Why Beauty is Truth (47 page)

Who was right? It turned out the two theories were identical, as Schrödinger discovered in 1926. They were two distinct mathematical representations of the same underlying concepts—just as Euclidean methods and algebra are two equivalent ways of looking at geometry. At first Heisenberg could not believe this, because the essence of his matrix approach was the existence of discontinuous jumps as an electron changed its state. The entries in his matrices were the associated changes of energy. He couldn't see how waves, as continuous entities, could model discontinuities. In a letter to the Austrian-Swiss physicist Wolfgang Pauli, he
wrote, “The more I think about the physical portion of Schrödinger's theory, the more repulsive I find it . . . What Schrödinger writes about the visualizability of his theory ‘is probably not quite right,' in other words, it's crap.” But really this disagreement was a rerun of a much older debate, in which Bernoulli and Euler had disagreed about solutions of the wave equation. Bernoulli had a formula for the solutions, but Euler could not see how this formula, which looked continuous, could represent discontinuous solutions. Nevertheless, Bernoulli was right, and so was Schrödinger. His
equations
might be continuous, but many features of their solutions could be discrete—including the energy levels.

Most physicists preferred the wave-mechanical picture because it was more intuitive. Matrices were a bit too abstract. Heisenberg still preferred his lists, because they consisted of observable quantities, and it seemed impossible to detect one of Schrödinger's waves experimentally. In fact, the Copenhagen interpretation of quantum theory, dramatized as Schrödinger's cat, stated that any attempt to do so would “collapse” the wave into a single, well-defined spike. So Heisenberg became more and more concerned about what aspects of the quantum world can be measured, and how. You can measure every entry in his lists. You can't do that for one of Schrödinger's waves. Heisenberg considered this difference a powerful reason for sticking to matrices.

Following this line of thought, he discovered that in principle you can measure a particle's position as accurately as you wish—but there is a price to be paid, because the more accurately you know the position, the less accurately you can know the momentum. Conversely, if you can measure the momentum very accurately, you lose track of the position. The same trade-off occurs for energy and time. You can measure one or the other, but not both. Not if you want high-accuracy measurements.

This wasn't a problem with experimental procedure; it was an inherent feature of quantum theory. He wrote out his reasoning in a letter to Pauli in February 1927. The letter eventually inspired a paper, and Heisenberg's idea acquired the name “uncertainty principle.” It was one of the first examples of an inherent limitation in physics. Einstein's assertion that nothing can move faster than light was another.

In 1927, Heisenberg became Germany's youngest professor, at the University of Leipzig. In 1933, the year Hitler rose to power, Heisenberg won the Nobel Prize for physics. This made him a highly influential figure, and his willingness to stay in Germany during the Nazi regime made
many believe that Heisenberg was himself a Nazi. As far as can be established, he wasn't. But he was a patriot, and that led him to associate with Nazis and be complicit in many of their activities. There is some evidence that Heisenberg tried to stop the ruling powers kicking Jews out of university positions, but to no effect. In 1937, he found himself described as a “white Jew” and was under threat of being sent to a concentration camp, but after a year he was cleared of suspicion by Heinrich Himmler, head of the SS. Also in 1937, Heisenberg married Elisabeth Schumacher, the daughter of an economist. Their first children were twins; eventually they had seven.

During World War II, Heisenberg was one of the leading physicists involved in Germany's search for nuclear weapons—the “atomic bomb.” He worked on nuclear reactors in Berlin, while his wife and children were dispatched to the family's summer home in Bavaria. His role in Germany's atom bomb project has proved highly controversial. When the war ended he was detained by the British and held for six months for questioning in a country house near Cambridge. The transcripts of his interrogations, recently made public, have exacerbated the controversy. Heisenberg does say at one point that he was solely interested in making a nuclear reactor (“engine”) and did not want to be involved in a bomb. “I would say that I was absolutely convinced of the possibility of our making a uranium engine, but I never thought we would make a bomb, and at the bottom of my heart I was really glad that it was to be an engine and not a bomb. I must admit that.” The truth of this claim is still hotly debated.

After the war, and his release from British custody, Heisenberg went back to work on quantum theory. He died of cancer in 1976.

Most of the great German creators of quantum theory came from an intellectual background—they were the sons of doctors, lawyers, academics, or other professionals. They lived in expensive homes, played music, and took part in the local social life and culture. The great English creator of quantum mechanics had a very different and much sadder childhood, with an autocratic and distinctly eccentric father who was largely estranged from his own parents and family, and a mother who was so browbeaten that she and two of her children ate in the kitchen while her husband and their younger son ate in total silence in the dining room.

The father was Charles Adrien Ladislas Dirac, born in the Swiss canton of Valais in 1866, who ran away from home at the age of 20. Charles arrived in Bristol in 1890 but did not become a British citizen until 1919. In 1899, he married Florence Hannah Holten, a sea captain's daughter, and their first child, Reginald, was born the next year. Two years later a second son, Paul Adrien Maurice, was added to the growing family; four years after that they had a daughter, Beatrice.

Charles did not tell his parents that he had married, or that they had become grandparents, until 1905, when he visited his mother in Switzerland. By then, his father had been dead for ten years.

Charles worked as a teacher at the Merchant Venturer's Technical College in Bristol. He was generally considered a good teacher, but he was also renowned for a total absence of human feelings and very strict discipline. He was, in short, a martinet, but so were many teachers.

Paul, a natural introvert, was made even more so by his father's curious isolation and lack of any social life. Charles insisted that Paul speak to him only in French, presumably to encourage him to learn that language. Since Paul's French was dreadful, he found it simpler not to speak at all. Instead, he spent his time wondering about the natural world. The antisocial dining arrangements in the Dirac household also seem to have stemmed from the rule that conversation should be held entirely in French. It was never clear whether Paul actively hated his father or just disliked him, but when Charles died, Paul's main comment was “I feel much freer now.”

Charles was proud of Paul's intellectual abilities and very ambitious for his children—by which he meant that they should do what he had planned for them. When Reginald said he wanted to become a doctor, Charles insisted that he become an engineer. In 1919, Reginald obtained a very poor engineering degree; five years later, while working on an engineering project in Wolverhampton, he killed himself.

Paul lived at home with his parents, and also studied engineering at the same college as his brother. His favorite subject was mathematics, but he chose not to study that. Possibly he didn't want to go against his father's wishes; but he was also under the erroneous impression, still widespread today, that the only career for someone with a mathematics degree is school-teaching. No one had told him that there were alternatives—among them, research.

Salvation came in the form of a newspaper headline. The front page of the
Times
for 7 November 1919 shrieked, REVOLUTION IN SCIENCE.
NEW THEORY OF THE UNIVERSE. NEWTONIAN IDEAS OVERTHROWN. Halfway down the second column was the subheading SPACE “WARPED.” Suddenly everyone was talking about relativity.

Recall that one of the predictions of general relativity is that gravity bends light, by twice the amount that Newton's laws would predict. Frank Dyson and Sir Arthur Stanley Eddington had mounted an expedition to Príncipe Island in West Africa, where a total eclipse of the Sun was due. Simultaneously, Andrew Crommelin, of Greenwich Observatory, led a second expedition to Sobral, in Brazil. Both parties observed stars near the edge of the Sun during the period of totality, and found slight displacements in the stars' apparent positions, consistent with Einstein's predictions, but not with Newtonian mechanics.

Einstein, an overnight celebrity, sent his mother a postcard: “Dear Mother, joyous news today. H. A. Lorentz telegraphed that the English expeditions have actually demonstrated the deflection of light from the Sun.” Dirac was hooked. “I was caught up in the excitement produced by relativity. We discussed it very much. The students discussed it among themselves, but had very little accurate information to go on.” Public knowledge of relativity was confined largely to the word; philosophers claimed that they had known for years that “everything is relative,” and dismissed the new physics as old hat. Unfortunately, they only displayed their ignorance, and the ease with which they had fallen for misleading terminology.

Paul went to some lectures on relativity by Charlie Broad, then a philosophy professor at Bristol, but their mathematical content was insignificant. Eventually, he bought a copy of Eddington's
Space, Time and Gravitation
and taught himself the necessary mathematics and physics. Before leaving Bristol, he knew both special and general relativity inside out.

Paul was good at theory, terrible at laboratory work. In later years, physicists spoke of the “Dirac effect”: he had only to walk into a laboratory for nearby experiments to go wildly wrong. An engineering profession would have been a disaster. He found himself with a first-class degree, but unemployed at a time when jobs were scarce because of the postwar economic depression. Luckily, he was offered the chance to study mathematics at Bristol University, all tuition paid, and he leaped at it. There he specialized in applied mathematics.

In 1923, Paul became a postgraduate research student at the University of Cambridge, where his shyness was a real handicap. He wasn't interested in sports, made few friends, and had nothing whatsoever to do with women. He spent most of his time in the library. In 1920, he had spent the summer working at the same factory as his brother Reginald. The two would often pass in the street but never stopped to talk, so ingrained was the habit of silence between family members.

Paul quickly rose to prominence; within six months he had written his first research paper. Other papers followed in a rapid stream. Then, in 1925, he encountered quantum mechanics. On a long autumn walk in the Cambridgeshire countryside he found himself thinking about Heisenberg's “lists.” These are matrices, and matrices do not commute, something that had initially bothered Heisenberg. Dirac was aware of Lie's idea that in these circumstances the important quantity is the commutator
AB
–
BA
, not the product
AB
, and he was struck by the intriguing thought that a very similar concept occurs in the Hamiltonian formalism of mechanics, where it is called a Poisson bracket. But Dirac couldn't remember the formula.

The thought kept him awake much of the night, and the next morning he “hurried along to one of the libraries as soon as it was open, and then I looked up Poisson brackets in Whittaker's
Analytical Dynamics
, and I found that they were just what I needed.” His discovery was this: the commutator of two quantum matrices is equal to the Poisson bracket of the corresponding classical variables, multiplied by the constant i
h
/2π. Here
h
is Planck's constant,
i
is
and π is, well, π.