The Strangest Man Page 8

  52 Nahin (1987: 27, n. 23). Heaviside never completed his autobiography.

  53 Interview with Dirac, AHQP, 6 May 1963, p. 4. Another example of the kind of neat tricks that engineers use and that Dirac read about as an engineering student is featured in the appendix to one of his set textbooks (Thomälen, 1907).

  54 The two books that Dirac used to study stress diagrams were Popplewell (1907) (see especially Chapter 5) and Morley (1919) (see especially Chapter 6).

  55 Dirac (1977: 113).

  56 The ‘spoilsport’ taught Dirac in the autumn of 1920. Dirac’s reports are in Dirac Papers, 1/10/16 (FSU).

  57 Interview with Dirac, AHQP, 6 May 1963, p. 13. Dirac’s lack of a qualification in Latin was not a bar to his admission to postgraduate study at Cambridge, but it would have made him ineligible to study there as an undergraduate.

  58 Warwick (2003: 406 n.); Vint (1956).

  59 Letter from Charles Dirac, 7 February 1921, STJOHN.

  60 Dirac took the examination on 16 June 1921. The examination papers are in Dirac Papers, 1/10/11 (FSU).

  61 Letter from Dirac to the authorities at St John’s College, 13 August 1921, STJOHN.

  62 Boys Smith (1983: 23). A much higher estimate of the amount needed to live as a student in Cambridge at the time is given in Howarth (1978: 66): about £300.

  63 Letter from Charles Dirac, 22 September 1921, STJOHN.

  64 Unsigned letter from St John’s College to Charles Dirac, 27 September 1921, STJOHN. The signatory concludes his letter: ‘Perhaps before deciding [what to do] you would be so kind as to let me know the sum total of means that he would have at his disposal, I could then better advise what he can do.’

  Four

  Mathematics […] does furnish the power for deliberate thought and accurate statement, and to speak the truth is one of the most social qualities a person can possess. Gossip, flattery, slander, deceit all spring from a slovenly mind that has not been trained in the power of truthful statement.

  S.T. DUTTON, Social Phases of Education in the School and the Home, London, 1900

  What might have happened to Dirac if he had got one of the jobs he applied for, perhaps in the burgeoning aviation industry? Might the loss to physics have been offset by a commensurate gain for aeronautics? That these are questions of virtual history is due to the mathematician Ronald Hassé, who deftly steered Dirac’s career from engineering to science. Things could easily have worked out quite differently. In September 1921, when Dirac was at a loose end and looking for jobs, David Robertson suggested to Dirac that, rather than hang around doing nothing, he should do an electrical-engineering project.1 Dirac dabbled in some experiments, but, after a few weeks, Hassé wooed him back to the lecture theatres in the mathematics department, having arranged for him to do a full mathematics degree free of charge and for him to skip the first year’s work so he could complete it in two years.

  Dirac’s fellow mathematics students were struck by his punctuality. For the first lectures of the day, beginning at 9 a.m., he was always the first to arrive, silently occupying a seat in the front row and showing no interest whatever in his fellow students. He spoke only when spoken to and talked only in clipped, matter-of-fact sentences that bore no trace of emotion. One of the students later recalled that no one even knew the name of the ‘tall, pallid youth’ or showed much interest in him until the results of the Christmas examination results revealed that the new student ‘P. A. M. Dirac’ was top of the class.

  Some of the students resolved to make some enquiries about their mysterious colleague. They were surprised to learn that although he was eighteen months younger than anyone else in the class, he already had a degree in engineering. One of his characteristics was that although he was preternaturally silent, he did stir if he spotted a serious scientific error. In one such incident, after a lecturer had filled two and a half blackboards with symbols and left almost all the students frantically scribbling as they tried to keep up with him, he realised that he had made a mistake. He stood back from the blackboard and turned to Dirac: ‘I have gone wrong, can you spot it?’ After Dirac identified the error and explained how to put it right, the lecturer thanked him and resumed his exposition.2

  In Dirac’s first year of his new course, he studied pure mathematics – the branch of mathematics pursued with no concern for its applications – and applied mathematics, employed to solve practical problems. One of his lecturers was Peter Fraser, a farmer’s son from the Scottish Highlands, a bachelor who lived much of his life in a reverie and liked to tramp the countryside while contemplating the higher truths of mathematics. He did no original research and never wrote a research paper but channelled all his intellectual energy into his teaching. Dirac believed he was the best teacher he ever had.3

  Shortly before 9 a.m. on Mondays, Wednesdays and Fridays, Dirac was in his seat, awaiting the next episode of Fraser’s teaching of a special type of mathematics, known as projective geometry, largely a French invention derived from studies of perspective, shadows and engineering drawing. One of its founders was Gaspard Monge, a draughtsman and mathematician who much preferred to solve mathematical problems using geometric ideas rather than complicated algebra. In 1795, Monge founded the descriptive geometry that Dirac had used in the first technical drawings he made in Bishop Road School, representing objects in three orthogonal points of view. Jean-Victor Poncelet, an engineer in Napoleon’s army, built on Monge’s ideas to set out the principles of projective geometry when he was a prisoner in Russia in 1812. His ideas and their consequences were to become the mathematical love of Dirac’s life.

  When most students come across projective geometry, they find it an unusual branch of mathematics because it primarily taxes their powers of visualisation and does not feature complicated mathematical formulae. What matters in projective geometry is not the familiar concept of the distance between two points but the relationships between the points on different lines and on different planes. Dirac became intrigued by the techniques of projective geometry and by their ability to solve problems much more quickly than algebraic methods. For example, the techniques allow geometers to conjure theorems about lines from theorems about points, and vice versa – ‘that appealed to me very much’, Dirac stressed forty years later.4 To him, an impressionable young mathematician, this was a powerful demonstration of the power of reasoning to probe the nature of space.5

  Fraser also persuaded Dirac of the value of mathematical rigour – an uncompromising respect for logic, consistency and completeness – something he had, as an engineering student, been taught to wink at.6 In Dirac’s studies of applied mathematics, he learned how to describe electricity, magnetism and the flows of fluids using powerful equations that yielded neat solutions, all consistent with experimental observations. He also used Newton’s laws of mechanics to study the contrived examples that inform the education of every applied mathematician: rigid ladders resting against walls, spheres rolling down inclined planes, and beads sliding around circular hoops.7 Dirac filled several exercise books with his answers, most of them flawless. He did most of this work in his bedroom, his escape from the family he perceived to be unloving and a refuge from Betty’s yapping dog. Betty was developing into an unambitious, self-deprecating young woman, in awe of her brother Paul’s intelligence, content to while away hours doing nothing. Her father doted on her, as Bishopston local Norman Jones remembered sixty years later when he said that his main recollection of Charles Dirac was ‘seeing him always carrying an umbrella, struggling up the hill […] often with his daughter, of whom he was very fond’.8

  Dirac saw Felix only occasionally, at weekends, when he returned from his lodgings in the Black Country of the Midlands, near Wolverhampton. The brothers were still not on speaking terms.

  In the final year of his course, Dirac should have been given the choice of specialising in either pure or applied mathematics. He wanted to take the pure option but did not get his way. His fellow student on the honours mathematics deg
ree programme, Beryl Dent – the strong-minded daughter of a headmaster – had the upper hand because she was paying for her tuition, unlike Dirac. She expressed a firm preference for studying applied mathematics, and her wishes carried the day, perhaps partly because it was easiest for the lecturers to teach the same courses to the two students. So, for the first time since he began senior school, Dirac had to work alongside a young woman, but his relations with her were strictly formal; they seldom spoke.9

  Dirac spent the 1922–3 academic year with his head down, building on the applied mathematics that he had learned the year before. One bonus for him was that his course included a few lectures on the special theory of relativity, though he probably knew more about the subject than his lecturer.10 By the time he had finished, he had acquired considerable expertise in Newtonian mechanics. Although he knew that Einstein had found fault with Newton’s laws of mechanics, they worked extremely well for all real-world applications, so it made good sense to master them, as tens of thousands of other students – including Einstein himself – had done before.

  During his mathematics degree, Dirac encountered the ideas of William Hamilton, the nineteenth-century Irish mathematician and amateur poet. He was a friend and correspondent of William Wordsworth, who served science well by helping to persuade Hamilton that he would do better to spend his time on mathematics rather than on poetry. Among his discoveries, Hamilton was most enamoured with his invention of quaternions, mathematical objects that behave peculiarly when they are multiplied together. If two ordinary numbers are multiplied, the same result emerges regardless of their order of multiplication (for example 6 × 9 has the same value as 9 × 6). Mathematicians say that such numbers ‘commute’. But quaternions are different: if one quaternion is multiplied by a second, the result is different from the result obtained if the second is multiplied by the first. In modern language, quaternions are said to be ‘non-commuting’.11 Hamilton believed that quaternions have many practical applications, but the consensus was that they are mathematically interesting but scientifically infertile.

  Dirac also heard about Hamilton’s reformulation of Newton’s laws of mechanics. Hamilton’s approach largely dispensed with the idea of force and, in principle, enabled scientists to study any material thing – from a simple pendulum to cosmic matter in outer space – much more easily than was possible using Newton’s methods. The key to Hamilton’s technique was a special type of mathematical object that comprehensively describes the behaviour of the thing under study, the Hamiltonian, as it became known. Hamilton’s methods became another of Dirac’s fixations and were to become his favourite way of setting out the fundamental laws of physics.

  The mathematics degree did not present a sufficient challenge to keep Dirac occupied, so Hassé encouraged him to take as many of the undergraduate physics courses as his timetable allowed. Once again, Dirac chose to study fundamental subjects which were not covered in his syllabus. In one course, he studied the electron, the particle discovered twenty-five years before in the Cavendish Laboratory in Cambridge by J. J. Thomson, a man equally adept at investigating nature theoretically and – despite his ham-fistedness – experimentally. Several of Thomson’s colleagues thought he was joking when he argued that the electron was smaller than the atom and was a constituent of every atom; to many scientists, the idea that there could exist matter smaller than the atom was inconceivable. Yet he was proved right, and, by the time Dirac first became acquainted with the electron, textbooks routinely ascribed electric current to the flow of Thomson’s electrons.

  Dirac also attended lectures in atomic physics given by Arthur Tyndall, a kindly and articulate man with a keen eye for scientific talent. Tyndall introduced Dirac to what was to prove one of the central insights of twentieth-century physics: the idea that the laws of ‘quantum theory’, which describe nature on the smallest scale, are not the same as the scientific laws that describe everyday matter. Tyndall illustrated this by describing how the energy of light arrives not in continuous waves but in separate, tiny amounts called quanta. At first, this idea was not taken seriously, as virtually all scientists were convinced that light behaves as waves. Their faith rested on the unarguable success of the theory of light published several decades before by the Scottish physicist James Clerk Maxwell, the Cavendish Laboratory’s first professor. According to this theory, checked by many experiments, the energy of light and all other types of electromagnetic radiation is delivered not in lumps but continuously, like water waves lashing against a harbour wall.

  Quantum theory had been discovered – largely by accident – by Max Planck, the Berlin-based doyen of German physics. He happened on the idea of quanta when he was analysing the results of some apparently obscure desktop experiments that investigated the radiation bouncing around inside the reflecting walls of ovens at steady temperatures (the experiments aimed to help German industry improve the efficiency of lighting devices).12 The quantum emerged stealthily from the darkness of those ovens through the ingenuity of Planck, who brilliantly guessed a formula for the variation in the intensity of the radiation with its wavelength, at every temperature setting of the oven. In the closing weeks of 1900, Planck found he could explain the formula for the ‘blackbody radiation spectrum’ only if he introduced a concept that seemed completely contrary to Maxwell’s theory: the energy of light (and every other type of radiation) can be transferred to atoms only in quanta.

  The conservative Planck did not view this quantisation as a revolutionary discovery about radiation but as ‘a purely formal assumption’ needed to make his calculations work. Einstein first recognised the true importance of the idea in 1905, when he took the concept of radiation quanta literally and demonstrated that the reasoning Planck had used to derive his black-body radiation spectrum formula was hopelessly flawed. The challenge was to do better than Planck by finding a logical derivation of the formula.

  When Planck discovered the quantum of energy, he also realised that its size is directly determined by a new fundamental constant, which he denoted h and others dubbed Planck’s constant. It figures in almost every equation of quantum theory, but nowhere in the previously successful theories of light and matter, retrospectively labelled ‘classical theories’. The minuscule size of the constant means that the energy of a typical quantum of light is tiny; for example, a single quantum of visible light has only about a trillionth of the energy of the beat of a fly’s wing.

  In these lectures, Tyndall introduced Dirac to a new way of thinking about light, to new physics. But although Tyndall was admired for his clear presentations, quantum physics was then vague, provisional and messy, so it was impossible for him to present to Dirac the kind of tidy, well-reasoned course that he preferred, underpinned by clear principles and concise equations. This may explain why, if Dirac’s later recollections are correct, his first course in quantum theory made virtually no impact on him. His main interest remained relativity.

  Despite his earlier setback, Charles Dirac had not lost hope of sending Paul to Cambridge. Late in March, Ronald Hassé wrote to the applied mathematician Ebenezer Cunningham, one of the Fellows of St John’s College, reminding him of Dirac’s failure to win a local scholarship that would have enabled him to take up the place that he had won two years earlier. Hassé pointed out that he was ‘certain to get first class honours in June’, and that he was ‘an exceedingly good mathematician’, interested mainly in ‘general questions – relativity, quantum theory etc., rather than in particular details, and is, I think, very keen on the logical side of the subject’. Among his perceptive comments, Hassé did include some provisos about the young Dirac’s character: ‘He is a bit uncouth, and wants some sitting on hard, is rather a recluse, plays no games, is very badly off financially.’ Those minor points aside, Hassé warmly recommended that the college should accept Dirac if he could find the funds to eke out a living.13

  This time, Paul Dirac was successful. In August, after he heard that he had won a place at Cambridge, he a
sked to study relativity with Eddington’s Quaker colleague Cunningham, who had introduced a muddled version of Einstein’s special theory of relativity to the UK shortly before the Great War.14 At that time, Cunningham and Eddington were streets ahead of the majority of their Cambridge colleagues, who dismissed Einstein’s work, ignored it or denied its significance.15 But Cunningham was not available: he had given up supervising graduate students after the war, when he had been pilloried as a conscientious objector, most woundingly by authorities who prevented him from working in schools on the grounds that he ‘was not a fit person to teach children’.16 The supervisor chosen for Dirac was another mathematical physicist, Ralph Fowler, a generous-spirited man with the build of Henry VIII and the voice of a drill sergeant. He was not a master of relativity but the foremost quantum theorist in the country and an expert in linking the way materials behave to the en-masse behaviour of their atoms. For Dirac, wanting above all to study relativity, this was not encouraging news.

  Two scholarships – one of £70 per year from St John’s College, the other from the Government’s Department of Scientific and Industrial Research for £140 per year – were sufficient to fund Dirac’s first year in Cambridge, provided he lived frugally, as was his wont.17 The arrangements seemed to have fallen into place, but, in September, he received bitter news: the university required students to settle their bills at the beginning of term, but his government grant was going to arrive too late. He feared that he would again have to forgo his place, all for the sake of £5.

  But his father came to the rescue by handing him the money he desperately needed to be sure of solvency in Cambridge. Dirac was touched. This was a crucial act of compassion, he later said, and it minded him to forgive his father for the browbeatings round the dinner table and all the other earlier miseries.18 Charles Dirac did not seem so bad after all.