**In the pantheon** of heroic thinkers who effected the scientific revolution of the first half of the 20th century, some are household names, at least in households with bookshelves: Einstein with his hair on end and his tongue sticking out, Heisenberg and Bohr at odds in Michael Frayne's play *Copenhagen,* and Schrödinger with his dead-and-alive cat. The other names of that revolution are less often on the general public's lips, among them Max Planck, Enrico Fermi, Wolfgang Pauli, Max Born, Paul Dirac.

Of these the last named is perhaps the most obscure in the public eye, but he was far from the least important. The main reason for his anonymity to all but scientists is that his work was so abstract and technical, so mathematical, that its Nobel Prize-winning contribution to the formation of quantum mechanics in the 1920s and '30s is not easily explicable apart from a detailed telling of that discovery's story.

But it is also in part the result of Dirac's diffidence and taciturnity, his shunning of publicity, his introversion, which is, as Graham Farmelo shows in this copiously detailed and sympathetic biography, itself explainable by the fact that Dirac was probably to some degree autistic. Dirac himself attributed his marked social dysfunctionality to bullying by his Swiss-born father, who forced the young Dirac to dine alone with him and converse only in French (though they lived in the very English port city of Bristol, where Dirac was born in 1902). The father's angry responses to linguistic mistakes were so terrifying that Dirac lapsed into a frozen silence that only thawed later with a few special friends, or when he was enthusiastic about something, or when he lectured on quantum mechanics.

**Illustrative examples** of his manner occur in the many stories told about Dirac in physics circles, of which Farmelo makes good use in exploring his character. One relates how a visitor to his Cambridge college, sitting next to him at high table, attempted to engage him in conversation by asking whether he had seen any films recently. After considering this question for some time in his customary silence, Dirac replied, "Why do you want to know?"

Absence of small talk, intense focus on his work, an obsession with precision, and a remarkable degree of abstract mathematical genius, made the thin, otherworldly figure of Dirac stand apart even in the small and highly rarified world of early-20th-century physics. Yet his story, as Farmelo tells it, could be universally emblematic of creative genius in mathematics and science. It is a story of stunning achievements made when young, followed by long decades of fading powers and influence, increasing conservatism, dislike for the direction taken by the work of younger colleagues -- Dirac's story is Einstein's story, too, and the story of many lesser figures. Close to the end of his life Dirac said, in a sudden impassioned outburst to a visiting physicist whom he had not met before and who wished to discuss some ideas with him, "No! I have nothing to talk about! My life has been a failure!"

One of the best of many good things about Farmelo's account is the clarity with which, in describing Dirac's contributions to physics, it shows how his life was anything but a failure. To understand the story one needs a nodding acquaintance with quantum mechanics, as follows.

**The idea of the quantum** was first proposed by Max Planck in the year 1900 as a merely heuristic device to solve the problem of why the glow emitted by a heating-up object changes from red through orange to blue. He suggested that we can give a systematic account of this phenomenon if we think of radiated energy as coming in discrete packages, to which he gave the name "quanta" (Latin for "amounts"). In the third of his three amazingly seminal papers of 1905, Einstein used Planck's idea to solve the puzzle of why metals produce electricity when light is shined on them, a phenomenon known as the photoelectric effect. This involved treating light as consisting of particles -- photons -- thus appearing to controvert the then prevailing and experimentally well-grounded view that light is a wave.

In 1912, Prince Louis de Broglie extended the idea of wave-particle duality from photons to other particles. The suggestion was greeted with scepticism until Niels Bohr conclusively showed that it explains why atoms are stable while yet being able to absorb and emit energy. If energy comes in lumps at discrete intervals on the energy gradient, so that the wavelength of an electron always has to have a whole-number value, then if it absorbs or emits energy it can only do so by jumping to another whole-number wavelength. When the mathematical basis of this picture was worked out, mainly by Heisenberg and Schrödinger, what resulted was an interpretation of electrons as in effect probability smears around the nuclei of atoms (these consisting of protons and neutrons), thus explaining why they seem to be both wave-like and particle-like. When electrons gain or lose energy they move instantaneously from one "position" around the nucleus to another, with no staging posts between.

It was for this reason that Heisenberg formulated the "Uncertainty Principle" stating that one cannot simultaneously measure both the position and momentum (the mass multiplied by velocity) of a subatomic particle. This has profound consequences for the notion of causality and the idea in classical physics that if you knew everything about a physical system at a given moment, including the laws governing it, you would be able to deduce its past and future states. Quantum mechanics showed that this cannot be done; the neat world of classical physics had been overturned.

And since a quantum state consists not in a definite set of values but in a range of probabilities -- for example: a particle does not have a definite path until it is calculated -- it further seems to imply that reality only has a determinate character when it is measured (thus making physicists indispensable to reality). This is the idea behind the "Schrödinger's Cat" example: the cat in the physics box in which a quantum event will settle whether a noxious gas is released or is not released, will be both dead and alive (both states will be in "superposition") until the box is opened and someone looks in, at which point it will become definitely either dead or alive.

**The philosophical weirdness** of this way of thinking about reality was happily accepted by Bohr and his colleagues at his institute in Copenhagen -- hence it is known as the "Copenhagen Interpretation" -- but Einstein and not a few others found it impossible to accept, and thought that it meant there is something fundamentally wrong with quantum mechanics. Dirac's response to the weirdness of the quantum world as revealed by the powerful mathematics in which it was expressed was to say, "Shut up and calculate!" -- that is, forget the philosophical angst, if the mathematics if beautiful, that is enough.

Indeed, mathematical beauty was Dirac's guiding principle, and it was what led him to the discoveries that earned him the Nobel Prize in 1933 and the placement of his portrait next to Einstein's portrait on a wall in Princeton's Institute of Advanced Studies. His chief discovery was the equation -- the Dirac Equation -- that describes the behaviour of matter particles, but he also both made and anticipated a number of further discoveries, showing what a remarkable scientific imagination he had.

**The Dirac Equation** describes the wave function of electrons relativistically, and it led to his prediction of the existence of positrons (the antiparticle of the electron) and by extension the concept of anti-matter. He is the founder of quantum electrodynamics (and the coiner of this name), the part of quantum theory that deals with the interaction of photons with electrons and positrons. Satisfyingly for Dirac's love of precision and mathematical beauty, the theory (described by Richard Feynman as "the jewel of physics") makes extremely accurate predictions of certain atomic quantities.

No account of Dirac's work -- and his work was his life -- is complete without mentioning that in it he joined Einstein's special theory of relativity to quantum mechanics, predicted the existence of magnetic monopoles, founded operator theory, and devised the notation (known as "bra-ket" notation) that became standard in the field. In addition he anticipated, in sketch form, certain ideas in cosmology, and even today's dominant theory of the fundamental structure of matter, string theory. One of his enduring services to science was his brilliant book *The Principles of Quantum Mechanics, *whose clarity, logic, lucidity, and innovations of formalism made it an essential text. His remark that his life had been a failure could therefore not be further from the truth.

Farmelo explains all the science relevant to understanding Dirac, and does it well; equally good is his careful and copious account of a personal life that was dogged by a sense of tragedy - his hatred for his father, his lifelong grief over his elder brother's suicide, and his complicated relationship with a demanding mother, lay at the centre of it. But Dirac bloomed in the all-male atmosphere of Cambridge, which Farmelo describes as perfect for someone with his degree of autism: he was fed, looked after, closeted, allowed to spend six days a week doing nothing but mathematics undisturbed, on the seventh taking solitary walks in the surrounding fenlands.

**But then Dirac married** someone whom Farmelo wittily describes as his anti-particle, an ebullient and warmhearted Hungarian divorcée, Manci, who had two children. This was perhaps the most astonishing thing Dirac ever did outside physics, but evidently it was a wise decision -- and not lightly taken, as Farmelo's quotations from several years' worth of Dirac's dry and unemotional letters shows. When Manci wrote, "Do you have any feelings for me?" Dirac replied, "Yes, some." The letters briefly ceased to be reticent after the honeymoon, during which Dirac seems to have fallen in love with his wife: sex unleashed passion for a while. He was very lucky to have married a divorcée; the idea of his marrying a virgin does not bear thinking about.

Despite being a Cambridge don in the years before 1939, and despite having much sympathy with communism and a great friend in the Russian physicist Peter Kapitsa whom he frequently visited in the Soviet Union, Dirac was not one of the Cambridge spies. Although he contributed to the war work of developing an atom bomb, he did so on the margins of the work, refusing to become more deeply engaged. He was in most ways apolitical, a disengaged theoretician, whose single political act was to join the campaign to persuade Stalin to let Kapitsa leave the Soviet Union.

At one point Dirac and Einstein had offices in the same corridor of the Institute for Advanced Studies at Princeton, but they scarcely interacted. Yet when news of Einstein's death reached Dirac years later, Farmelo tells us, he wept; the only time his wife ever saw him cry.

The long years of unproductivity and marginalization that Dirac experienced when his creative period was over -- its prime years occurred between the mid-1920s and the early 1930s, when he was in his 20s -- might have made Dirac feel a failure at last, not least because he never succeeded in solving his great puzzle concerning the interaction of electrons and photons; but if he could read Farmelo's absorbing and accessible account of his life he would see that it had magic in it, and triumph: the magic of revelations about the deep nature of reality, and the triumph of having moved human understanding several steps further toward the light.

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