Paul Dirac

 From: The Strangest Man by Graham Farmelo

The theories give ‘a radically new way of looking at Nature’. The first of Einstein’s theories is usually dubbed the ‘special theory‘ because it deals only with observers who move in straight lines at constant speeds with respect to one another; for example, passengers on two trains moving smoothly on parallel tracks. Einstein based his theory on just two simple assumptions; first, that when each of the observers measures the speed of light in a vacuum, they will always find the same value, regardless of their speed; and, second, that measurements made by the observers will lead them to agree on all the laws of physics. Einstein’s great insight was to see that if these assumptions were followed to their logical conclusion, a new understanding of space, time, energy and matter emerged. 

A casualty of Einstein’s theory was the widely accepted belief that the universe is pervaded by an ether, which Broad argued had become superfluous: there was supposed to be a peculiar kind of matter, called Ether, that filled all Space. On these theories the Ether was supposed to produce all kinds of effects on ordinary matter, and it became a sort of family pet with certain physicists. As physics has advanced, less and less has been found for the Ether to do.

Contrary to the theory, the existence of such a substance would imply that there is a uniquely privileged frame of reference, so relativity implies that the ether is an unnecessary assumption and may well not exist, unless experiments say otherwise. Einstein also noted that measurements of space and time are not, as almost everyone else thought, independent but are inextricably linked, leading to the idea of a unified space-time, a concept introduced by his former teacher Hermann Minkowski, a German mathematician. Finally, Einstein Showed that an inevitable consequence of this new way of thinking was his equation E = mc2, implying that the mass of a small coin is equivalent to the vast energy needed to run a city for days or indeed to raze it. An apocalyptic vision of this power had already been presented by H. G. Wells, shortly before the outbreak of the First World War, in his novel The World Set Free. 

For most purposes, the predictions of Einstein’s special theory were extremely similar to the corresponding ones made by Newton’s theory. The two sets of predictions, however, were noticeably different at speeds approaching the speed of light in a vacuum: Einstein claimed that, under these conditions, his theory was more accurate, though it would be several decades before the superiority was convincingly demonstrated by experimenters. In the meantime, Einstein’s reasoning made it possible to amend the description of anything given by Newton’s theory and produce a ‘relativistic’ version = one that agreed with the principles of the Special theory of relativity. Two years later, Dirac took up a new hobby, aiming to produce relativistic versions of Newtonian theories = an activity he pursued like an engineer upgrading tried-and-tested designs to ones that perform to a higher specification: ‘There was a sort of general problem one could take, Whenever one saw a bit of physics expressed in a non-relativistic form, to transcribe it to make it fit in with special relativity. It was rather like a game, which I indulged in at every opportunity.’

Einstein’s second theory of relativity applied to all observers, including ones who are accelerating; for example, observers who fall freely under the action of gravity. In this ‘general theory of relativity’, Einstein proposed a geometric picture of gravity, replacing Newton’s concept that an apple and every other mass is subject to a force of gravity by a radically new way of describing the situation. According to Einstein, every mass exists in a curved space-time = roughly analogous to a curved sheet of rubber and the motion of the mass at every point in space-time is determined by the curvature of space-time at that point. Because the theory is relativistic, information cannot be transmitted faster than light, and all energies contribute to mass (via E = mcz) and therefore to gravity. It turns out that, in the Solar System, where almost all matter has comparatively low density and travels much more slowly than light, the predictions of Einstein’s theory of gravity are in extremely good agreement with Newton’s. But, in some situations, they can be distinguished, and one of the most straightforward ways of doing so involved measuring the bending of starlight by its gravitational attraction to the Sun during a solar eclipse: Einstein’s theory predicted that this deflection would be twice Newton’s value. This was the prediction that Eddington and his colleagues believed they had verified in their solar-eclipse experiments. 

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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 hrst 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). 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 assump tion’ 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. 

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In Cambridge, he had iound again that the concept of beauty was in vogue. The popularity of the concept was at least partly due to the enduring success of Principia ethica, published in 1903 by the philosopher George Moore, one of Charlie Broad’s colleagues in Trinity College. Writing With a refreshing absence of jargon, Moore made the incisive sugges. tion that ‘the beautiful should be defined as that of which the admiring contemplation is good in itself’. Soon the talk of intellectuals, Principia ethical was admired by Virginia Woolf and her colleagues in the Bloomsbury Group and declared by Maynard Keynes to be ‘better than Plato’. Over a century before, Immanuel Kant had rendered the subject of beauty too complex and intimidating for most philosophers, but Moore made it accessible again in a way that commanded respect.54 Although Principia ethica did not consider the aesthetics of science, Moore’s common-sense approach to beauty probably influenced his scientific colleagues at Trinity, including Rutherford and the college’s most eminent pure mathematician, G. H. Hardy: both often talked about the beauties of their subject. Kapitza, too, looked on experimental physics not as ‘business’, as it was to several of his colleagues, but as a kind of ‘aesthetic enjoyment’.55 ' 

Although Dirac was not interested in philosophy, this fascination with the nature of beauty had powerful resonances for him. Like many theoreticians, he had been moved by the sheer sensual pleasure of working with Einstein’s theories of relativity and Maxwell’s theory. For him and his colleagues, the theories were just as beautiful as Mozart’s jupiter Symphony, a Rembrandt self-portrait or a Milton sonnet. The beauty of a fundamental theory in physics has several characteristics in common with a great work of art: fundamental simplicity, inevitability, power and grandeur. Like every great work of art, a beautiful theory in physics is always ambitious, never trifling. Einstein’s general theory of relativity, for example, seeks to describe all matter in the universe, throughout all time, past and present. From a few clearly stated principles, Einstein had built a mathematical structure whose explanatory power would be ruined if any of its principles were changed. Abandoning his usual modesty, he .described his theory as ‘incomparably beautiful’.

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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 difhcult 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 rep resent the electron’s position; instead, position is replaced by num\ bers 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 elec‘ tron’s energy levels and represents the likelihood that the electron will jump between that pair of energy levels. 80, 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.

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.” 50, 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 eonsistent 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 hnd the lock a way of interpretng these quantities, a way of linking them to experimentally observed reality.