Quantum mechanics as built in 1925 to 1927 had a glaring gap. It ignored Einstein's special relativity, which governs anything moving close to the speed of light. Repairing that gap led to one of the strangest predictions in the history of science, a new kind of matter, and to a theory of fields that would take another twenty years to get working.

The man and the equation

Paul Dirac was the son of a Swiss father and an English mother, trained first as an electrical engineer in Bristol before turning to mathematics at Cambridge. He was famously quiet and literal, and he trusted mathematical beauty as a guide to truth. In 1928 he set out to make the quantum theory of the electron fit special relativity. An earlier attempt, now called the Klein-Gordon equation, had produced meaningless results such as negative probabilities.

Dirac looked for an equation of a simpler mathematical form, and found that it worked only if the electron's wave had four parts instead of one. The result, published in 1928 as "The Quantum Theory of the Electron," was a triumph. The electron's spin and magnetic strength came out of the equation without being put in by hand. Uhlenbeck and Goudsmit's spin, which had seemed an add-on, turned out to follow from relativity and quantum theory together.

A sea of negative energy

The equation had a troubling feature. It allowed solutions with negative energy, and an electron should be able to fall into them forever, giving off radiation. In 1929 and 1930 Dirac proposed that every negative energy state is already filled, a vast unseen sea, so that ordinary electrons cannot fall in. A missing electron in the sea, a hole, would behave like a particle with positive charge.

At first Dirac suggested the holes might be protons, the only known positive particle. Hermann Weyl and Robert Oppenheimer showed this could not work: the hole had to have the same mass as the electron. In 1931 Dirac accepted the consequence and predicted a new particle, an anti-electron.

The next year Carl Anderson at Caltech was photographing cosmic ray tracks in a cloud chamber, a device in which charged particles leave trails of droplets. One track curved the wrong way in a magnetic field for its weight. A lead plate in the chamber showed which way the particle had traveled, and its behavior fit a positive particle with an electron's mass. Anderson called it the positron. Patrick Blackett and Giuseppe Occhialini in Cambridge soon confirmed it, catching electron and positron pairs being created together.

Fields, not just particles

Another strand began in 1927, when Dirac treated the electromagnetic field itself as quantized, with photons as its quanta. This explained how atoms spontaneously emit light, which earlier theory could not. Jordan, Eugene Wigner, Heisenberg and Pauli extended the approach to electrons and other matter, treating every particle as a ripple in a field. Particles could now be created and destroyed.

Enrico Fermi used this idea in 1933 for beta decay, in which a nucleus gives off an electron. His theory held that the electron and a neutrino are created at the moment of decay, not stored in the nucleus beforehand. The journal Nature turned down his paper as too speculative. In 1935 Hideki Yukawa in Japan predicted a new particle carrying the force that binds the nucleus, a heavier cousin of the photon. A particle of this kind, the pion, was found in 1947, and in 1949 Yukawa became the first Japanese Nobel laureate.

Infinities and their taming

Quantum field theory had a serious flaw. Simple calculations worked, but more precise ones produced infinite answers, for example for the electron's interaction with its own field. Through the 1930s many physicists feared the whole approach was doomed. Then in June 1947 a small meeting on Shelter Island, New York, heard new results. Willis Lamb reported a tiny shift in hydrogen's energy levels that Dirac's theory said should not exist. On a train afterward, Hans Bethe estimated the shift by absorbing the infinite parts into the electron's observed mass, an approach later called renormalization.

Julian Schwinger and Richard Feynman in America, and Sin-Itiro Tomonaga, working almost in isolation in wartime and postwar Japan, built full versions of the theory. In 1949 Freeman Dyson showed their approaches were equivalent. The three shared the 1965 Nobel Prize. Dirac himself never accepted renormalization as a real solution.

Worked example: anatomy of a prediction

Follow the positron from idea to discovery. Dirac starts with a demand for consistency: quantum theory and relativity must agree. His equation delivers spin as a bonus, a sign it is on the right track, but also negative energy states that seem absurd. Instead of discarding the equation, he asks what the strange solutions might mean. His first answer, the proton, is wrong, and colleagues show why. His second answer, a new particle with the electron's mass, is right.

Anderson was not searching for Dirac's particle. He was studying cosmic rays, and the evidence came from a careful look at an odd track. The story shows a pattern that recurs in science: taking a successful theory's strange implications seriously, being willing to be wrong in public, and letting independent experiment decide.