Dirac’s electron: a successful equation with an unfinished story

Spin had been added before it was explained

Electron spin and its magnetic coupling accounted for the doublets in atomic spectra, but they had been added by hand to a nonrelativistic theory. Could they follow from a relativistic quantum equation itself? Dirac’s 1928 paper makes that requirement explicit.

The goal was to fit quantum evolution and relativity into one mathematical structure, which is more than correcting a classical orbit for high speed. The resulting theory explained important properties of the electron and also exposed difficulties that the first paper did not resolve.

Read the evidence: P. A. M. Dirac (1928), The Quantum Theory of the Electron, Proc. R. Soc. A 117, 610–624 — UC San Diego scan

A first-order equation needs a four-component state

The equation shown is the modern form for a free particle, with no electromagnetic field. The state has four components, and alpha and beta are dimensionless matrices, not extra coordinates. Their algebra lets a first-order time evolution satisfy the relativistic relation between energy and momentum.

The rest mass, the speed of light and the reduced Planck constant set the units, and the gradient acts on position. On its own the equation is not a complete theory of electron creation, radiation or experiments with many interacting particles. Its four components are not four classical particles either.

i\hbar\frac{\partial\psi}{\partial t}=\left(-i\hbar c\,\boldsymbol{\alpha}\cdot\boldsymbol{\nabla}+\beta mc^2\right)\psi

Read the evidence: P. A. M. Dirac (1928), The Quantum Theory of the Electron, Proc. R. Soc. A 117, 610–624 — UC San Diego scan

The spin factor of two becomes a baseline

With minimal coupling and before radiative corrections, the Dirac electron has a spin magnetic factor of two. The theory ties intrinsic angular momentum to magnetic response more tightly than a separately added classical spinning charge could. That gives experiments a definite value to challenge.

Kusch and Foley’s atomic comparisons later found a small departure from that value; their 1948 abstract summarizes several atomic-state comparisons. The discrepancy is a measured problem in the magnetic response. It is no reason to discard every successful relativistic prediction.

Read the evidence: P. Kusch and H. M. Foley (1948), The Magnetic Moment of the Electron, Phys. Rev. 74, 250 — author abstract

Negative energies were a difficulty, not a finished discovery

The first 1928 paper explicitly sets aside one difficulty with the negative-energy solutions. The modern particle–antiparticle interpretation was not yet in place, and retelling the paper as if it were would mislead. A branch of solutions in the mathematics and a particle seen in a detector are different kinds of evidence.

Anderson’s 1933 report analyzes cosmic-ray tracks of a particle with positive charge and a mass much smaller than the proton’s. Track curvature, ionization and energy loss constrain that identification. The experimental case has to be judged on those tracks; the existence of Dirac’s equation cannot supply it after the fact.

Read the evidence: P. A. M. Dirac (1928), The Quantum Theory of the Electron, Proc. R. Soc. A 117, 610–624 — UC San Diego scan C. D. Anderson (1933), The Positive Electron, Phys. Rev. 43, 491 — author abstract

What the hydrogen comparison can test

A nonrelativistic Coulomb model, the Dirac Coulomb model and a theory with radiative and nuclear corrections are three levels of approximation. They need not agree on which levels coincide or how far they split. The experiment’s resolution decides which of those differences is actually tested.

The hydrogen Lab gives a nonrelativistic reference; it does not solve the four-component equation. The coupled-spin Lab uses an effective Hamiltonian and is not a precision calculation of the electron anomaly. Both are useful for comparison, but neither stands in for the historical theory.

Read the evidence: P. A. M. Dirac (1928), The Quantum Theory of the Electron, Proc. R. Soc. A 117, 610–624 — UC San Diego scan P. Kusch and H. M. Foley (1948), The Magnetic Moment of the Electron, Phys. Rev. 74, 250 — author abstract

The new questions the theory opened

The electron-moment article follows the comparison of magnetic factors, and the Lamb article a different spectroscopic challenge to the Dirac baseline. They concern different observables, and agreement in one cannot stand in for a measurement of the other.

The theory mattered both for the relations it got right and for the sharp questions it made possible. Quantum electrodynamics and the antiparticle experiments that followed have their own histories of evidence, which this article does not try to cover.

Read the evidence: P. Kusch and H. M. Foley (1948), The Magnetic Moment of the Electron, Phys. Rev. 74, 250 — author abstract C. D. Anderson (1933), The Positive Electron, Phys. Rev. 43, 491 — author abstract

Continue with the models

These teaching Labs produce simulated data, which are separate from the historical observations discussed above.

Dirac trembling motion

Does the oscillation survive one energy branch?

Try: Compare Let it tremble with One energy branch. Scrub through the evolution, then measure position and velocity, each on its own set of fresh preparations.
Read: Compare the sample means with the model’s expectation values. Remove the interference between the energy branches and the free mean velocity stops oscillating. The two spinor components and the two energy branches are different things.
Scope: Speeds are dimensionless, with the unit speed set to one. This one-dimensional, two-component analogue complements the article’s four-component theory of the electron. It does not observe a real electron’s trajectory, solve a hydrogen spectrum or calculate a magnetic anomaly.

\begin{aligned}H_0&=p\sigma_x+\mu\sigma_z\\v&=\sigma_x\end{aligned}

Model reference: Gerritsma et al. · Dirac simulation (2010)

Klein wave-packet tunnelling

Can raising the barrier reopen a channel?

Try: Compare the preset where the gap blocks the packet with the one where the barrier is raised. At the same final time, collect position samples, then move the observation gate.
Read: Compare the fraction of samples to the right of the gate with the model’s occupancy at that finite time. The gate is only an analysis threshold; moving it keeps the samples already collected.
Scope: Here a smooth, finite barrier shifts the one-dimensional Dirac branches. The cited trapped-ion experiment used a linear potential, and the Lab does not reproduce its data or hardware. Occupancy at a finite time differs from the asymptotic transmission coefficient. This is a single-particle calculation, so it has no pair creation.

E_\pm(p)=V\pm\sqrt{p^2+\mu^2}

Model reference: Gerritsma et al. · Klein simulation (2011)

Sources and reading notes

We read Dirac’s first 1928 paper in a scan hosted by UC San Diego. For Anderson’s 1933 report and the 1948 Kusch–Foley paper we used the author abstracts. The equation’s early achievements are kept apart from the later interpretation of its negative-energy solutions. The article does not retrace the positron discovery or calculate radiative corrections.