Dirac by Farmelo

In Eddington’s words, “The fascinating point is that as the development process proceeds, actual numbers are exuded from the symbols.’ 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.** 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’. 

 It would take several years before quantum mechanics crystallised into a complete theory. During that time it was a work in progress by about 50 physicists. In retrospect they resembled a group of construction workers who had agreed on a common project to build a new theory of the behaviour of matter though not on how to accomplish it.

 Quantum mechanics was still only a rudimentary fury. Much remained to be clarified about the interpretation of its mathematical symbols what did they really mean? It was impossible to say any more about the motion of subatomic particles?

 Dirac had recently heard  That an alternative version of quantum theory had appeared one that looked completely different from Heisenberg's. The author of the new version was the Austrian theoretician Erwin Schrodinger working in zurich. He was 38 years old. Schrodinger had developed his quantum theory independently of Heisenburg and a few weeks later.  Show dingir discovered an equation that describe the behaviour of quanta of matter in terms of their associated waves and then applied the theory in a series of dazzling papers. The great virtue of schroedinger's theory was that it was easy to use. For the many scientists intimidated by the abstract mathematics in Heisenberg's approach schrodinger offered the balm of familiarity his theory was based on an equation that closely resembled those most physicists had mastered as undergraduates when they were studying water and sound waves . Better still in schroedinger's theory the atom could be at least to some extent visualised.

Within a few weeks of mastering Schrodinger’s equation, Dirac used it to make one of his most famous contributions to science, |t concerned the most basic particles that exist in nature, usually described as ‘fundamental’ because they are believed to have no constituents at all. Classic examples are photons and electrons. Today, two established experimental facts form the bedrock of studies about fundamental particles. First, for each type of fundamental particle, every single one of them in the universe is the same and identical to all other particles of the same type every electron in every atom on Earth is indistinguishable from every electron in galaxies millions of light years away, just as all the trillions of photons given out each second from a light bulb are the same as the photons given out by the most distant star. For electrons and photons, if you have seen one, you have seen them all. Second, the types of fundamental particles fall into one of two classes, much as almost all human beings can be classified as males or females. The first class is exemplified by the photon, the second by the electron. In 1926, no one knew that there were two such classes. 

The differences between the behaviours of electrons and photons exemplify the sharp contrast in behaviour between the two known classes of particle. For a collection of electrons, say in an atom, each available energy state can usually accommodate no more than two electrons. The situation is quite different for photons: each energy state can host any number of them. One way to visualise this difference is to imagine a pair of bookcases with horizontal shelves arranged vertically above one another in ascending order of energy the higher the shelf, the higher the energy to which it corresponds. The shelves of the ‘electron bookcase’ represent the energy states available to electrons, while the shelves of the ‘photon bookcase’ correspond to the states available to photons. For the ‘electron bookcase’, each shelf can accommodate at most two books: once the shelf is occupied, it is full and no others can join it. The ‘photon bookcase’ is different because its shelves can each house any number of books. It is as if electrons are unsociable, whereas photons are gregarious. 

Pauli first realised the aversion of electrons to their own company in 1925 when he suggested his exclusion principle. This explained the puzzle of why all the electrons in an atom do not all orbit the nucleus in the same, lowest-energy orbit: it is because the electrons simply are not allowed to fit into the same state they are forced by the exclusion principle to occupy higher-energy states. This is why the different types of atom manifest as different chemical elements ~ behave so differently. In common experience, neon is a gas and sodium is a metal, yet the atoms of neon gas are very similar to the sodium atoms: outside their nuclei, they differ only in that a sodium atom contains one more electron than a neon atom, That additional electron determines the differences between the two elements, and the Pauli exclusion principle explains why sodium’s extra electron does not simply join the others and form an almost identical type of atom; rather, it occupies a higher-energy quantum state that is responsible for the differences between the behaviour of the two elements. For the same reason, if there were no exclusion principle, the world around us would have none of the huge variety of forms, textures and colours that we take for granted. Not only would our senses have nothing to perceive, they would not exist. Nor, indeed, would human beings or even life itself. 

And

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 granuJar: 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 Schrodinger waves as if they were mathematical toys, wondering what would happen if they behaved not as ordinary numbers but as non-commuting quantities.>* 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 bout the apparent conflict between two theories of light: a wave theory seemed to account for propagation, while a particle theory way heeded to explain the interactions with matter. The new theory avoided the embarrassment of having to choose between the wave 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 wag 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.’ 

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. 

At the end of January, as he was preparing to leave Copenhagen, Dirac posted his paper to the Royal Society. It turned out that he was the first to introduce the mathematics of creation and annihilation into quantum theory.

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Robert Oppenheimer, who had fled Cambridge and was flourishing in Max Born’s Department of Theoretical Physics as a Ph.D. student of rare ability, self-confidence and superciliousness. Ever the intellectual peacock, Oppenheimer ensured that his colleagues knew he was thinking about more than physics: his eclectic reading list included F. Scott Fitzgerald’s collection of short stories Winter Dreams, Chekhov’s play Ivanov and the works of the German lyric poet Johann Holderlin.! He was also composing verse, a hobby that puzzled Dirac. ‘I don’t see how you can work on physics and write poetry at the same time,’ he remarked during one of their walks. ‘In science, you want to say something nobody knew before, in words everyone can understand. In poetry, you are bound to say something that everybody knows already in words that nobody can understand.’ For decades to come, Oppenheimer liked to recount this anecdote over cocktails, no doubt having polished Dirac’s original phrasing to give it the bite of one of Wilde’s paradoxes.” 

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JANUARY 1927-SPRING 1927 

Meanwhile, the debates about the interpretation of quantum theory had not abated, least of all in Copenhagen, where Heisenberg was struggling to understand the theoretical limits of what can be known about a quantum. He achieved this brilliantly with his uncertainty principle, which made him into the nearest the quantum fraternity had to a household name. 

The principle emerged only after anguished and protracted gestation, which apparently began with a letter from Pauli during the previous October.?> Heisenberg believed that the correct way to think about the quantum world was in terms of particles, and that the more popular wave-based ideas were merely useful supplementaries. Somehow, Heisenberg wanted to find a way of making definite statements about the measurements that could be made on quantum particles, especially about the limitations on what experimenters can know about them. Heisenberg had talked with Einstein about this, and, when Dirac was in Copenhagen developing transformation theory, he had also discussed it with him.?° 

The nub of what became known as Heisenberg’s uncertainty principle is that the knowledge experimenters have of a quantum’s position limits what they can know about its speed, at the same instant. The more they know about a quantum’s position, the less they can know about its speed. So, for example, if experimenters know an electron’s location with perfect precision, then it follows that they can know nothing whatsoever about its speed at the same moment; on the other hand, if they know the exact value of the electron’s speed, they will be totally ignorant of its position. There is, Heisenberg argued, no way round this: regardless of the accuracy of the measuring apparatus or the extent of the experimenters’ ingenuity, the principle puts fundamental limitations on knowledge. It turns out that even the most accurate knowledge imaginable of the location of an ordinary object puts only negligible constraints on knowledge of its speed (likewise with the location and speed reversed), so the principle is unimportant in everyday life. This is the root of the physicists’ joke about the motorist who tries to con the traffic police by pleading not guilty of speeding on the grounds ‘I knew exactly where I was, so I had no idea how fast I was travelling’: the plea would be perfectly admissible if it were made by a sentient electron. 

In his paper, Heisenberg explained his principle by picturing what happens when an experimenter uses a photon of light to probe the behaviour of an electron demonstrating that the very act of probing disturbs the electron.

The metaphor of nature as a colossal clockwork mechanism, popular since Newton's day, had long been apt for most purposes. But no longer. Quantum mechanics was based fundamentally on mathematical abstractions and could not be visualised using concrete images ~ that is why Dirac refused to discuss quantum mechanics in everyday terms, except in later life, when he began to use analogies between the behaviour of quanta and the way ordinary matter behaves. Yet Dirac often remarked that he did not think about nature in terms of algebra, but by using visual images. Since he was a boy, he had been encouraged to develop visual imagination in his art and technicaldrawing classes, which were an ideal grounding for his studies of projective geometry. None of the other pioneers of quantum mechanics had been given an education in which geometric visualisation played such a prominent part. Five decades later, when he looked back on his early work in quantum mechanics, Dirac declared that he had used the ideas of projective geometry, unfamiliar to most of his physicist colleagues: 

{Projective geometry] was most useful for research, but I did not mention it in my published work [. . .} because I felt that most physicists were not familiar with it. When I had obtained a particular result, I translated it into an analytic form and put down the argument in terms of equations.