Why Beauty is Truth (48 page)

This was a dramatic discovery. It told physicists how to turn classical mechanical systems into quantum ones. The mathematics was very beautiful, linking two deep but previously unconnected theories. Heisenberg was impressed.

Dirac's contributions to quantum theory are many, and I will select just one of the high points, his relativistic theory of the electron, which dates from 1927. By then, the quantum theorists knew that electrons have “spin”—somewhat analogous to the spin of a ball about an axis, but with strange features that make the analogy very rough. If you take a spinning ball and rotate the system through a full 360°, both ball and spin get back to where they started. But when you do the same to an electron, the spin
reverses.
You have to rotate through 720° before the spin gets back to its original value.

This is actually very similar to quaternions, whose interpretation as “rotations” of space has the same quirk. Mathematically, rotations of space form the group SO(3), but the relevant group for both quaternions and electrons is SU(2). These groups are almost the same, but SU(2) is twice as big, built—in a certain sense—out of two copies of SO(3). It is called a “double cover,” and the result is to expand a 360° rotation into one through twice that angle.

Dirac didn't use quaternions, and he didn't use groups either. But over the Christmas season at the end of 1927 he came up with “spin matrices,” which play the same role. Mathematicians later generalized Dirac's matrices to “spinors,” which are important in the representation theory of Lie groups.

The spin matrices allowed Dirac to formulate a relativistic quantum model of the electron. It did everything he had hoped for—and a little more. It predicted solutions with negative energy as well as the expected ones with positive energy. Eventually, after some false starts, this puzzling feature led Dirac to the concept of “antimatter”—that every particle has a corresponding antiparticle, with the same mass but the opposite charge. The antiparticle of the electron is the positron, and it was unknown until Dirac predicted it.

The laws of physics remain (almost) unchanged if you replace every particle with its antiparticle—so that operation is a symmetry of the natural world. Dirac, who was never terribly impressed by group theory, had discovered one of the most fascinating symmetry groups in nature.

From 1935 onward, until his death in Tallahassee in 1984, Dirac placed enormous value on the mathematical elegance of physical theories, and used that principle as a touchstone for his research. If it wasn't beautiful, he believed, it was wrong. When he visited Moscow State University in 1956, and followed tradition by writing words of wisdom on a blackboard to be kept for posterity, he wrote, “A physical law must possess mathematical beauty.” And he talked of a “mathematical quality” in nature. Yet he never seemed to think of group theory as beautiful, possibly because physicists mostly approach groups through massive calculations. Only mathematicians, it seemed, were attuned to the exquisite beauty of Lie groups.

Beautiful or not, group theory soon became essential reading for any budding quantum theorist, thanks to the son of a leather merchant.

At the turn of the nineteenth century, leather was big business, and it still is. But in those days a small businessman could make a good living by tanning and selling leather. A good example was Antal Wigner, the director of a tannery. He and his wife Erzsébet were of Jewish descent, but did not practice Judaism. They lived in what was then Austro-Hungary, in the city of Pest. Conjoined with neighboring Buda, this town became today's Budapest, the capital of Hungary.

Jen
Pál Wigner, the second of their three sons, was born in 1902, and between the ages of five and ten he was educated at home by a private tutor. Soon after Jen
started school, he was diagnosed with tuberculosis and sent to an Austrian sanatorium to recover. He stayed there for six weeks before it turned out that the diagnosis had been wrong. Had it been right, he would almost certainly not have survived to adulthood.

Made to lie on his back much of the day, the boy did mathematical problems in his head to pass the time. “I had to lie on a deck chair for days on end,” he later wrote, “and I worked terribly hard on constructing a triangle if the three altitudes are given.” The altitudes of a triangle are the three lines that pass through a corner and hit the opposite side at right angles. Finding the altitudes, given the triangle, is easy. Going in the opposite direction is decidedly more difficult.

After Jen
left the sanatorium he continued to think about mathematics. In 1915, at Budapest's Lutheran High School, he met another boy who would become one of the world's leading mathematicians: Janós (later John) von Neumann. But the two never became more than loose acquaintances, because von Neumann tended to keep to himself.

In 1919, the Communists overran Hungary and the Wigners fled to Austria, returning to Budapest later in the same year when the Communists were kicked out. The entire family converted to Lutheranism, but this had little effect on Jen
, he later said, because he was “only mildly religious.”

In 1920, Jen
finished school near the top of his class. He wanted to become a physicist, but his father wanted him to join the family leather-tanning business. So instead of taking a physics degree, Jen
studied chemical engineering, which his father thought would help to advance the business. For his first year at university, the young man went to the Budapest Technical Institute; then he switched to the Technische Hochschule in Berlin. He ended up spending most of his time in the chemical laboratory, which he enjoyed, and precious little time in theoretical classes.

Still, Jen
had not given up on physics. The University of Berlin was not far away, and who might be there but Planck and Einstein, along with lesser luminaries. Jen
took advantage of their proximity and went to the immortals' lectures. He completed his doctorate, with a thesis on the formation and breakup of molecules, and duly joined the tannery. Predictably, this proved a bad idea: “I did not get along very well in the tannery . . . I did not feel at home there . . . I did not feel that this was my life.” His life was mathematics and physics.