by Harry Cliff
After months of guessing, testing, and discarding possible equations, Dirac finally came upon a promising-looking candidate. Not only did it agree with both relativity and quantum mechanics, it also naturally accounted for a hitherto mysterious property of the electron, the fact that it behaves as if it is spinning.*5 Solving his equation, Dirac found two solutions, one describing the electron with its spin pointing up, the other with its spin pointing down. The spin of the electron emerged almost miraculously from Dirac’s unification of quantum mechanics and special relativity. If the electron’s spin hadn’t already been discovered in experiments, his equation would have predicted it.
Dirac was elated. Not only had he pulled off one of the greatest feats in the history of theoretical physics, he had discovered an equation of near-incomparable beauty. The concept of mathematical beauty is a bit hard to define, although many mathematicians will recognize it when they see it, a bit like you might recognize beauty in the smooth, clean lines of a sailing ship. Dirac’s equation had a piercing simplicity, simultaneously resolving several intractable problems while employing an absolute minimum of extraneous bells and whistles, like a razor-sharp blade cutting through a dense tangle of undergrowth. The equation has what some theorists would describe as a feeling of inevitability, the sense that it is so simple, so elegant, and yet so powerful that it could not have possibly been otherwise. I am now going to commit a cardinal sin of popular science writing and show you the equation I am babbling on about:
(iγμ∂μ–m)ψ = 0
Isn’t that a gorgeous thing? Even if the sight of algebra makes you feel dizzy, I hope I can impress upon you just how lean and mean this equation is.*6 It has only three bits—the first term, iγμ∂μ, describes how the electron changes through space and time, the m is its mass, and finally ψ is the electron’s wave function (the mathematical object that tells you the probability of finding the electron in a particular place or state)—and yet despite this simplicity it describes every electron that ever was or ever will be.
Dirac kept his monumental discovery quiet for more than a month, suffering occasional bouts of intense panic at the thought that his beautiful equation might unravel when it was forced to confront experimental reality. Fearing the result, he kept putting off checking whether it could accurately predict the energy levels of the hydrogen atom, a test he knew the equation had to pass. However, when he did eventually bring himself to do the math, he found that it not only got the answer right, it actually matched the experimental data even more closely than ordinary quantum mechanics.
When Dirac finally allowed his equation out into the wild at the start of 1928, it set the physics world on fire. His rivals in continental Europe’s theoretical powerhouses reacted with a mixture of wonder and dismay. Pascual Jordan, who had been working on the same problem, was left totally demoralized, while Heisenberg spoke of an English physicist who was so clever there was no point trying to compete with him.
However, at the back of Dirac’s mind was a gnawing anxiety; he suspected that there was something deeply wrong with his equation. He had discovered that it had not two but four different solutions. The first two were all fine and dandy, describing the established spin up and spin down states of the electron, but the other two seemed to describe something profoundly disturbing—electrons with negative energy (not to be confused with negative charge).
The idea of negative energy electrons makes about as much sense as a pond with a negative number of ducks. At first, Dirac was tempted to sweep those solutions under the carpet, but he soon realized that they couldn’t be ignored so easily. If these negative energy states existed, then ordinary positive energy electrons should be able to fall into them, like a pool ball tumbling into a pocket.
The problem was, no one had ever seen an electron falling into a negative energy state. Determined to save his beautiful equation, Dirac proposed a rather brazen solution: The reason we never see electrons falling into negative energy states is that the negative energy states are already full. An electron trying to jump from a positive to a negative energy state finds its way blocked by an existing electron, like a ball kept out of a pocket by a stack of previously potted balls.
In principle, this means that the entire universe is filled by an infinite sea of negative energy electrons. This raises an obvious question: Why don’t we notice them? Surely it’d be pretty obvious if we spent our lives wading through an infinite number of electrons? Not necessarily, said Dirac. As long as these negative energy electrons were perfectly evenly distributed throughout space, then they would fade into the background.
This electron-sea solution didn’t end Dirac’s woes. What if, for example, a photon were to crash into one of these negative energy electrons and kick it up into a positive energy state? Suddenly, we’d see an electron appear out of nowhere as it emerged from below the waterline. Meanwhile, an electron-shaped hole would be left in the sea, spoiling the perfect uniformity that kept it hidden. However, rather than think of this as a hole in an infinite sea of negatively charged, negative energy electrons, Dirac realized that the hole would behave like a positive energy electron—but one with a positive charge.
Therein lay the rub; according to all experiments to date, there was no such thing as a positively charged electron. Every electron that had ever been seen had been negatively charged. At first, Dirac tried to show that these positively charged holes might in fact be protons, but you’d expect the hole to have the same mass as the electron, and a proton weighs almost two thousand times as much. Worse still, if protons really were holes in the negative energy sea, then electrons should be able to fall into them, annihilating both electron and proton and leading to the instantaneous destruction of every atom in the universe.
Despite these problems and the gloom of many of his colleagues, Dirac’s conviction of the beauty and rightness of his equation was unshakeable. By 1931, after all attempts to get rid of the negative energy states had failed, he was ready to make his most audacious prediction: positively charged electrons really must exist in nature.
What happened next still gives me goose bumps. A year later, thousands of miles away in California, a positively charged electron turned up in a cloud chamber photograph taken by the young experimental American physicist Carl Anderson, who had been studying cosmic rays raining down from the heavens. Hot on the heels of Anderson’s paper, the Cavendish physicists Patrick Blackett and Giuseppe Occhialini spotted more positive electrons, this time popping into existence accompanied by ordinary negatively charged electrons when a cosmic ray smashed into an atom in their cloud chamber. Dirac, a frequent visitor to the Cavendish Laboratory, which in those days was still ruled by the booming Ernest Rutherford, was soon poring over the photographs with Blackett, making calculations and checking the results against his equation. It didn’t take long to realize that these positive electrons were the very same particles that Dirac had predicted must exist.
Dirac had achieved something truly miraculous. Using the power of pure thought, he had conjured the existence of a form of matter that had never been seen before in nature. By bringing together quantum mechanics and special relativity and following his nose, he had opened a window into the world of antimatter, a mirror image of the ordinary stuff that makes up the visible universe. We now know that every matter particle has an antiversion, with precisely the same properties but opposite charge. Dirac’s positive electron is now known as the positron or the antielectron. Meanwhile, the proton has a negatively charged version, the antiproton, and there are also antineutrons, antimuons, antiquarks, and antineutrinos. The fact that Dirac managed to predict such a fantastical thing just by thinking really, really hard has surely got to rank as one of the most incredible feats in the history of science.
What’s more, the discovery of antimatter destroyed the idea that matter is eternal. Matter particles could now be created by smacking one particle into another
with enough energy to make a new particle-antiparticle pair. And the reverse is possible too—if a particle is unlucky enough to meet up with its antiparticle, they annihilate each other, disappearing into oblivion with a flash of radiation.
Of course, this does raise a question: If matter and antimatter are always created and destroyed together, how come the universe is made only of ordinary matter? As we’ll see, this rather troublesome conundrum will come back to bite us.
There is one bit of Dirac’s work that hasn’t stood the test of time: the idea that antiparticles are holes in a negative energy sea. Within a few years, physicists found a way to do away with Dirac’s sea altogether by describing electrons and positrons in the same way as photons—as vibrations in quantum fields. The boundaries between fields and particles, light and matter, had finally dissolved.
Today, we physicists think of all particles this way. For every particle we’ve met along our journey so far there is a corresponding quantum field. Photons are little ripples in the electromagnetic field, electrons and positrons, likewise, are ripples in something called the “electron field.” Up quarks are little ripples in the up quark field, and so on and so on. When two protons smash into each other at the LHC, they set the quantum fields of nature ringing like bells, sending a cascade of ripples outward through our detectors, each a different musical note in a quantum mechanical symphony. We interpret these ripples as particles, but what we believe we are really seeing are transitory wobbles in quantum fields.
In fact, you might even say that there’s no such thing as a particle. As far as we can tell, the real building blocks of the universe are quantum fields: invisible, fluid-like substances that we can’t see or taste or touch, and yet are all around us, stretching from deep within the smallest atom of your being to the farthest reaches of the cosmos. Quantum fields—not chemical elements, nor atoms, nor electrons, nor quarks—are the real ingredients of matter. We are walking, talking, thinking bundles of tiny self-perpetuating disturbances sloshing about in intangible quantum fields.
Of course, things aren’t quite that simple. While it would be lovely and comforting if we could just think of an electron as a little ripple in the electron field, that is really only half the story. It turns out that even an object as simple as an electron is a fantastically complex thing, not merely a ripple in the electron field but a baroque mixture of every quantum field in nature. While this makes calculations in quantum field theory fiendishly difficult, it also opens up opportunities to explore nature in ways that would be totally impossible in either quantum mechanics or special relativity alone. In particular, experiments that study the electron in exquisite detail have the potential to teach us both about the electron itself and even quantum fields that we have never seen before. One such experiment is going on right now under the bustling streets of London.
DRESSING THE ELECTRON
Squeezed into a pokey basement lab at Imperial College in central London is an experiment that can pull off the same trick as the Large Hadron Collider for a thousandth of the price. Just meters beneath the thunder of London traffic and the ceaseless hammering of footsteps from the thousands of tourists and schoolchildren who throng to the museums of South Kensington, a small team of physicists are making one of the most delicate measurements of a fundamental particle ever attempted.
Their mission is to measure the shape of the electron. The idea that a fundamental particle can have a shape might seem strange, particularly given that we just said that particles are shape-shifting ripples in quantum fields, but park that thought for just a moment. The really surprising thing is that by measuring the electron’s shape with fantastic precision, the team at Imperial can search for hints of quantum fields that we’ve never seen before, potentially uncovering evidence for particles with masses so huge that even the LHC wouldn’t be powerful enough to produce them.
How on earth can measuring the shape of a puny little electron tell us anything about particles with gigantic masses? Well, it all comes down to the fact that particles are really just ripples in quantum fields, a fact that has dramatic consequences for the properties of an electron. To properly understand what the physicists at Imperial are up to we need to have a serious think about what an electron really is, and perversely, perhaps, the best way to get started is to consider what quantum field theory tells us about empty space, or what physicists refer to as “the vacuum.”
Imagine taking a little region of space and sucking out all the atoms, all the particles, every last stray photon and neutrino. What’s left? If there are no particles then presumably the answer is nothing at all. Actually, quantum field theory tells us that this little region of “empty” space is still an amazingly crowded place; it’s chock-full of quantum fields. There might not be any particles left, but the fields that they are ripples in are always there. In the standard model of particle physics there are dozens of fields (the exact number depends on how you choose to count them, but for the sake of argument let’s say there are twenty-five) including the electron field, the neutrino fields, the quark fields, the electromagnetic field, the gluon fields, and more besides. All these fields exist everywhere, even in the vacuum. Empty space is far from empty.
Now let’s say we dump enough energy into the electron field to create a little quantized ripple—a single electron. Since the electron has an electric charge, it has a direct effect on all the quantum fields hanging around in the vacuum. The most obvious thing that happens is that the charge of the electron warps the shape of the electromagnetic field in the area around the electron. Close to the electron the electromagnetic field becomes strong, while farther away from the electron it becomes weaker, eventually fading away to (almost) zero. In principle, this distortion in the electromagnetic field contains some energy, so when we think of an electron we should really consider both the ripple in the electron field plus the distortion it creates in the electromagnetic field.
But it doesn’t stop there. Since the electromagnetic field is what communicates the electromagnetic force, it is “connected” to every other quantum field that has an electric charge. This means that the distortion the electron creates in the electromagnetic field causes yet more distortions in a whole bunch of other fields, including the electrically charged quark fields. Now quarks have this property we call “color,” which means they interact with the gluon fields (the fields of the strong force), and so the distortions in the quark fields cause further distortions in the gluon fields. There is even a back reaction where the distortion in the electromagnetic field causes a further distortion in the original electron field. And so it goes on and on. The upshot of all this is that an electron is not simply a ripple in the electron field; it is a ripple in the electron field plus distortions in every quantum field we have ever discovered. What we might call the bare electron—the pure ripple in the electron field—is dressed up in an elaborate gown woven from every quantum field in nature.
The way that quantum fields dress up the electron (and every other particle for that matter) makes calculating even simple processes in quantum field theory fiendishly complicated, but on the other hand it gives us a fantastic opportunity to search for the influence of quantum fields we’ve never seen before. As we’ll see in the coming chapters, there are lots of good reasons to believe that there are more quantum fields than the twenty-five-odd we’ve discovered so far. A good example is dark matter, a mysterious substance that astronomers and cosmologists have shown must be about five times more common than the ordinary atomic stuff that you and I, and every star and planet in the sky, are made from. It’s usually assumed that dark matter is some kind of particle, in which case the vacuum should also contain an extra quantum field, the quantum field that dark matter particles are ripples in.
The brute force approach to search for dark matter particles at the LHC is to smack two protons into each other really, really hard and hope that the collision has enough energy
to set off a vibration in the dark matter field. If we’re lucky, and the amount of energy needed to set the dark matter field wobbling—in other words, the mass of the dark matter particle—is within reach of the LHC, then we should be able to detect evidence of dark matter particles zipping out from the collisions. However, if the mass of the dark matter particle is higher than the maximum energy of the LHC, we won’t be able to get a vibration going in the dark matter field, and dark matter will remain a mystery.
There is, however, another way. It relies on how quantum fields dress up fundamental particles. If a quantum field for dark matter exists and it interacts with at least one of the other quantum fields in the standard model, then in principle it should also contribute to the elaborate quantum field gown worn by the electron. If you think of this gown as a fabric woven from threads of different quantum fields, then a few of the threads in the electron’s gown would be made of the dark matter quantum field. Since what we measure in experiments is the bare electron, plus its outfit, then if we make superbly precise measurements of the electron, we might be able to detect the subtle effects of new quantum fields woven into its quantum mechanical glad rags.
This is exactly what the team at Imperial are trying to do. I first read about their work back in 2011 shortly after the LHC had fired up for its inaugural run. At first glance, what they were claiming seemed impossible; in a small lab in central London with a budget of millions, not billions, they were ruling out the existence of the same quantum fields that thousands of physicists at the world’s largest experiment were in the middle of searching for. Since then I had always wanted an excuse to pay a visit and get a glimpse of the miraculous machine that surely lurked beneath the streets of South Ken.
