approximately half. The number the theory demanded, produced by a system in which the alternative explanation had been physically eliminated.

    What changed

    The classical position was that the fields carry the physics and the potentials carry the bookkeeping. After 1986 that cannot be maintained. There is an experiment in which the fields are zero everywhere the particle goes, and something happens anyway.

    What replaced it is not the opposite claim. A is not an ordinary field with a definite value at each point; it remains gauge-dependent, and any description that treats its pointwise value as measurable is wrong. What the experiments establish is narrower and stranger: that there exists a gauge-invariant quantity built from the potentials, not reducible to the fields, which is not local — it belongs to a closed path rather than to a place — and which nature is demonstrably sensitive to.

    The potentials turned out not to be scaffolding. They turned out to be carrying something the field description had no way to express.

    Sources

    • W. Ehrenberg and R. E. Siday, “The Refractive Index in Electron Optics and the Principles of Dynamics,” Proc. Phys. Soc. London, vol. B62, p. 8, 1949.
    • Y. Aharonov and D. Bohm, “Significance of Electromagnetic Potentials in the Quantum Theory,” Phys. Rev., vol. 115, pp. 485–491, 1959.
    • A. Tonomura et al., “Observation of Aharonov-Bohm Effect by Electron Holography,” Phys. Rev. Lett., vol. 48, pp. 1443–1446, 1982.
    • A. Tonomura et al., “Evidence for Aharonov-Bohm Effect with Magnetic Field Completely Shielded from Electron Wave,” Phys. Rev. Lett., vol. 56, pp. 792–795, 1986.
    • N. Osakabe et al., “Experimental Confirmation of Aharonov-Bohm Effect Using a Toroidal Magnetic Field Confined by a Superconductor,” Phys. Rev. A, vol. 34, pp. 815–822, 1986.
    • G. Rousseaux, R. Kofman, and O. Minazzoli, “The Maxwell-Lodge effect: significance of electromagnetic potentials in the classical theory,” Eur. Phys. J. D, vol. 49, pp. 249–256, 2008.