There was never any hope that the two would collaborate, as became plain early in Dirac’s stay when Bohr called him into his office to help him write a paper. This was Bohr’s usual practice: he often dragooned one of his young colleagues into spending a few days as his scribe. The only reward was the honour of being asked and a daily lunch with the Bohrs in their apartment. But the process was not without its frustrations: no sooner would a sentence escape Bohr’s lips than he would qualify, amend or delete it in favour of another form of words that might, or might not, be a closer approximation to his intended meaning. So, the tortuous process of dictation continued, never quite reaching a coherent conclusion. Dirac had better things to do than to spend hours disentangling Bohr’s fractured locutions and rendering them into prose of exemplary clarity. ‘At school’, Dirac announced soon after the first session with Bohr began, ‘I was always taught not to start a sentence until I knew how to finish it.’ His employment as Bohr’s amanuensis lasted about half an hour.14
In the evenings, most of the young physicists at the institute liked to relax in the cinema or in their lodgings with a plate of hot dogs and a few beers. But Dirac preferred to spend his nights taking long, solitary walks around the city. He would set out from his lodgings after dinner, take a tram to its terminus and walk the Copenhagen streets back to his digs, thinking about the problems of quantum physics.15 He probably did not know that he was following in the footsteps of the nineteenth-century philosopher Søren Kierkegaard, pioneer of Christian existentialism and almost as famous among his fellow Danes for his eccentricities as his ideas.16Kierkegaard chewed over his ideas in his apartment, pacing back and forth for hours, and during the ‘people bath’ he took each day in the streets of his native city. For two decades from the mid-1830s, the people of Copenhagen saw the hunch-backed aristocrat walking around in his broad-rimmed hat, his umbrella folded under his arm. ‘I have walked myself into my best thoughts,’ he said, a remark precisely echoed by the elderly Dirac.17 But they reacted differently to the people they passed in the street. Dirac said nothing to his fellow pedestrians, but Kierkegaard would startle some of them by interrogating them about some subject on his mind, following in the tradition of Socrates, whom he called ‘the virtuoso of the casual encounter’.18
During the day, Dirac spent most of his time working in the library, occasionally pausing to read the latest publications in the adjoining ‘journal room’ and to attend a seminar. To Christian Møller, one of the young Danish physicists at the institute, Dirac appeared distracted and aloof:
Often he sat alone in the innermost room of the library in the most uncomfortable position and was so absorbed in his thoughts that we hardly dared to creep into the room, afraid as we were to disturb him. He could spend the whole day in the same position, writing an entire article, slowly and without ever crossing anything out.19
In the library, Dirac was cooking up what would turn out to be one of his most famous insights, the connection between the Heisenberg and Schrödinger versions of quantum theory. Everyone knew that the theories seemed to give the same results, yet they looked as different as Japanese and English. Dirac found the rules that allow the two languages to be translated into each other, laying bare the relationship between them and giving new clarity to the Schrödinger equation. It turned out that the Schrödinger waves were not quite as mysterious as they seemed but were simply the mathematical quantities involved in transforming a description of a quantum – an electron, or any other tiny particle – based on its energy values to one based on possible values of its position. Dirac’s theory also accommodated Born’s interpretation of Schrödinger’s waves and showed how to calculate the probability of detecting a quantum. He began to realise that the knowledge an experimenter can have about the behaviour of a quantum is also limited. He wrote that ‘one cannot answer any question on the quantum theory which refers to the numerical values for both [the initial position and momentum values of a quantum]’, and he pointed out cryptically that one would expect to be able to answer questions in which only one of those initial values
is known. He was within a split whisker of what would become the most famous principle in quantum mechanics, the uncertainty principle, soon to be snatched from under his nose by Heisenberg.
In the course of working out his theory, Dirac introduced a new mathematical construction that made no sense within conventional mathematics. The object, which he called the delta function, resembles the outer edge of the finest of needles, pointing vertically upwards from its base.20 Away from that base, the numerical value of the delta function is zero, but its height is such that the area enclosed between the perimeter and the base is exactly one unit. Dirac knew but did not care that pure mathematicians would regard the function as preposterous as it did not behave according to the usual rules of mathematical logic. He conceded that the function was not ‘proper’ but added blithely that one can use it ‘as though it were a proper function for practically all purposes in quantum mechanics without getting incorrect results’. It was not until the late 1940s that mathematicians accepted the function as a concept of unimpeachable respectability.
In an interview in 1963, he remarked that it was his study of engineering that led him to his new function:
I think it was probably that sort of training that first gave me the idea of the delta function because when you think of load in engineering structures, sometimes you have a distributed load and sometimes you have a concentrated load at the point. Well, it is essentially the same whether you have a concentrated load or a distributed one but you use somewhat different equations in the two cases. Essentially, it’s only to unify these two things which sort of led to the delta function.21
But Dirac’s recollections may have been wrong. It may well be that he first read about the delta function from Heaviside, who introduced the function with his customary belligerence in one of the books Dirac read as an engineering student in Bristol.22Today, the function is associated with Dirac’s name, but he had not been the first to invent it – that appears to have been done in 1822 by Heaviside’s favourite mathematician, the Frenchman Joseph Fourier, though several others later discovered it independently.23
Bohr was indifferent to mathematical rigour, so he would not have been perturbed by the delta function when he read about it in the draft Dirac submitted to him, following the understanding that Bohr had to approve each paper submitted from the institute. However, Bohr and Dirac were soon in disagreement, like two poets in dispute over the syntax of a stanza. Bohr cared about every word and repeatedly requested detailed changes.24 For Dirac, the words were there to give the clearest possible expression to his thoughts, and, once he had found the right words, he saw no need to change them. He would have agreed with T. S. Eliot: ‘It means what it says and if I had wanted to say it any other way, I should have done so.’
Dirac was usually quick to attribute his success to luck, but not in this case – he referred to the paper as ‘my darling’.25 He later remarked that he was pleased to have solved the particular problem he set out to tackle, of laying bare the relationship between Heisenberg’s theory and Schrödinger’s. The main quality needed in its solution was technical skill and application; in his view, no special inspiration was involved. Another reason why Dirac was so fond of his ‘darling’ was probably that it was a success for his method of developing quantum mechanics by analogy with classical mechanics. During his reading about Hamilton’s approach to classical mechanics, he had read how ‘transformation theory’ related different descriptions of the same phenomenon – by using this idea to find the connection between Heisenberg’s theory and Schrödinger’s, Dirac had shed light on both.
If he hoped that the paper would establish him as the leader in the field, he was soon to be disappointed. In the late autumn, before he had the proofs of his paper, he heard that Pascual Jordan had solved exactly the same problem. Although Dirac’s approach and presentation were more elegant and easier to use, the two paper
s covered substantially the same ground and featured much the same conclusions. So although Dirac had made another distinguished contribution to quantum mechanics – his second within a year – he had yet to beat all his colleagues to a key innovation in the theory. He had, however, acquired some distinguished admirers, though most of them were struggling to understand his peculiar combination of logic and intuition. One of them was Albert Einstein, who told a friend: ‘I have trouble with Dirac. This balancing on the dizzying path between genius and madness is awful.’26
One evening in Dirac’s lodgings shortly before Christmas, the telephone rang. It was Professor Bohr, Dirac’s landlady told him, as she passed the receiver to him. This was a new experience for him – he had never before used a telephone.27 Knowing that Dirac was about to spend the holiday alone, Bohr was calling to ask if he would like to spend Christmas with him and his family. Dirac accepted, though he did not tell his parents. They had been shivering in an unseasonably cold autumn and recovering from the upheaval of having mains electricity installed. Dirac’s mother persisted with her doomed campaign to persuade him to do less work and to eat more (‘I hope you will take it easy & get nice and plump like Shakespeare’s Hamlet’) and, for the first time, confided in her son that she was unhappy and tired of the domestic routine. Desperate for a measure of independence, when Charles was out, she and the unemployed Betty sneaked out together to evening classes in French.28
The Dirac family was also preparing itself for its saddest Christmas: a year before, they had had three children at home for the holiday; now they would have only one. On 22 December, the ailing Charles wrote his son a letter, one of only two that Dirac kept from his father, possibly the only letters Dirac received from him in adult life.29 No longer communicating with Dirac only in French, Charles wrote the four-page letter entirely in English and on black-bordered notepaper that signalled his continuing mourning for Felix.
My dear Paul
It will be a lonely time here without you – the first time since you came to us – not so very long ago it seems, but my thoughts are with you to wish you all the happiness a father can wish his only son.
If you can any time spare a few moments to give me some details of your life there and your work – nothing could please me more, except seeing you again. I should like to feel sure you take sufficient care of yourself – and do not let your studies make you forget your health.30
Charles goes on to say that he would like to buy his son a Christmas present, perhaps ‘a set of chessmen’, and he offers to do ‘anything at all’ he can to help him. He signs off ‘Many kisses from your loving Father’. The note is a window on his grief, his loneliness, his desperation to be closer to his unresponsive ‘only son’.
At midnight on Christmas Eve, Charles and Betty went to a service at a local church, where Felix’s death had first been marked. Later, on Christmas Day, Dirac’s mother wrote Dirac a fragmentary letter showing that she was as lonely as the man she was living with:
All we do, as you know, is work & then more work. […] I am trying to get Pa to have [the front room] re-papered. He ought to after 13 years […] He and Betty went up to Horfield Church at 12-midnight for a Service […] This is the first Xmas Day you have been away from home. It is lonely without you.
She then asked him an unusual favour:
Would you like to send me a few pounds for a diamond ring? I want one so very much. I could wear it in the evenings & think what a darling you are. It is so monotonous doing housework all day long. I get so fed up with it.
Pa has pupils all the year round & gives me £8 a year for clothes and everything. It is worse than a servant.31
For the first time in her correspondence, she showed Dirac that he was not just her favourite son but her most intimate confidant and even a substitute for a gift-bearing lover. As her subsequent letters showed, she was in desperate straits, trapped in an unfulfilling marriage to a man who was highly regarded in the community but whom she regarded as an unsympathetic and insensitive brute. In the coming years, her life would unfold like an Ibsen tragedy.
Another of the out-of-the-blue ideas that Dirac apparently conceived in Copenhagen is now the basis of all modern descriptions of the fundamental constituents of the universe. Such descriptions are based on the nineteenth-century concept of a ‘field’, which had superseded Newton’s vision that nature’s basic particles move under the influence of forces exerted by other such particles, often over long distances. Physicists replaced the notion that the Sun and the Earth exert gravitational forces on each other by the more effective picture that the Sun, the Earth and all the other matter in the universe collectively give rise to a gravitational field which pervades the entire universe and exerts a force on each particle, wherever it is located. Likewise, an all-pervasive electromagnetic field exerts a force on every electrically charged particle. Maxwell’s theory of electromagnetism and Einstein’s theory of gravity are examples of classical ‘field theory’, each featuring a field that varies smoothly throughout space and time, not mentioning individual quanta. Such classical theories describe the universe in terms of a smooth, underlying fabric. Yet, according to quantum theory, the universe is fundamentally granular: it is ultimately made of tiny particles such as electrons and photons.
Loosely speaking, the texture of the underlying fields should, according to classical ideas, be rather like a smooth liquid, whereas quantum theory suggests that it would be like a vast collection of separate grains of sand. To find a quantum version of Maxwell’s classical electromagnetism was one of the theoreticians’ most pressing problems, and Dirac’s next innovation was to solve it.
Quite what put him on to the solution is something of a mystery. Although he was probably aware of the first steps taken a few months before by Jordan, Dirac later said that he first hit on the idea when he was playing with Schrödinger waves as if they were mathematical toys, wondering what would happen if they behaved not as ordinary numbers but as non-commuting quantities.32 The answer began a new way of describing the quantum world.
Dirac found a way of mathematically describing the creation and destruction of photons, both commonplace processes. Particles of light are continually created in vast numbers all over the universe in stars and also here on Earth, when an electric light is switched on, a match is struck, a candle is lit. Likewise, photons are continually destroyed – annihilated – for example, when they disappear into human retinas and when leaves convert sunlight to life-giving energy. Neither of these processes of creation and annihilation can be understood using Maxwell’s classical theory, which has no way of describing things that appear out of nowhere or disappear into oblivion. Nor did ordinary quantum mechanics have anything to say in detail about the processes of emission or absorption. Yet Dirac showed that this wizardry can be described in a new type of theory, a compact mathematical description of the creation and destruction of photons. He associated each creation with a mathematical object, a creation operator, which is closely related to but quite distinct from another object associated with annihilation, an annihilation operator.
In this picture, at the heart of modern quantum field theory, the electromagnetic field pervades the entire universe. The appearance of every photon is simply an excitation of this field at a particular place and time, described by the action of a creation operator. By a similar token, the disappearance of a photon is the de-excitation of the field, described by an annihilation operator.
Dirac had begun to set out a quantum version of Maxwell’s unified field theory of electricity and magnetism. He had learned about that theory only three years before, in Cunningham’s lectures in Cambridge, and was now standing on Maxwell’s shoulders. So far as Dirac was concerned, his theory put an end to the hand-wringing about the apparent conflict between two theories of light: a wave theory seemed to account for propagation, while a particle theory was needed to explain the interactions with matter. The new theory avoided the embarrassment of having to choose between the wav
e and particle descriptions and replaced the two sharply contrasting pictures with a single, unified theory. Evidently pleased with himself, Dirac wrote that the pictures were in ‘complete harmony’. But he was not interested in sharing the good news with his parents, who read on their weekly postcard their son’s familiar message: ‘There is not much to say now.’33
In his paper, Dirac applied his theory and compared his results with the successful predictions Einstein had made a decade before, in 1916. Einstein had used old quantum ideas to calculate the rate at which atoms can emit and absorb light, producing formulae that appeared to describe these processes successfully. The question Dirac had to answer was: does the new theory compare favourably with Einstein’s?
Einstein’s theory had accounted for the interaction of light and matter in terms of three fundamental processes. Two of them were familiar enough: the emission and absorption of a photon by an atom. But Einstein also predicted a previously unknown way of ‘persuading’ an atom to jump from one energy level to a lower one, by stimulating it with another photon whose energy is exactly equal to the difference between the two energy levels. The result of this process of ‘stimulated emission’ is that two photons emerge from the atom: the original one and another one given out when the atom jumps to the lower energy level. This process takes place in the ubiquitous laser – there is at least one in every CD and DVD player and in every bar-code reader – and so is the most common technological application of Einstein’s science. Dirac’s theory produced exactly the same formulae as Einstein’s and had the other advantages that it was more general and mathematically more coherent. As he probably realised, he had gone one better than Einstein.
The Strangest Man Page 17