by Marcus Chown
The cosmic ray experiment had a soul-destroyingly low success rate. The arrival of particles from space was unpredictable, so it was never possible to know when one was flying through the cloud chamber. Anderson’s only option was to activate the piston at random moments, illuminate its interior with a flash of light, take a photograph and hope for the best. Not surprisingly, the majority of the photographs were blank. In fact, after a year of operation, during which Anderson took some 1,300 photographs, only fifteen turned out to contain anything of interest, a success rate of barely 1 per cent. Thomas Edison, who famously called genius ‘1 per cent inspiration and 99 per cent perspiration’, had hit the nail on the head.
In a handful of the photographs, however, it was possible to see the track of a lightweight, low-momentum particle, curving tightly in the enormous 15,000-gauss magnetic field. It could only be an electron, created in the cloud chamber by the collision of a cosmic ray with an atomic nucleus. But Anderson noticed something curious: in addition to showing an electron corkscrewing one way in the magnetic field, pretty much every photograph showed the track of a particle corkscrewing in the opposite direction. That could mean only that it was a positively charged particle, since a magnetic field bends positively charged particles the opposite way to negatively charged ones (and electrons always carry a negative charge). The thickness of the tracks revealed that the charge carried by the particles was of exactly the same magnitude as that of an electron.
The idea that a positively charged electron existed was too ridiculous to contemplate. Something must have gone awry with Anderson’s experiment.
Millikan, though a meticulous experimenter, was prone to eccentricity and poor judgement. He had the half-baked idea that cosmic rays were gamma rays generated in deep space by the birth of atoms.7 If he was right, the high-energy photon of a gamma ray should simply give a single electron an almighty kick. However, this was not at all compatible with Anderson’s photographs, which showed the creation of roughly equal numbers of negative and positive particles, often speeding away from a common point.
The only positively charged particle known at the time was the proton, but that was about two thousand times more massive than an electron and the tight curvature of Anderson’s tracks indicated that the mystery particle was considerably lighter than that.
On inspecting the tracks, Millikan asked his assistant whether a gamma ray could have interacted with the cloud chamber’s glass base, sending an electron speeding back upwards? His reasoning was that the magnetic field would bend the path of an electron travelling upwards in the same way as it would a positively charged particle travelling downwards.
There was an easy way to find out. Anderson inserted a horizontal lead sheet across the middle of his cloud chamber. Inevitably, a particle ploughing through it would lose speed, causing it to spend more time in the magnetic field so that its track would become more curved. The part of the track that was more tightly curved showed the particle at a later time, and therefore indicated its direction of travel.
As Anderson modified his experiment, the 1932 Olympics were in full swing in nearby Los Angeles. Due to the Great Depression, only half as many athletes could afford to attend as had competed in the 1928 Olympics in Amsterdam. As part of the games’ money-saving effort, Pasadena’s Rose Bowl Stadium had been converted into a velodrome; it was not far from Caltech and, if Anderson had a spare moment, he intended to watch some of the cycling. Despite the Olympic excitement, however, on the Caltech campus, in the shadow of the San Gabriel Mountains, all was quiet. The students were away and many of the faculty had gone on vacation to escape the sweltering July heat.
Once again, Anderson knuckled down and took lots of photographs, most of which turned out to be useless. But on 2 August 1932, he obtained a heart-stopping image. He was staring at it now as he penned the introduction to his paper.
Crossing the middle of the photograph was a thick black horizontal line, the shadow of the lead plate. Above the line, the track of the particle – no thicker than a human hair – was more tightly curved than below it, confirming that the particle was indeed travelling upwards not downwards – a rare event that historians of science would debate much in the future. But it was not the fact that the particle was travelling upwards that made the picture heart-stopping; it was the nature of the track itself. It was curving the wrong way.
Millikan dismissed the track as a freak, and Anderson himself had nagging doubts. But taken at face value, the track could be interpreted only as the trajectory of a light particle like an electron, but carrying a positive rather than a negative electric charge. For a moment, Anderson hesitated over the page he was writing. Then, for the first time, he wrote down the word he had coined for the new particle. He called it a ‘positron’.
Anderson’s paper, which he intended to submit to the American journal Science, was entitled ‘The Apparent Existence of Easily Deflectable Positives’. ‘It seems necessary to call upon a positively charged particle having a mass comparable with an electron,’ he wrote. It was a controversial statement. But what could he do? He had no choice but to accept what his experiment was telling him. There on his photograph, as clear as day, was the unmistakable signature of a positively charged electron.
In 1932, only three fundamental constituents of matter were known: the electron, the proton and the neutron, whose discovery had been announced in February that year by James Chadwick at the University of Cambridge.8 Between them, the three particles provided the basic building blocks of an atom, in which electrons circled a tight ball of protons and neutrons like planets around the Sun. It was a neat and appealing picture of the structure of matter, and the last thing anyone wanted was for another particle to mess things up. Nobody needed a positive electron. Nature had no place for it. Or did it?
Cambridge, Late November 1927
When he first wrote down the equation that described the electron, Paul Dirac was stunned and awed by its beauty,9 but he was also terrified. He felt like a tightrope walker who had achieved a miraculous balancing feat but might be sent plunging to his death by the slightest puff of air. His equation was a piece of magic whose beauty was the mysterious hallmark of something that was right, but what if he was deluding himself? What if there was an ugly fact out there waiting to kill it stone dead?10 He had to take deep breaths to ward off a panic attack.
Tall, gangly and reminiscent of a stick insect, Dirac was the strangest of strange men. For six days each week he worked hard, and then on Sundays he cleared his head with a long walk in the countryside outside Cambridge, where he would climb trees while dressed in a suit and tie. Literal to the point of obtuseness, Dirac could have out-Spocked Mr Spock. When a student in one of his classes put up their hand and said, ‘Professor Dirac, I don’t understand the equation on the blackboard,’ he replied, ‘That’s a comment not a question,’ and looked into the middle distance.11 One of Dirac’s friends, the Russian physicist Peter Kapitza, tried to get Dirac interested in Russian literature and gave him a copy of Crime and Punishment. When he had finished it, Kapitza was eager to know what he had thought.
Dirac’s one and only comment was, ‘In one of the chapters the author has the sun rising twice on the same day.’ Dirac could spend hours in the company of others without feeling the slightest obligation to utter a single word, and if he did decide to speak, his conversation would often be limited to ‘yes’ or ‘no’. But although Dirac seemed baffled by the world of everyday social interactions – though not by the cartoon world of Mickey Mouse, which was one of his peculiar obsessions – he was not baffled by the abstract realm of fundamental physics. He was a high priest of quantum theory and Einstein’s theory of relativity.
By the autumn of 1927, the problem of how to unite quantum theory and relativity in a description of the electron had been occupying Dirac for many months. It had been occupying many physicists, in fact. It was, after all, the obvious problem.
Quantum theory was the description of the microscop
ic realm of atoms and their constituents. It was fantastically successful and predicted the results of many experiments extremely accurately. But in addition to its success, it provided a window onto a bizarre, counter-intuitive, Alice-in-Wonderland world that lurked just beneath the skin of reality. It was a place where a single atom could be in many places at once; where things happen for no reason whatsoever; and where two atoms could influence each other instantaneously, even if on opposite sides of the universe.
Much of this quantum weirdness arose from a single extraordinary observation: the fundamental building blocks of matter – electrons, protons and photons – could behave both as localised, bullet-like particles and as spread-out waves like ripples on a pond. They are like nothing in the familiar everyday world. The first hint of this microscopic madness had come in 1900, but it had not been until the mid-1920s that physicists formulated a fundamental theory from which it was possible to make precise predictions of the behaviour of the atomic world.
The high point of quantum physics was the Schrödinger equation, devised by Austrian physicist Erwin Schrödinger in 1925. It melded together the particle and wave behaviours, and described how ‘quantum waves’ spread through space, their ‘amplitude’ (strictly speaking, their ‘amplitude-squared’) at any location determining the ‘probability’ of finding a particle there.12 The Schrödinger equation, however, had a problem: it was not compatible with the other great development of early-twentieth-century physics, relativity.
Einstein’s special theory of relativity, published in 1905, recognised that the speed of light is the rock on which the universe is founded and that space and time are but shifting sands. In fact, at speeds approaching that of light, space and time blur into each other, revealing that they are aspects of the same thing: space–time. If it were possible for someone to fly past you at close to the speed of light, their time would slow so that they would appear to be moving through treacle and their space would shrink in the direction of their motion so that they appeared flattened like a pancake.†
These counter-intuitive effects are noticeable only at speeds approaching that of light, which at a million times faster than a passenger jet is way beyond anything we experience in the everyday world. The Schrödinger equation is therefore perfectly adequate for describing a hydrogen atom, in which an electron orbits the solitary proton in the nucleus at less than 1 per cent of the speed of light. However, the electric force which binds electrons to an atomic nucleus gets stronger the more protons there are in a nucleus. In the heaviest atoms, such as uranium, the force can whirl electrons around at speeds approaching that of light.‡ The Schrödinger equation is inadequate for describing such particles; an equation that was compatible with special relativity was needed, and this is what Dirac had been pursuing.
The challenge was to generalise Schrödinger’s equation for an electron – to find an overarching formula of which Schrödinger’s relation would turn out to be merely a special case when the speed was much less than that of light. There is no recipe for generalising an equation in physics – it involves intuition, guesswork and courageous leaps of faith. It is like being in an unknown land at midnight without a torch or map and trying to guess the topology of the landscape. There were clues, however. Since Einstein had shown that space and time were aspects of the same thing, Dirac knew that the equation he was seeking must treat space and time equally. It must also incorporate another key aspect of special relativity: the idea that mass is a form of energy.
A cornerstone of Einstein’s theory is that light is uncatchable: its velocity, for some unknown reason, plays the role of infinite speed in our universe. The only way light can be uncatchable is if a material body resists being pushed to the speed of light; such resistance, or ‘inertia’, is the very definition of mass. A body must therefore become more massive as it approaches the speed of light. Since the only thing that obviously increases as the body accelerates is its ‘energy of motion’, the unavoidable conclusion is that energy of motion has mass. In fact, as Einstein realised, all forms of energy, and not just energy of motion, have an equivalent mass.
But just as energy has mass, mass has energy. Einstein’s most remarkable discovery was arguably that energy is locked away inside matter, even when it is at rest. Mass-energy is the most compact and concentrated form of energy, with the amount available stupendously large and given by the most famous formula in science: E = mc2, where c is the speed of light.
The expectation might be that the total energy of a particle travelling at an appreciable fraction of the speed of light is equal to its rest energy plus its energy of motion. However, according to Einstein, it is more complicated than that: it turns out that the square of a particle’s total energy is equal to the square of its rest energy plus the square of its energy of motion. It is therefore necessary to take the square root of this expression to obtain the energy. Immediately, however, this creates a problem. Just as the square root of nine can be both minus three and three, the square root of the expression for the ‘relativistic’ energy can be negative. This was a nonsensical result that Dirac wanted to avoid at all cost, so he set out to find an equation that directly yielded the energy of a particle and not its energy squared.
The technical question was: How could he obtain an expression for the energy of an electron as a sum of a multiple of its rest energy and another multiple of its energy of motion? This task turned out to be impossible if the two multiples are numbers. For anyone else this would have been an impasse, but Dirac’s genius was to realise that it was possible to obtain an expression for the energy if each of the multiples, rather than being a simple number, was a ‘two-dimensional number’ – a table of values with two rows and two columns.
Mathematicians have special rules for adding and multiplying together such ‘matrices’. A key property is that multiplying matrix A by matrix B does not necessarily give the same result as multiplying matrix B by matrix A, which is not an uncommon property of ‘operations’ in the everyday world. Take a die. If it is rotated ninety degrees clockwise about a vertical axis and then ninety degrees top to bottom about a horizontal axis, its final orientation is not the same as if it is turned ninety degrees top to bottom, then ninety degrees clockwise.13 Since a die keeps track of what happens when it is rotated and the matrices Dirac required to describe a relativistic electron did the same, it provided a hint that an electron can in some sense rotate; that is, it has ‘spin’.
Such a property had been revealed in experiments and had completely baffled theorists. Electrons flying through a magnetic field were deflected in two distinct ways, as if they were miniature magnets that could point either in the direction of the field and be deflected one way or in the opposite direction and be deflected the other way.14 Magnetic fields are generated by electric currents, which are simply electric charges in motion. And the only way the charge on an elementary particle like an electron could be moving is if the electron is spinning.
However, calculations showed that, for an electron to generate the strength of magnetism revealed in experiments, it would have to be spinning faster than the speed of light, which, according to Einstein, was impossible. Physicists were forced to accept that an electron behaves as if it is spinning, even though it is not. Its intrinsic ‘quantum spin’ is a property with no analogue in the everyday world, but nevertheless it has real effects. If a large number of electrons ran into you, they would impart their intrinsic spin to you and you would find yourself spinning like a pirouetting ice skater.
The fact that the matrices Dirac used to describe the electron used two columns of paired numbers implied a ‘two-ness’ to the spin, which was what had been observed. Although spin appeared nowhere in the Schrödinger equation, it emerged quite naturally from the mathematics of Dirac’s matrices.
Dirac worked in a study at St John’s College in Cambridge that had no pictures on the wall nor any ornaments or other frivolity; were it not for an ancient settee against one wall, it would have bee
n indistinguishable from an empty classroom. He worked best in the early mornings, seated at a simple folding desk, with his head down, scribbling on scraps of paper with a pencil and occasionally pausing to rub out an error or to check something in one of his handful of reference books. The silence of his study was interrupted only by the creaking of his door as his man-servant, or ‘gyp’, crept in to add coal to his fire or to bring him tea and biscuits.
By late November, Dirac had tried and discarded many mathematical formulations; it was then that he conjured a description of the electron which simultaneously respected the constraints of both theories, squaring what had seemed an impossible circle.15 He could hardly believe that he had finally found what he had been looking for. What convinced him was that the formula he had concocted appeared to have something of the divine about it. It was economical, elegant and beautiful. He, a human being, had invented it, but it could have been a thought from the Creator that had wafted down from heaven and landed on his page.
Dirac’s equation described not only a particle with the mass of an electron but one with exactly the same spin and magnetic field as had been found in experiments. The definitive test, however, would be to apply it to nature’s simplest atom, hydrogen. The ‘energy levels’ of its single electron had been pinned down by experiments to a high degree of precision, though Dirac’s fear of falling from a great height was so great that he could not bring himself to make predictions with his equation. Instead, he carried out only an approximate calculation; to his relief, his predictions chimed with reality, but he dared go no further.
