by Harry Cliff
Our apple pie ingredient list has shrunk. A lot. We started with a whole cupboard full of ingredients—oxygen, carbon, hydrogen, sodium, nitrogen, phosphorous, calcium, chlorine, iron, and more besides—but are now left with just three: electrons, up quarks, and down quarks. That’s cheating a little bit, because to bind these matter particles together to form atoms we also need the electromagnetic and strong forces. So add to that list photons and gluons. But still, that’s a wonderfully economical list of basic ingredients given you can use them to make literally anything, apple pies included.
Quarks and electrons are particles, a term that I’ve admittedly used rather too casually so far, assuming that you’ve probably been picturing a little spherical thing, maybe a bit like a marble. As we’ve delved deeper into the structure of matter, that sort of mental image has served us pretty well; in many ways particles really do behave like little hard balls that stick together to create nuclei and atoms. The objects that come flying out from the collisions at the Large Hadron Collider travel through our detectors like microscopic*1 bullets. When we create images of these collisions—usually only for publicity purposes these days; there are far too many to actually examine them all by eye—we draw out the path each particle took as if it really were a well-defined little nugget, as the word suggests.
This nuggetty picture of matter has a long pedigree; you can trace it back to John Dalton’s atomic theory, and if you’re trying to show off, all the way back to the ancient Greek philosophers Democritus and Leucippus, who were the first to argue that matter was made of indivisible, hard, particle-like things. However, the modern conception of a particle is a far cry from what Dalton or the ancient atomists imagined. The word “particle” has become like an iceberg; its everyday meaning is just the visible bit above the waterline, while below the surface lurks a vast mass of properties, concepts, and half-understood phenomena that have built up over decades of experimenting and theorizing. Even particle physicists are only dimly conscious of the full meaning of the word “particle” most of the time. I for one really do think of particles like little marbles when I’m doing my day job. It’s a mental picture that works just fine most of the time. But it’s wrong.
This simplistic view misses the true complexity, beauty, and downright weirdness of what modern particle physics tells us the world is ultimately made from. This deeper picture is only revealed when we start to think really hard about particles. In the process, we will discover that particles aren’t the fundamental building blocks of nature at all. Instead, a new set of objects emerges that are far stranger and less tangible than anything we experience in our everyday lives. These objects are still only partly understood, even by the world’s brainiest theorists, but seem to be the true ingredients of our universe.
CREATION AND ANNIHILATION
A few years ago, I worked on a small exhibition at the Science Museum in London, taking up a couple of big showcases in a corner of the bustling Exploring Space gallery. Little noticed among the various bits of physics ephemera by the throngs of screaming children who tear through the gallery like flocks of excitable, high-viz-clad geese was a loosely bound bundle of papers—Paul Dirac’s original PhD dissertation. The title, handwritten in charmingly uneven capital letters, is simply “Quantum Mechanics.” That’s one punchy title for a PhD.*2
Paul Dirac was one of the most brilliant theoretical physicists of the twentieth century, probably coming second only to Albert Einstein. To give you a sense of his prodigious ability, within just three months of reading the paper where the German theorist Werner Heisenberg first laid out the foundations of quantum mechanics, Dirac had produced a brand-new version of the theory, reframing and extending Heisenberg’s ideas in a more elegant mathematical language. All at the age of just twenty-three. It’s people like that who make you realize how little you’ve accomplished.
Dirac was also one of physics’s strangest figures, and if you’ve met many physicists, you’ll know there’s some stiff competition in that department. He was socially awkward, literal minded to a fault, and so uncommunicative that his colleagues defined the unit of a Dirac as one word spoken per hour. Like a visitor from another planet, he struggled to understand the many common human pastimes, particularly poetry—which he summarized as stating the obvious incomprehensibly—and worst of all, dancing. Among the many Dirac stories recounted by his colleagues was the time he asked Heisenberg why he enjoyed dancing while they were on a scientific jolly together in Japan. After Heisenberg replied that it was a pleasure to dance with nice girls, Dirac sat in thought for several minutes before responding, “Heisenberg, how do you know beforehand that the girls are nice?”
However, despite his inability to get his head around ordinary human behavior, Dirac was almost without rival when it came to understanding the behavior of the smallest ingredients of nature. During the first few years of his scientific career he would lay the foundations upon which all of modern particle physics is built. His first step was in making sense of what happens when a photon is born.
Photons are created and destroyed all the time. Every time you flick a light switch or idly tap your phone, you create billions and billions of photons, which then get destroyed almost immediately as they crash into your eyes, the walls of the room, or whatever else gets in their way. Similarly, when an electron orbiting an atom falls from a higher energy level to a lower one, a photon is created that carries off the difference in energy between the two levels. The question is, what is actually happening when a photon is created?
To answer this question, we need to go back to the prequantum view of light, a view that was based on the nineteenth-century concept of the electromagnetic field. We owe the idea of fields in large part to the English scientist Michael Faraday, who spent years getting hands-on with electromagnetic phenomena, experimenting with magnets, coils of wire, and dynamos in his basement lab at the Royal Institution in London. In the process, he became convinced that the electric and magnetic forces that he was playing with were communicated by invisible and yet undeniably physical entities—electric and magnetic fields.
Formally speaking, a field is a pretty abstract concept: a mathematical object with a numerical value at every point in space. However, fields are far more than just mathematical abstractions. If you’ve ever picked up two bar magnets and pushed their north poles toward each other, you’ll have felt a powerful force pushing back. Jostle the magnets around a bit and this force changes in strength and direction, as if you were feeling out the edges of some invisible, repulsive thing. You can look as hard as you like at the gap between the two magnets and you won’t see anything except empty space, and yet you can feel that there’s something there. What you’re feeling is a magnetic field, and once you’ve felt it, it’s impossible to deny that it’s real.
Faraday found that he could even make magnetic fields visible. Sprinkling iron filings onto a piece of waxed paper placed on top of a magnet, he produced beautiful images tracing out the field’s otherwise invisible influence. You can still see Faraday’s stunning field maps if you turn up at the Royal Institution on Albemarle Street in London and ask nicely. As the son of a poor blacksmith’s apprentice, Faraday hadn’t received much in the way of a formal mathematical education, so he instead developed powerful visual representations of his electric and magnetic fields based on lines of force, which he sketched flowing outward from the north pole of a magnet and back in through the south, or from a positive electric charge to a negative one. You were probably made to draw diagrams like that in school; I certainly was. He imagined these lines as real, physical objects that would move or even vibrate when magnets or electric charges moved, in the same way that you can send a wave rippling down a rope by suddenly flicking one of its ends.
It was the Scottish physicist James Clerk Maxwell who took Faraday’s intuitive understanding of electromagnetic phenomena and translated it into mathematical language. In the process he discovered an equa
tion describing a wave of intertwined electric and magnetic fields, dancing through space together. Astonishingly, when he calculated the wave’s speed, he found it was exactly the same as the speed of light. Maxwell’s theory seemed to show that light was a wave in a unified electromagnetic field.
By the time Dirac was working as a young scientist in the late 1920s, Maxwell’s electromagnetic theory of light had been tremendously successful, not least as the basis of wireless communication and radio broadcasting. However, Maxwell and Faraday’s electromagnetic field was a continuous object, and it was tricky to see how it could be reconciled with quantum theory, which described light as the flow of individual photons. The challenge was to get the two descriptions of light to play nicely with each other.
Dirac’s breakthrough came during a six-month stay at Niels Bohr’s Institute for Theoretical Physics in Copenhagen in the autumn of 1926, which followed hot on the heels of his triumphant PhD dissertation. While Bohr cultivated an open, relaxed atmosphere where lively discussion was encouraged, Dirac preferred to work alone, locking himself away in the library during the daytime and taking long solo walks around the city after dark. When he did attend the discussion sessions, he would sit listening in silence, responding when prompted with a monosyllabic yes or no. His colleagues, Bohr included, didn’t know what to make of the strange Englishman.
It was probably during one of his solitary days in the institute’s library that Dirac started to ponder the thorny issue of making photons. As a physicist working at the beating heart of the quantum revolution, you might have expected Dirac to take light quanta as his starting point and try to build the electromagnetic field from a multitude of tiny photons, in much the same way an ocean is made up of vast numbers of individual water molecules. But that isn’t what he did. Instead, Dirac took the electromagnetic field as the fundamental thing. It was photons that were made of the electromagnetic field, not the other way around. A photon, said Dirac, was nothing more than a discrete, transitory little ripple in the ever-present electromagnetic field.
Dirac had just invented a brand-new physical entity, a “quantum field”—a strange amalgam of Faraday’s electromagnetic field and Einstein’s photons. In many ways the quantum electromagnetic field looks a lot like Faraday’s ordinary nonquantum or classical version. Both are invisible and yet fill all of space, can transmit electric and magnetic forces, and, if you wobble them around in the right way, can sustain waves that travel through the field in the form of light. However, there is a crucial difference between Dirac’s quantum field and the old classical one. Whereas you can create a wave of any size you like in the classical electromagnetic field, in quantum field theory there is a fundamental minimum amount of waviness that you can have. This is what we call a “photon.”
To try to understand this a bit better, imagine two friends, let’s call them Alice and Bob,*3 standing a few meters apart, each holding one end of a tightly stretched length of elasticated bungee cord. In this analogy, the one-dimensional bungee cord is standing in for the admittedly three-dimensional electromagnetic field, but let’s not overcomplicate things. Now imagine that Alice starts to move her end of the cord up and down at a rate of, say, three wobbles per second, while Bob keeps his end still. As she moves her hand, waves start to ripple along the length of rope until they reach Bob at the other end. Now, as this is an ordinary, classical bungee cord, Alice can choose to move her hand up and down by any amount she likes; she can make little waves that are 5 centimeters high or wave her hand around wildly and make waves that are as high as she is, or any size in between. This is pretty similar to how light waves are created in a classical electromagnetic field; you just need to substitute Alice’s hand for a charged particle like an electron.
However, now let’s say we give Alice and Bob a quantum bungee cord (to be clear, there is no such thing, but go with it). Because the cord now obeys the laws of quantum field theory, Alice discovers something strange. She can no longer create waves of any height she likes. If she moves her hand up and down, still at a frequency of three wobbles per second, by say 5 centimeters, the bungee cord remains mysteriously still. Try as she might, she cannot make a wave 5 centimeters high, or indeed 6 centimeters, 7 centimeters, or 8 centimeters. However, when she moves her hand up and down by 10 centimeters suddenly a wave pings along the cord, perhaps startling Bob out of his daydream at the other end. There seems to be a fundamental minimum amplitude that a wave can have on this quantum bungee cord. In the electromagnetic field we’d call this smallest possible wave a photon, so I guess in this analogy a quantum of the bungee cord might be called a “bungeeon.”
The same is true for the quantum electromagnetic field. For a given frequency of light, you can only add energy to the electromagnetic field in discrete little lumps. The field can have no photons, one photon, two photons, or a quadrillion photons rippling about in it, but you can’t have a bit of a photon. They must come in whole numbers—or, to put it more scientifically, the electromagnetic field is “quantized.”
Dirac described the process of creating and annihilating photons in rather more abstract terms, inventing mathematical objects called “creation operators” and “annihilation operators.” As the names suggest, the creation operator injects one photon into the electromagnetic field, while the annihilation operator takes one out. Using this mathematical language, Dirac was able to calculate how likely an atom was to absorb or emit a photon in certain circumstances, finding an answer that agreed perfectly with a more ad-hoc calculation performed by Einstein ten years earlier.
Dirac’s quantum field theory was a triumph; not only had he gone one better than Einstein, he also believed he had laid to rest all the handwringing over wave-particle duality.*4 There was no longer any need to sometimes think of photons as waves and sometimes as particles; instead they could be understood as vibrations of a single unified object, the quantum electromagnetic field.
However, Dirac’s theory was only half the story. Perhaps you can think of photons as little ripples in an electromagnetic field, but what about the particles of matter? Electrons and protons seemed to be rather different beasts. Sure, they exhibit the same wave-particle duality as photons, but as far as anyone could tell, it was impossible to create or destroy them. Unlike photons, which blink in and out of existence willy-nilly, electrons and protons seemed to be eternal.
To understand the birth and deaths of matter particles we need to bring in the other great revolutionary theory of the early twentieth century, special relativity. Just as quantum mechanics had upended the laws governing atoms and particles, special relativity redefined what we mean by space and time, with some delightfully counterintuitive results. At its core is the principle proposed by Einstein that the laws of physics—and, crucially, the speed of light—are always the same regardless of how fast you are moving. It turns out that to make this work, you have to be willing to let go of a universal definition of space and time that everyone can agree on. Instead (for reasons too thorny to get into here) space and time become relative, with the distances that we measure between objects or the number of ticks of a clock between two events depending on how fast we are moving relative to one another.
The versions of quantum mechanics doing the rounds in the mid-1920s were inconsistent with special relativity. In other words, two observers moving at different speeds would disagree on what the laws of quantum mechanics were. It was clear that this meant quantum mechanics was, at best, incomplete, but fusing it with special relativity proved to be no easy task.
Over the summer of 1926, about six different physicists thought they had found an equation that might do the trick. It is known as the Klein-Gordon equation (after two of its discoverers, Oskar Klein and Walter Gordon) and it appears to describe the quantum behavior of an electron traveling close to the speed of light in a way that is consistent with the edicts of special relativity. In particular, it includes special relativi
ty’s most famous consequence—the equivalence of mass and energy as captured by E = mc2—by including the mass-energy of the electron as a term in the equation.
Niels Bohr, for one, thought the problem of getting a relativistic equation for the electron was solved. Dirac, though, was unconvinced. For one thing, the wave functions of the Klein-Gordon equation couldn’t easily be interpreted as the probability of finding a particle in a particular location as it could in ordinary quantum mechanics. Dirac was sure he could do better.
Back in Cambridge in the autumn of 1927 after a few months spent in the charming medieval German university town of Göttingen with the quantum power trio of Max Born, Werner Heisenberg, and Pascual Jordan, Dirac attacked the problem with quiet determination. No longer a lowly PhD student but a fellow of the grand St. John’s College, Dirac now had his own comfortable, albeit rather spartan, rooms in the college’s picturesque grounds on the banks of the River Cam. As usual, he worked alone, scribbling out pages of algebra at his little desk from early in the morning until dusk, only taking breaks on Sundays when he would go for long walks in the Cambridgeshire countryside, climbing the occasional tree, all dressed in a full three-piece suit.
Dirac knew that he was unlikely to be able to derive the relativistic equation for the electron from some profound universal principle in the way Einstein had derived relativity. Instead, as is often the case in physics, he would have to make a series of educated guesses. There were, however, some features he knew the equation needed to have that he could use as guides. First off, it had to be consistent with special relativity, which meant it needed to look the same regardless of how fast an observer was moving and also to include the mass-energy of the electron. Secondly, at speeds much lower than the speed of light, the equation should look like ordinary bog-standard quantum mechanics. Finally, to make sure the wave function of the electron could be straightforwardly interpreted in terms of probabilities, he was convinced that the equation needed to be “first order” in space and time—in other words, contain space and time just as they are, rather than squared (second order) as they appeared in the Klein-Gordon equation.
