The Quantum Theory of the Electron
Force the electron's wave equation to obey relativity — and spin, then antimatter, fall out of the mathematics.
Demand that one equation respect both quantum mechanics and Einstein's relativity at once — and it answers back with a universe that must contain antimatter.
The idea, unpacked
By 1928 physicists knew the electron "spins" — it carries a tiny built-in magnetism — but they had simply added that fact to their equations by hand. Dirac wanted an equation that obeyed quantum mechanics and special relativity together, and insisted it take a particular clean shape. When he found the one that fit, the electron's spin and its magnetism were already inside it, unasked for.
The same equation also had a stubborn second answer. Just as asking "what number squared gives four?" yields both +2 and −2, Dirac's energy equation gave a positive answer and an equal negative one. He couldn't discard the negative without breaking the maths — so he eventually concluded it described a real, opposite twin of the electron. That was antimatter.
Where it came from
Paul Dirac was twenty-five, a famously silent man at St John's College, Cambridge, who worked by chasing mathematical beauty. He disliked the relativistic equation others were using and set out to build a better one. He wrote up the result over the Christmas of 1927 and submitted it on the 2nd of January 1928.
The negative-energy answers haunted him for years. In 1931 he made a bold claim: somewhere out there must exist an "anti-electron", a particle with the electron's mass but opposite charge. In 1932 Carl Anderson, photographing cosmic rays, caught exactly such a track — the positron. Dirac shared the 1933 Nobel Prize, having predicted a new form of matter from little more than an insistence on the right equation.
Why it mattered
This was the first true marriage of quantum mechanics and relativity, and the template for how modern physics works. Dirac had pulled a previously unimagined kind of matter out of pure mathematics, and then watched the world confirm it. The lesson — that a deep enough equation can tell you what must exist before anyone has seen it — runs straight through to the prediction of the Higgs boson decades later.
A square root has two answers
Ask a calculator for the square root of four and the polite answer is 2. But −2 works just as well: (−2)×(−2) = 4 too. Energy, in Dirac's relativistic equation, sits under a square root the same way, so for any motion there is a positive energy and a mirror-image negative one. Throwing the negative away isn't allowed — it's a genuine solution. Nature's way of using it is to make an antiparticle. Slide the tool below and watch the energy line acquire its twin beneath zero.
Where it sits
Schrödinger's 1926 equation, already in this Library, describes a slow electron beautifully but ignores relativity. Dirac's equation is its relativistic successor, and the gateway to quantum field theory and the Standard Model. It belongs to a lineage that runs from Einstein's 1905 relativity, through the quantum revolution, to the modern habit — seen again with the Higgs — of letting a trusted equation predict particles the world has yet to find.
The new quantum mechanics, when applied to the problem of the structure of the atom with point-charge electrons, does not give results in agreement with experiment. The discrepancies consist of "duplexity" phenomena, the observed number of stationary states for an electron in an atom being twice the number given by the theory.
It appears that the simplest Hamiltonian for a point-charge electron satisfying the requirements of both relativity and the general transformation theory leads to an explanation of all duplexity phenomena without further assumption.