The Answers That Would Not Go Away — Dirac's Equation and the Positron, 1928–1932

Last time, particles slipped out of the nucleus through a wall they could not climb (Part 2 is here). Part 3 is about a single equation written in that same year, 1928. The task was to rewrite quantum mechanics so that it agreed with Einstein's theory of relativity. The equation that came out of it brought along two answers nobody had asked for. One was welcomed at once. The other was an embarrassment — until it turned into the prediction of a particle no one had ever seen.

Fast electrons were still out of reach

The quantum mechanics that took shape in 1926 had one plain weakness. It could not deal properly with things moving at close to the speed of light; in other words, it did not agree with relativity. The electron in a hydrogen atom moves at about 1/137 of the speed of light, so for most calculations this did no harm. But the "fine structure" of hydrogen — the way its lines of light turn out, on a close look, to be split into several — comes from exactly this small effect of relativity (it was Sommerfeld's work, in Part 4 of the previous series).

Then there was spin. The electron has a built-in property that makes it behave like a tiny magnet. In 1927 Pauli fitted this into quantum mechanics, but it was bolted on from outside. Why the electron should have such a property went unexplained.

A wave equation that agreed with relativity had in fact been written down in 1926, by Oskar Klein and Walter Gordon. But Paul Dirac, at Cambridge, was not satisfied with its form. The way it handled time, he thought, did not fit the general framework of quantum mechanics.

Nobody added spin. It came out anyway

Dirac's paper reached the Royal Society on 2 January 1928. He was looking for the simplest equation that met the demands of both relativity and quantum mechanics. The price he paid was that the electron's wave could no longer be a single wave. It had to be a set of four.

Then something unexpected happened. The extra freedom that came with the set of four turned out to be the electron's spin. At the start of his paper Dirac asks why Nature should have chosen an electron with spin instead of being satisfied with a simple point charge. The answer was: because that is what you get when relativity and quantum mechanics are made to agree. Nothing had needed bolting on in the first place.

In March of the same year the physicist Charles G. Darwin solved the equation for the hydrogen atom exactly, with no approximation. What came out was the very formula Sommerfeld had obtained in 1916 (Gordon solved the same problem).

It is a strange coincidence. Sommerfeld knew nothing of spin, and he had drawn his formula from a picture that no longer holds — an electron travelling round an elliptical orbit. The formula was right all the same. The previous series kept running into right answers reached for the wrong reasons; here is one more.

As many unwanted answers as wanted ones

Of the set of four waves, only half were needed to account for spin. What was the other half? In Dirac's own words, there were as many "unwanted" solutions as wanted ones. They were states in which the electron's energy is negative.

An electron with negative energy would behave backwards. As Dirac later put it, the faster it moves the less energy it has, and you must put energy in to bring it to rest. Nobody had ever seen such an electron.

In the old physics, such answers could simply be thrown out as meaningless. Energy changes only smoothly, little by little, so an electron on the positive side could never get across to the negative side. But in quantum mechanics an electron can jump to a state far from the one it is in. As long as the negative states are there in the equation, an ordinary electron ought to be able to fall into them — and there is no bottom to fall to.

In 1929 Klein worked out what happens when an electron hits a sufficiently high wall. It ought to be turned back every time; instead, the calculation said, some electrons pass on into the wall. It is stranger than the tunnelling of Part 2, and here too the negative-energy states were showing their face. Heisenberg wrote to Pauli that Dirac's equation had become the "saddest chapter of modern physics". The equation that had got spin and the fine structure so beautifully right was saying, at the same time, that a stable world is impossible.

A full house, and one empty seat

In December 1929 Dirac sent off a way out. The key was Pauli's exclusion principle, which we met in Part 4 of the previous series: no more than one electron can occupy a given state.

Dirac reasoned as follows. What if the negative-energy states are already packed full of electrons? If the house is full, an ordinary electron cannot fall in. And if every seat everywhere is filled in the same uniform way, we would never notice. What we call the vacuum — "nothing at all" — is the full house.

And if a single seat were empty? Something with negative charge and negative energy is missing, so from outside it looks as though a particle with positive charge and positive energy is sitting there. At first Dirac took the empty seat to be the proton, the nucleus of hydrogen. Only two particles were known at the time, the electron and the proton, and a single theory that explained both would have been ideal.

The idea soon ran into trouble. Robert Oppenheimer and Igor Tamm pointed out that if the proton were an empty seat, electrons would drop into it — wiping out both — far too quickly. The mathematician Hermann Weyl argued that the empty seat must have the same mass as the electron. The proton is about 1,800 times heavier.

In May 1931 Dirac changed his mind. In the introduction to a paper on another subject, he wrote that a hole would be "a new kind of particle, unknown to experimental physics", with the same mass as the electron and the opposite charge. He called it an "anti-electron", and added that light of high enough energy should be able to create an electron and an anti-electron together. The energy needed corresponds to the mass of two electrons: about 1 MeV (the unit of energy we met in Part 2 — read a bigger number as a harder push).

August 1932: a single track in a cloud chamber

At the California Institute of Technology, Carl Anderson was studying the particles that rain down from space — cosmic rays. His tool was the cloud chamber: a box in which an electrically charged particle leaves a trail of fine droplets along its path. Put the chamber inside a strong magnet and the trails curve, one way for positive charge and the other way for negative.

The trouble is that a trail alone does not tell you which way the particle was going. A positive particle moving to the right and a negative particle moving to the left leave trails of the same shape. So Anderson put a lead plate six millimetres thick across the middle of the chamber. A particle loses some of its push in passing through the plate, and curves more tightly afterwards. The tightly curved side is "after".

One photograph, taken on 2 August 1932, showed a track of 63 MeV before the plate and 23 MeV after it. That fixed the direction of travel, and the sense of the curve then said the charge was positive. A positive particle ought to be a proton — but a proton curving this tightly would stop within about five millimetres. The track ran on for more than five centimetres. Anderson found 15 such tracks among 1,300 photographs and concluded that the mass was less than twenty times that of the electron. In his paper the following year he called the particle the "positron".

The discovery is often summed up as "Dirac predicted it, Anderson confirmed it". What actually happened was more tangled. Anderson knew of Dirac's equation but apparently not of the anti-electron prediction, and he later said the discovery was "wholly accidental". The link to Dirac's theory was made by Patrick Blackett and Giuseppe Occhialini at Cambridge. They built a cloud chamber that took its photograph automatically, only when a particle passed through, and in 1933 they proposed that the positive and negative electrons they saw appearing in pairs could be understood through Dirac's theory. Even then, Bohr and Pauli resisted for a while the idea that the particle found was the same thing as Dirac's "hole".

What could be done at this point, and what could not

Dirac's equation achieved three things. It made spin a consequence of the theory instead of an addition to it. It gave the fine splitting of the hydrogen lines without approximation. And by refusing to throw away the "unwanted" answers and thinking them through, it pointed to a particle nobody had seen. In his Nobel lecture of December 1933 Dirac said that positive and negative charge stand in complete and perfect symmetry, and that negative protons probably exist as well. The antiproton was found in 1955. As for the positron, hospitals now use it every day, in PET scans, to make pictures of the inside of the body.

What it could not do is just as clear. First, the full-house picture itself. A vacuum packed with countless electrons that cannot be seen: the idea produced a correct prediction, but it never sat comfortably. In 1934 Pauli and Victor Weisskopf showed that particles of opposite charge appear without that picture at all. Today's theory is built with no full house.

Second, hydrogen. In 1947 Willis Lamb and Robert Retherford measured a small difference between two states that, by Dirac's equation, should lie at exactly the same height. The equation deals with one electron. It could not take in the whole of a world in which electrons trade light back and forth and particles are born and vanish.

Parts 1 and 2 looked at the gap between getting the answer and understanding it. This time the case runs the other way round. The full house was an easy picture to grasp, and it was later given up. Yet the prediction drawn from it came true. An easy picture is not necessarily a correct one. But a picture that will later be discarded can still be the scaffolding that leads to the right answer — as Sommerfeld's ellipses were.

Next time: electrons inside solids. A metal carries electricity well; a diamond carries almost none. Both are full of electrons, so why are they so different? Quantum mechanics went on to answer this everyday question too.