The Strangest Man Read online

  On the cool Tuesday evening of 28 July, the sweet summer air calm and damp after a day of wind and light showers of rain, Heisenberg addressed the Kapitza Club, his first presentation in Cambridge. He expected to be met with the university’s famous formality but, instead, found himself talking in a makeshift college room, with several members of his audience having to sit on the floor. It is not clear whether Dirac was awake throughout Heisenberg’s seminar or even if he attended it.27 Some of the physicists who attended vaguely remembered that Heisenberg spoke about the light emitted and absorbed by atoms and that he remarked in a coda that he had written an article about a new approach to atomic physics. Later, Heisenberg could be sure only that he did mention this article to his host Fowler, but no one in Cambridge – or even Heisenberg himself – appears to have realised that they had been part of history in the making.28

  Dirac returned home for the summer break having secured funding for another three years’ research from the Royal Commission for the Exhibition of 1851, which dispensed scholarships funded by the Exhibition’s unexpected profits. Dirac’s application had been recommended by Maynard Keynes and included encomia from Cunningham, Fowler and the physicist and astronomer James Jeans, who affirmed that Dirac had ‘ability of the highest order in mathematical physics’.29 Much was expected of the young Dirac, though he had published nothing of consequence since his brother’s suicide.

  Dirac probably had to fend off his mourning parents’ requests for him to return to Bristol. His father had already tried to persuade him to apply for the post of Assistant Lecturer in Mathematics at the university, but there can never have been any question that Dirac would accept such a post – he was starting to become aware of his academic worth.30 And he was still awaiting a challenge equal to his talent.

  Early in September 1925, a postman walked up the steep path to the front door of 6 Julius Road and delivered an envelope that changed Dirac’s life. The package, sent by Fowler, contained fifteen pages of the proofs of a paper sent to him by its author, Werner Heisenberg, who had made several corrections to it in his slanting handwriting.31 This article, written in German, contained the first glimpse of a completely new approach to understanding atoms. Most supervisors would have kept the proofs to themselves, to get a head start on their fellow researchers. Fowler, however, sent the proofs to Dirac with a few words scribbled on the top right-hand corner of the front page: ‘What do you think of this? I shall be glad to hear.’

  The paper, technical and complex, would not have been easy reading for Dirac, whose training at the Merchant Venturers’ had given him only a modest command of German. He could, however, see that this was not just another run-of-the-mill exercise in the mathematics of quantum theory. Bohr’s theory featured quantities such as the position of the electron and the time it takes to orbit its nucleus, but Heisenberg believed that this was a mistake, as no experimenter would ever be able to measure them. He made this point when he summarised the aim of his theory in the article’s introductory sentence: ‘The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable.’32 Heisenberg knew that it would be extremely difficult to come up with a complete atomic theory built along the lines he envisaged in a single flourish. That would have been too big a task. Instead, he attempted something simpler, by trying to set out a theory of an electron moving not in three dimensions of ordinary space but in just one dimension, that is, in a straight line. Such an electron exists only in the mind of the theoretical physicist, but if this prototype theory worked, then maybe it would be possible to extend it and produce a more realistic version of the theory, one that could be applied to atoms.

  Heisenberg considered how classical theory describes his electron, moving back and forth, and how quantum theory might account for it, bearing in mind that the two theories must merge smoothly, according to the correspondence principle. The new theory looked completely different from its classical counterpart. For example, there is no mention in the quantum theory of single numbers to represent the electron’s position; instead, position is replaced by numbers in a square array, an example of what mathematicians call a matrix. Each number in this array is a property of a pair of the electron’s energy levels and represents the likelihood that the electron will jump between that pair of energy levels. So, each number can be deduced from observations of the light given out by the electron when it jumps between them. In this way, Heisenberg demonstrated how to build an entirely new atomic theory solely in terms of measurable quantities.

  This picture looks bizarre to anyone coming to it for the first time. With astonishing boldness, Heisenberg had abandoned the assumption that electrons can be visualised in orbit around a nucleus – an assumption no one had previously thought to question – and replaced it by a purely mathematical description of the electron. Nor was this description easy to accept: for example, if it were to apply to ordinary matter, an object’s precise location would not be measured with a ruler but would be given in terms of an array of numbers that give the chances of its making transitions to other energy states. This was no one’s idea of common sense. In making an imaginative leap like this, Heisenberg was behaving rather like a painter who had switched from Vermeer’s classically descriptive style to one based on the abstractions of Mondrian. But whereas painters can use abstraction simply as a technique for producing an attractive image that may or may not refer to real things, abstraction for physicists is a way of representing things en route to the most accurate possible account of material reality.

  Dirac initially found Heisenberg’s approach too complicated and artificial, so he put the paper aside, dismissing it as being ‘of no interest’.33 About ten days later, however, Dirac returned to it and was struck by a point that Heisenberg made in passing, almost halfway through the paper. Heisenberg wrote that some of the quantities in the theory have a peculiar property: if one quantity is multiplied by another, the result is sometimes different from the one obtained if the sequence of multiplication is reversed. This was exemplified by the quantities he used to represent position and momentum of a piece of matter (its mass multiplied by its velocity): position multiplied by momentum was, strangely, not the same as momentum multiplied by position. The sequence of multiplication appeared to be crucial. Heisenberg later remarked that he mentioned this point as an embarrassing aside, hoping that it would not put off the paper’s reviewers and encourage them to think the theory was too far-fetched to be worth publishing. Far from being disconcerted, Dirac saw that these strange quantities were the key to a new approach to quantum physics. Several years later, his mother told an interviewer that Dirac was so excited that he broke his rule of saying nothing about his work to his parents and did his best to explain non-commutation. He did not try again.34

  Unlike Heisenberg, who had never come across non-commuting quantities before, Dirac was well acquainted with them – from his studies of quaternions, from the Grassmann algebra he had heard about at Baker’s tea parties, and from his extensive studies of projective geometry, which also features such relationships.35 So, Dirac was not only comfortable with the appearance of such quantities in the theory, he was excited by them, although at first he did not understand their significance, nor did he know how to build on Heisenberg’s ideas. What Dirac did notice was that Heisenberg had not constructed his theory to be consistent with special relativity so, true to form, Dirac played his favourite game of trying to produce a version of Heisenberg’s theory that was consistent with relativity, but he soon gave up.36 At the end of September, Dirac prepared to return to Cambridge, convinced that the non-commuting quantities in the theory were the key to the mystery. To make progress, he needed to find the lock – a way of interpreting these quantities, a way of linking them to experimentally observed reality.

  One person who, unknown to Dirac, shared his excitement about the theory was Albert Einstein, who wrote to a friend: ‘Heisenberg has
laid a big quantum egg.’37

  At the beginning of October, Dirac began his final year as a postgraduate student. With Fowler’s encouragement, he set aside his books of intricate calculations based on the Bohr theory, well aware that – if Heisenberg’s theory was right – those calculations were all but worthless.

  It was during a Sunday walk in the countryside, soon after term began, that Dirac had his first great epiphany. Long afterwards, he could not recall the exact date, though he clearly remembered those first exciting hours of discovery.38 He was, as usual, trying to forget about his work and let his mind wander in the tranquillity of the flat Cambridgeshire countryside. But on that day, the non-commuting quantities in Heisenberg’s theory kept intruding into his conscious mind. The crucial point was that two of these quantities, say A and B, give different results according to the order in which they are multiplied: AB is different from BA. What is the significance of the difference AB – BA?

  Out of the blue, it occurred to Dirac that he had come across a special mathematical construction, known as a Poisson bracket, that looked vaguely like AB – BA. He had only a faint visual recollection of the construction, but he knew that it was somehow related to the Hamiltonian method of describing motion. This was characteristic of Dirac, as he was much more comfortable with images than with algebraic symbols. He suspected that the bracket might provide the connection he was seeking between the new quantum theory and the classical theory of the atom – between the non-commuting quantities in Heisenberg’s theory and the ordinary numerical quantities in classical theory. Fifty-two years later, he remembered, ‘The idea first came in a flash, I suppose, and provided of course some excitement, and then of course came the reaction “No, this is probably wrong”. […] It was really a very disturbing situation, and it became imperative for me to brush up my knowledge of Poisson brackets.’

  He hurried home to see if he could find anything about the Poisson bracket from his lecture notes and textbooks, but he drew a blank. So he had a problem:

  There was just nothing I could do, because it was a Sunday evening then and the libraries were all closed. I just had to wait impatiently through that night without knowing whether this idea was any good or not, but still I think that my confidence grew during the course of the night. The next morning I hurried along to one of the libraries as soon as it was open […].39

  A few minutes after Dirac entered the library, he pulled from one of the shelves the tome that he knew would provide the answer to his question: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies by the Edinburgh University mathematics professor Edmund Whittaker. The index directed him first to page 299, where Whittaker set out the mathematical formula for the bracket. Sure enough, as Dirac had surmised, the Poisson bracket, which first appeared over a century before in the writings of French mathematician Siméon-Denis Poisson, had the form of two mathematical quantities multiplied together minus two related quantities multiplied together, the multiplication and minus signs making it appear similar to the expression AB – BA.40 In one of his greatest insights, Dirac saw that he could weave an entire carpet from this thread – within a few weeks of uninterrupted work he had set out the mathematical basis of quantum theory in analogy to the classical theory. Like Heisenberg, he believed that mental pictures of the tiniest particles of matter were bound to be misleading. Such particles cannot be visualised, nor is it possible to describe them using quantities that behave like ordinary numbers, such as position, speed and momentum. The solution is to use abstract mathematical quantities that correspond to the familiar classical quantities: it was these relationships that Dirac pictured, not the particles that they described. Using the analogy with the Poisson bracket, together with the correspondence principle, Dirac found connections between the abstract mathematical quantities in his theory, including the crucial equation connecting the symbols associated with the position and momentum of a particle of matter:

  position symbol × momentum symbol – momentum symbol × position symbol = h ×(square root of –1)/(2×π)

  where h is Planck’s constant and π is the ratio of the circumference to the diameter of every circle (its value is about 3.142). The square root of minus one – the number that, when multiplied by itself gives minus one – plays no role in everyday life but is common in mathematical physics. So there was nothing new on the right-hand side of the equation. The most mysterious part of the equation was on the left-hand side, especially for those unwise enough to think of the position and momentum symbols as anything other than abstractions: they are not numbers or measurable quantities but symbols, purely mathematical objects.

  To all but mathematical physicists of the most austere disposition, Dirac’s description looked remote from reality, but, in the right hands, it was possible to manipulate his abstract symbols to make concrete predictions. In Eddington’s words, ‘The fascinating point is that as the development process proceeds, actual numbers are exuded from the symbols.’41 By this, Eddington meant that the underlying symbolic language yielded, after mathematical manipulation, numbers that experimenters could check. The value of the theory depended on whether these predictions agreed with the readings on counters, dials and detecting screens. If the theory did that successfully and was logically consistent, it must be judged a success, according to Dirac, no matter how peculiar it looked.

  Fowler appreciated that his student had done something special. Dirac’s theory, much more ambitious than Heisenberg’s prototype description of the artificial case of an electron jiggling about in a straight line, sought to describe the behaviour of all quantum particles in all circumstances throughout all time. He knew, however, that the most important priority was to demonstrate that his theory could account for the most important general observations that experimenters had made about atoms. In a few lines of algebra, Dirac demonstrated that energy is conserved in his theory – as it is in the everyday world – and that when an atomic electron jumps from one energy level to another, it gives out a quantum of light whose energy is equal to the difference between the two levels. This indicated that the theory was able to reproduce Bohr’s successes, without having to assume that electrons are in orbit, like planets round a star, doomed to cascade into the nucleus. For Dirac, it was meaningless to use such graphic images – quantum particles can be described only using the precise, rarefied language of symbolic mathematics.

  Although Dirac had been inspired by Heisenberg’s paper, the two men had sharply different approaches to their subject. Heisenberg proudly referred to his paper as ‘the great saw’, a tool to cut off the limb on which the old Bohr theory rested.42 Dirac, on the other hand, sought to build a bridge between Newtonian mechanics and the new theory. His dream was that all the mathematics that Hamilton and others had used to recast Newton’s theory of mechanics would have exact counterparts in the new theory. If Dirac was right, physicists would be able to use the infrastructure of ‘classical mechanics’ – the stuff of hundreds of textbooks – in the construction of the new theory, which had been named the year before by Heisenberg’s senior colleague, Max Born: ‘quantum mechanics’.

  By early November, Dirac had written his paper and had given it an ambitious title that would catch the attention of even the most casual browser: ‘The Fundamental Equations of Quantum Mechanics’. Fowler was delighted. Only a few months before, he had described his student’s ability to ‘push forward the mathematical development of his ideas’ and to ‘view old problems in a fresh and simpler way’.43 Now he could alter the focus of his praise of Dirac from his potential to his achievement. Fowler’s highest priority now was to ensure that the paper was published as quickly as printing schedules allowed; if one of Dirac’s competitors managed to submit a similar paper before him, then, according to the unwritten rules of the scientific community, Dirac would be regarded as an ‘also ran’. Like sport, science is supposed to be an activity in which the winner takes all. Fowler had recently been elected a Fellow of the UK’s academy of
science, the Royal Society, qualifying him to send manuscripts for publication in its proceedings in the confident expectation that they would be accepted without delay.

  For most physicists in Cambridge, the discovery of quantum mechanics was a non-event. Apart from his discussions with Fowler, Dirac made no effort to draw his colleagues into the new revolution in physics that he knew was afoot. Word was beginning to spread, however, that he was a ‘first-rate man’ in the making, though his wispy, almost wordless presence gave no clue to the depth and subtlety of his thinking. It appears to have been at about this time that his colleagues invented a new unit for the smallest imaginable number of words that someone with the power of speech could utter in company – an average of one word an hour, ‘a Dirac’. On the rare occasions when he was provoked into saying more than yes or no, he said precisely what he thought, apparently with no understanding of other people’s feelings or the conventions of polite conversation.

  During a meal in St John’s Hall, he crushed a fellow student who was devoting his time to workaday problems in classical physics: ‘You ought to tackle fundamental problems, not peripheral ones.’44 This was Rutherford’s credo, too, though his approach was more down to earth. Rutherford was wary of the theorists’ effusions about their latest hieroglyphics until the results were useful to experimenters. Quantum mechanics had yet to do that. Most physicists found it implausible that nature could be so perverse as to favour a theory that required thirty pages of algebra to explain the simplest atom’s energy levels, rather than Bohr’s theory, which explained them in a few lines. For Rutherford and his boys, the real sensation that autumn was not the revelations about quantum mechanics but the discovery that electrons have spin. Made at the University of Leiden by two Dutchmen, this discovery took everyone by surprise. In terms of the Bohr picture of the atom, it was easy to envisage crudely what was going on: the orbiting electron is spinning, just as the Earth spins like a top around its north–south axis. Though soon to be taken for granted, many leading physicists thought the idea that the electron has spin was ridiculous.45