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:
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:
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:
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:
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.
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