The Origin of Mass

The discovery of E = mc2 marked a turning point in the way physicists viewed energy, for it taught us to appreciate that there is a vast latent energy store locked away inside mass itself. It is a store of energy much greater than anyone had previously dared imagine: The energy locked away in the mass of a single proton is approaching 1 billion times what is liberated in a typical chemical reaction. At first sight it seems we have the solution to the world's energy problems, and to a degree that may well be the case in the long term. But there is a fly in the ointment, and a big one too: It is very hard to destroy mass completely. In the case of a nuclear fission power plant, only a very tiny fraction of the original fuel is actually destroyed; the rest is converted into lighter elements, some of which may be highly toxic waste products. Even within the sun, fusion processes are remarkably ineffective at converting mass into energy, and this is not only because the fraction of mass that is destroyed is very small: For any particular proton, the chances of fusion ever taking place are exceedingly remote because the initial step of converting a proton into a neutron is an incredibly rare occurrence—so rare, in fact, that it takes around 5 billion years on average before a proton in the core of the sun fuses with another proton to make a deuteron, thereby triggering the release of energy. Actually, the process would never even occur if it weren't for the fact that the quantum theory reigns supreme at such small distances: In the pre-quantum worldview, the sun is simply not hot enough to push the protons close enough together for fusion to take place—it would have to be around 1,000 times hotter than its current core temperature of 10 million degrees. When the British physicist Sir Arthur Eddington first proposed that fusion might be the power source of the sun in 1920, he was quickly made aware of this potential problem with his theory. Eddington was quite sure that hydrogen fusion into helium was the power source, however, and that an answer to the conundrum of the low temperature would soon be found. "The helium which we handle must have been put together at some time and some place," he said. "We do not argue with the critic who urges that the stars are not hot enough for this process; we tell him to go and find a hotter place."

So ponderous is the conversion of protons into neutrons that, "kilogram for kilogram," the sun is several thousand times less efficient than the human body at converting mass to energy. One kilogram of the sun generates only 1/5,000 of a watt of power on average, whereas the human body typically generates somewhat more than 1 watt per kilogram. The sun is of course very big, which more than makes up for its relative inefficiency.

As we have been so keen to emphasize in this book, nature works according to laws. So it will not do to get too excited about an equation that tells us, as E = mc2 does, about what might possibly happen. There is a world of a difference between our imagination and what actually happens, and although E = mc2 excites us with its possibilities, we must still understand just how it is that the laws of physics allow mass to be destroyed and energy released. Certainly the equation itself does not logically imply that we have a right to convert mass to energy at will.

One of the wonderful developments in physics over the past hundred years or so has been the realization that we appear to need only a handful of laws to explain pretty much all of physics—at least in principle. Newton seemed to have achieved that goal when he wrote down his laws of motion way back in the late seventeenth century, and for the next two hundred years there was little scientific evidence to the contrary. On that matter, Newton was rather more modest. He once said, "I was like a boy playing on the sea-shore, and diverting myself now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me," which beautifully captures the modest wonder that time spent doing physics can generate. Faced with the beauty of nature, it seems hardly necessary, not to mention foolhardy, to lay claim to having found the ultimate theory. Notwithstanding this appropriate philosophical modesty about the scientific enterprise, the post-Newton worldview held that everything might be made up of little parts that dutifully obeyed the laws of physics as articulated by Newton. There were admittedly some apparently minor unanswered questions: How do things actually stick together? What are the tiny little parts actually made of? But few people doubted that Newton's theory sat at the heart of everything—the rest was presumed to be a matter of filling in the details. As the nineteenth century progressed, however, there came to be observed new phenomena whose description defied Newton and eventually opened the doors to Einstein's relativity and the quantum theory. Newton was duly overturned or, more accurately, shown to be an approximation to a more accurate view of nature, and one hundred years later we sit here again, perhaps ignoring the lessons of the past and claiming that we (almost) have a theory of all natural phenomena. We may well be wrong again, and that would be no bad thing. It is worth remembering not only that scientific hubris has often been shown to be folly in the past, but also that the perception that we somehow know enough, or even all there is to know, about the workings of nature has been and will probably always be damaging to the human spirit. In a public lecture in 1810, Humphry Davy put it beautifully: "Nothing is so fatal to the progress of the human mind as to suppose our views of science are ultimate; that there are no new mysteries in nature; that our triumphs are complete; and that there are no new worlds to conquer."

Perhaps the whole of physics as we know it represents only the tip of the iceberg, or maybe we really are closing in on a "theory of everything." Whichever is the case, one thing is certain: We currently have a theory that is demonstrably proven, after a vast and painstaking effort by thousands of scientists around the world, to work across a very broad range of phenomena. It is an astonishing theory, for it unifies so much, yet its central equation can be written on the back of an envelope.

Formula 7.1

We'll call this central equation the master equation, and it lies at the heart of what is now known as the Standard Model of Particle Physics. Although it is unlikely to mean much to most readers at first sight, we can't resist showing it above.

Of course, only professional physicists are going to know what's going on in detail in the equation, but we did not show it for them. First, we wanted to show one of the most wonderful equations in physics—in a moment we will spend quite some time explaining why it is so wonderful. But also it really is possible to get a flavor of what is going on just by talking about the symbols without knowing any mathematics at all. Let us warm up by first describing the scope of the master equation: What is its job? What does it do? Its job is to specify the rules according to which every particle in the entire universe interacts with every other particle. The sole exception is that it does not account for gravity, and that is much to everyone's chagrin. Gravity notwithstanding, its scope is still admirably ambitious. Figuring out the master equation is without doubt one of the great achievements in the history of physics.

Let's be clear what we mean when two particles interact. We mean that something happens to the motion of the particles as a result of their interaction with each other. For example, two particles could scatter off each other, changing direction as they do so. Or perhaps they might spin into orbit around each other, each trapping the other into what physicists call a "bound state." An atom is an example of such a thing, and in the case of hydrogen, a single electron and a single proton are bound together according to the rules laid down in the master equation. We heard a lot about binding energy earlier in the previous chapter, and the rules for how to calculate the binding energy of an atom, molecule, or atomic nucleus are contained in the master equation. In a sense, knowing the rules of the game means we are describing the way the universe operates at a very fundamental level. So what are the particles out of which everything is made, and just how do they interact with each other?

The Standard Model takes as its starting point the existence of matter. More precisely, it assumes the existence of six types of "quark," three types of "charged lepton," of which the electron is one, and three types of "neutrino." You can see the matter particles as they appear in the master equation: They are denoted by the symbol } (pronounced "psi"). For every particle there should also exist a corresponding antiparticle. Antimatter is not the stuff of science fiction; it is a necessary ingredient of the universe. It was British theoretical physicist Paul Dirac who first realized the need for antimatter in the late 1920s when he predicted the existence of a partner to the electron called the positron, which should have exactly the same mass but opposite electrical charge. We have met positrons before as the byproducts of the process whereby two protons fuse to make the deuteron. One of the wonderfully convincing features of a successful scientific theory is its ability to predict something that has never before been seen. The subsequent observation of that "something" in an experiment provides compelling evidence that we have understood something real about the workings of the universe. Taking the point a little further, the more predictions a theory can make, then the more impressed we should be if future experiments vindicate the theory. Conversely, if experiments do not find the thing that is predicted, then the theory cannot be right and it needs to be ditched. There is no room for debate in this kind of intellectual pursuit: Experiment is the final arbiter. Dirac's moment of glory came just a few years later when Carl Anderson made the first direct observations of positrons using cosmic rays. For their efforts, Dirac shared the 1933 Nobel Prize and Anderson the 1936 prize. Esoteric though the positron might appear to be, its existence is today used routinely in hospitals all over the world. PET scanners (short for "positron emission tomography") exploit positrons to allow doctors to construct three-dimensional maps of the body. It is not likely that Dirac had medical imaging applications in mind when he was wrestling with the idea of antimatter. Once again it seems that understanding the inner workings of the universe turns out to be useful.

There is one other particle that is presumed to exist, but it would be to rush things to mention it just yet. It is represented by the Greek symbol Z (pronounced "phi") and it is lurking on the third and fourth lines of the master equation. Apart from this "other particle," all of the quarks, charged leptons, and neutrinos (and their antimatter partners) have been seen in experiments. Not with human eyes, of course, but most recently with particle detectors, akin to high-resolution cameras that can take a snapshot of the elementary particles as they fleetingly come into existence. Very often, spotting one of them has won a Nobel Prize. The last to be discovered was the tau neutrino in the year 2000. This ghostly cousin of the electron neutrinos that stream out of the sun as a result of the fusion process completed the twelve known particles of matter.

The lightest of the quarks are called "up" and "down," and protons and neutrons are built out of them. Protons are made mainly of two up quarks and one down, while neutrons are made from two downs and one up. Everyday matter is made of atoms, and atoms consist of a nuclear core, made from protons and neutrons, surrounded at a relatively large distance by some electrons. As a result, up and down quarks, along with the electrons, are the predominant particles in everyday matter. By the way, the names of the particles have absolutely no technical significance at all. The word "quark" was taken from Finnegans Wake, a novel by Irish novelist James Joyce, by American physicist Murray Gell-Mann. Gell-Mann needed three quarks to explain the then known particles, and a little passage from Joyce seemed appropriate:

Three quarks for Muster Mark!

Sure he has not got much of a bark

And sure any he has it's all beside the mark.

Gell-Mann has since written that he originally intended the word to be pronounced "qwork," and in fact had the sound in his mind before he came across the Finnegans Wake quotation.

Since "quark" in this rhyme is clearly intended to rhyme with "Mark" and "bark," this proved somewhat problematic. GellMann therefore decided to argue that the word may mean "quart," as in a measure of drink, rather than the more usual "cry of a gull," thereby allowing him to keep his original pronunciation. Perhaps we will never really know how to pronounce it. The discovery of three more quarks, culminating in the top quark in 1995, has served to render the etymology even more inappropriate, and perhaps should serve as a lesson for future physicists who wish to seek obscure literary references to name their discoveries.

Despite his naming tribulations, Gell-Mann was proved correct in his hypothesis that protons and neutrons are built of smaller objects, when the quarks were finally glimpsed at a particle accelerator in Stanford, California, in 1968, four years after the original theoretical prediction. Both Gell-Mann and the experimenters who uncovered the evidence were subsequently awarded the Nobel Prize for their efforts.

Apart from the matter particles that we have just been talking about, and the mysterious Z, there are some other particles we need to mention. They are the W and Z particles, the photon and the gluon. We should say an introductory word or two about their role in affairs. These are the particles that are responsible for the interactions between all the other particles. If they did not exist, then nothing in the universe would ever interact with anything else. Such a universe would therefore be an astonishingly dull place. We say that their job is to carry the force of interaction between the matter particles. The photon is the particle responsible for carrying the force between electrically charged particles like the electrons and quarks. In a very real sense it underpins all of the physics uncovered by Faraday and Maxwell and, as a bonus, it makes up visible light, radio waves, infrared and microwaves, X-rays, and gamma rays. It is perfectly correct to imagine a stream of photons being emitted by a lightbulb, bouncing off the page of this book and streaming into your eyes, which are nothing more than sophisticated photon detectors. A physicist would say that the photon mediates the electromagnetic force. The gluon is not as pervasive in everyday life as the ubiquitous photon, but its role is no less important. At the core of every atom lies the atomic nucleus. The nucleus is a ball of positive electric charge (recall that the protons are all electrically charged, while the neutrons are not) and, in a manner analogous to what happens when you try to push two like poles of a magnet together, the protons all repel each other as a result of the electromagnetic force. They simply do not want to stick together and would much rather fly apart. Fortunately, this does not happen, and atoms exist. The gluon mediates the force that "glues" together the protons inside the nucleus, hence the silly name. The gluon is also responsible for holding the quarks together inside the protons and neutrons. This force has to be strong enough to overcome the electromagnetic force of repulsion between the protons, and for that reason it is called the strong force. We are really not covering ourselves in glory in the naming-stakes.

The W and Z particles can be bundled together for our purposes. Without them the stars would not shine. The W particle in particular is responsible for the interaction that turns a proton into a neutron during the formation of the deuteron in the core of our sun. Turning protons into neutrons (and vice versa) is not the only thing the weak force does. It is responsible for hundreds of different interactions among the elementary particles of nature, many of which have been studied in such experiments as those carried out at CERN. Apart from the fact that the sun shines, the W and Z are rather like the gluon in that they are not so apparent in everyday life. The neutrinos only ever interact via the W and Z particles and because of that they are very elusive indeed. As we saw in the last chapter, many billions of them are streaming through your head every second, and you don't feel a thing because the force carried by the W and Z particles is extremely weak. You've probably already guessed that we've named it the weak force.

So far we have done little more than trot off a list of which particles "live" in the master equation. The twelve matter particles must be added into the theory a priori, and we don't really know why there are twelve of them. We do have evidence from observations of the way that Z particles decay into neutrinos made at CERN in the 1990s that there are no more than twelve, but since it seems necessary to have only four (the up and down quarks, the electron, and the electron neutrino) to build a universe, the existence of the other eight is a bit of a mystery. We suspect that they played an important role in the very early universe, but exactly how they have been or are involved in our existence today is something to be added to the big unanswered questions in physics. Humphry Davy can rest easy for the moment.

As far as the Standard Model goes, the twelve are all elementary particles, by which we mean that the particles cannot be split up into smaller parts; they are the ultimate building blocks. That does seem to go against the grain of common sense—it seems perfectly natural to suppose that a little particle could, in principle, be chopped in half. But quantum theory doesn't work like that—once again our common sense is not a good guide to fundamental physics. As far as the Standard Model goes, the particles have no substructure. They are said to be "pointlike" and that is the end of the matter. In due course, it might well turn out that an experiment reveals that quarks can be split into smaller parts, but the point is that it does not have to be like that; pointlike particles could be the end of the story and questions of substructure might be meaningless. In short, we have a whole bunch of particles that make up our world and the master equation is the key to understanding how they all interact with each other.

One subtlety we haven't mentioned is that although we keep speaking of particles, it really is something of a misnomer. These are not particles in the usual sense of the word. They don't go around bouncing off each other like miniature billiard balls. Instead they interact with each other much more like the way surface waves can interact to produce shadows on the bottom of a swimming pool. It is as if the particles have a wavelike character while remaining particles nonetheless. This is again a very counterintuitive picture and it arises out of the quantum theory. It is the precise nature of those wavelike interactions that is rigorously (i.e., mathematically) specified by the master equation. But how did we know what to write down when we wrote the master equation? According to what principles does it arise? Before tackling these obviously very important questions, let's look a little more deeply at the master equation and try to gain some insight into what it actually means.

The first line represents the kinetic energy carried by the W and Z particles, the photon and the gluon, and it tells us how they interact with each other. We didn't mention that possibility yet but it is there: Gluons can interact with other gluons and W and Z particles can interact with each other; the W can also interact with the photon. Missing from the list is the possibility that photons can interact with photons, because they do not interact with each other. It is fortunate that they don't, because if they did it would be very difficult to see things. In a sense it is a remarkable fact that you can read this book. The remarkable thing is that the light coming from the page does not get bounced off-track on the way to your eyes by all the light that cuts across it from all the other things around you, things you could see if you turned your head. The photons literally slip past, oblivious to each other.

The second line of the master equation is where much of the action is. It tells us how every matter particle in the universe interacts with every other one. It contains the interactions that are mediated by the photons, the W and Z particles, and the gluons. The second line also contains the kinetic energies of all the matter particles. We'll leave the third and fourth lines for the time being.

As we have stressed, buried within the master equation are, bar gravity, all the fundamental laws of physics we know of. The law of electrostatic repulsion, as quantified by Charles Augustin de Coulomb in the late eighteenth century is in there (lurking in the first two lines), as is the entirety of electricity and magnetism, for that matter. All of Faraday's understanding and Maxwell's beautiful equations just appear when we "ask" the master equation how the particles with electric charge interact with each other. And of course, the whole structure rests firmly on Einstein's special theory of relativity. In fact, the part of the Standard Model that explains how light and matter interact is called quantum electrodynamics. The "quantum" reminds us that Maxwell's equations had to be modified by the quantum theory. The modifications are usually very tiny and lead to subtle effects that were first explored in the middle of the twentieth century by Richard Feynman and others. As we have seen, the master equation also contains the physics of the strong and weak forces. The properties of these three forces of nature are specified in all of their details, which means that the rules of the game are laid out with mathematical precision and without ambiguity or redundancy. So, apart from gravity, we seem to have something approaching a grand unified theory. It is certainly the case that no one has ever found any evidence anywhere in any experiment or through any observation of the cosmos that there is a fifth force at work in the universe. Most everyday phenomena can be explained pretty thoroughly using the laws of electromagnetism and gravity. The weak force keeps the sun burning but otherwise is not much experienced on Earth in everyday life, and the strong force keeps atomic nuclei intact but extends barely outside of the nucleus, so its immense strength does not reach out into our macroscopic world. The illusion that such solid things as tables and chairs are actually solid is provided by the electromagnetic force. In reality, matter is mainly empty space. Imagine zooming in on an atom so that the nucleus is the size of a pea. The electrons might be grains of sand whizzing around at high speeds a kilometer or so away— the rest is emptiness. The "grain of sand" analogy is stretching the point a little, for we should remember that they act rather more like waves than grains of sand, but the point here is to emphasize the relative size of the atom compared to the size of the nucleus at its core. Solidity arises when we try to push the cloud of electrons whizzing around the nucleus through the cloud of a neighboring atom. Since the electrons are electrically charged, the clouds repel and prevent the atoms from passing through each other, even though they are largely empty space. A big clue to the emptiness of matter comes when we look through a glass window. Although it feels solid, light has no trouble passing through, allowing us to see the outside world. In a sense, the real surprise is why a block of wood is opaque rather than transparent!

It is certainly impressive that we can shoehorn so much physics into one equation. It speaks volumes for Wigner's "unreasonable effectiveness of mathematics." Why should the natural world not be far more complex? Why do we have a right to condense so much physics into one equation like that? Why should we not need to catalog everything in huge databases and encyclopedias? Nobody really knows why nature allows itself to be summarized in this way, and it is certainly true that this apparent underlying elegance and simplicity is one of the reasons why many physicists do what they do. While reminding ourselves that nature may not continue to submit itself to this wonderful simplification, we can at least for the moment marvel at the underlying beauty we have discovered.

Having said all that, we are still not done. We haven't yet mentioned the crowning glory of the Standard Model. Not only does it include within it the electromagnetic, strong, and weak interactions, but it also unifies two of them. Electromagnetic phenomena and weak interaction phenomena at first sight appear to have nothing to do with each other. Electromagnetism is the archetypal real-world phenomenon for which we all have an intuitive feel, and the weak force remains buried in a murky sub-nuclear world. Yet remarkably the Standard Model tells us that they are in fact different manifestations of the same thing. Look again at the second line of the master equation. Without knowing any mathematics, you can "see" the interactions between matter particles. The portions of the second line involving W, B, and G (for gluon) are sandwiched between two matter particles, }, and that means that here are the bits of the master equation that tell us how matter particles "couple" with the force mediators but with a punch line. The photon lives partly in the symbol " W" and partly in "B," and that is where the Z lives too! The W particle lives entirely in "W." It is as if the mathematics regards the fundamental objects as W and B, but they mix up to conjure the photon and the Z. The result is that the electromagnetic force (mediated by the photon) and the weak force (mediated by the W and Z particles) are intertwined. In experiments, it means that properties that can be measured in experiments on electromagnetic phenomena should be related to properties measured in experiments on weak phenomena. That is a very impressive prediction of the Standard Model. And it was a prediction: The architects of the Standard Model, Sheldon Glashow, Steven Weinberg, and Abdus Salam, shared a

Nobel Prize for their efforts, for their theory was able to predict the masses of the W and Z particles well before they were discovered at CERN in the 1980s. The whole thing hangs together beautifully. But how did Glashow, Weinberg, and Salam know what to write down? How did they come to realize that " W and B mix up to produce the photon and the Z"? To answer that question is to catch a glimpse of the beautiful heart of modern particle physics. They did not simply guess, they had a big clue: Nature is symmetrical.

Symmetry is evident all around us. Catch a snowflake in your hand and look closely at this most beautiful of nature's sculptures. Its patterns repeat in a mathematically regular way, as if reflected in a mirror. More mundane, a ball looks unchanged as you turn it around, and a square can be flipped along its diagonal or along an axis that slices through its center without changing its appearance. In physics, symmetry manifests in much the same way. If we do something to an equation but the equation doesn't change, then the thing we did is said to be a symmetry of the equation. That's a little abstract, but remember that equations are the way physicists express how real things relate to one another. A simple but important symmetry possessed by all of the important equations in physics expresses the fact that if we pick up an experiment and put it on a moving train, then, provided the train isn't accelerating, the experiment will return the same results. This idea is familiar to us: It is Galileo's principle of relativity that lies at the heart of Einstein's theory. In the language of symmetry, the equations describing our experiment do not depend on whether the experiment is sitting on the station platform or onboard the train, so the act of moving the experiment is a symmetry of the equations. We have seen that this simple fact ultimately led Einstein to discover his theory of relativity. That is often the case: Simple symmetries can lead to profound consequences.

We're ready to talk about the symmetry that Glashow, Weinberg, and Salam exploited when they discovered the Standard Model of particle physics. The symmetry has a fancy name: gauge symmetry. So what is a gauge? Before we attempt to explain what it is, let's just say what it does for us. Let's imagine we are Glashow or Weinberg or Salam, scratching our heads as we look for a theory of how things interact with other things. We'll start by deciding we are going to build a theory of tiny, indivisible particles. Experiment has told us which particles exist, so we'd better have a theory that includes them all; otherwise, it will be only a half-baked theory. Of course, we could scratch our heads even more and try to figure out why those particular particles should be the ones that make up everything in the universe, or why they should be indivisible, but that would be a distraction. In fact, they are two very good questions to which we still do not have the answers. One of the qualities of a good scientist is to select which questions to ask in order to proceed, and which questions should be put aside for another day. So let's take the ingredients for granted and see if we can figure out how the particles interact with each other. If they did not interact with each other, then the world would be very boring—every-thing would pass through everything else, nothing would clump together, and we would never get nuclei, atoms, animals, or stars. But physics is so often about taking small steps, and it is not so hard to write down a theory of particles when they do not in teract with each other—we just get the second line of the master equation with the W, B, and G bits scratched out. That's it— a quantum theory of everything but without any interactions. We have taken our first small step. Now here comes the magic. We shall demand that the world, and therefore our equation, have gauge symmetry. The consequence is astonishing: The remainder of the second line and the whole of the first line appear "for free." In other words, we are mandated to modify the "no interactions" version of the theory if we are to satisfy the demands of gauge symmetry. Suddenly we have gone from the most boring theory in the world to one in which the photon, W, Z, and gluon exist and, moreover, they are responsible for mediating all of the interactions between the particles. In other words, we have arrived at a theory that has the power to describe the structure of atoms, the shining of the stars, and ultimately the assembly of complex objects like human beings, all through the application of the concept of symmetry. We have arrived at the first two lines of our theory of nearly everything. All that remains is to explain what this miraculous symmetry actually is, and then those last two lines.

The symmetry of a snowflake is geometrical and you can see it with your eyes. The symmetry behind Galileo's principle of relativity isn't something you can see with your eyes, but it isn't too hard to comprehend even if it is abstract. Gauge symmetry is rather like Galileo's principle in that it is abstract, although with a little imagination it is not too hard to grasp. To help tie together the descriptions we offer and the mathematical underpinnings, we have been dipping into the master equation. Let's do it again. We said that the matter particles are represented by the Greek symbol } in the master equation. It's time now to delve just a little deeper. } is called a field. It could be the electron field, or an up-quark field, or indeed any of the matter particle fields in the Standard Model. Wherever it is biggest, that's where the particle is most likely to be. We'll focus on electrons for now, but the story runs just the same for all the other particles, from quarks to neutrinos. If the field is zero someplace, then the particle will not be found there. You might even want to imagine a real field, one with grass on it. Or perhaps a rolling landscape would be better, with hills and valleys. Where the hills are, the field is biggest, and in the valleys it is smallest. We are encouraging you to conjure up, in your mind's eye, an imaginary electron field. It might be surprising that our master equation is so noncommittal. It doesn't work with certainties and we cannot even track the electron around. All we can do is say that it is more likely to be found over here (where the mountain is) and less likely to be found over there (at base camp in the valley). We can put definite numbers on the chances of finding the electron to be here or there, but that is as good as it gets. This vagueness in our description of the world at the very smallest distance scales occurs because quantum theory reigns supreme there, and quantum theory deals only in the odds of things happening. There really does appear to be a fundamental uncertainty built into concepts such as position and momentum at tiny distances. Incidentally, Einstein really did not like the fact that the world should operate according to the laws of probability and it led him to utter his famous remark that "God does not play dice." Nevertheless, he had to accept that the quantum theory is extremely successful.

It explains all the experiments we have conducted in the subatomic world, and without it we would have no idea how the microchips inside a modern computer work. Maybe in the future someone will figure out an even better theory, but for now quantum theory constitutes our best effort. As we have been at pains to point out throughout this book, there is absolutely no reason why nature should work according to our common-sense rules when we venture to explain phenomena outside of our everyday experience. We evolved to be big-world mechanics, not quantum mechanics.

Returning to the task at hand, since quantum theory defines the rules of the game, we are obliged to talk of electron fields. But having specified our field and laid out the landscape, we are not quite done. The mathematics of quantum fields has a surprise lurking. There is some redundancy. For every point on the landscape, be it hill or valley, the mathematics says that we must specify not only the value of the field at a particular point (say, the height above sea level in our real-field analogy), corresponding to the probability that a particle will be found there, but we need also to specify something called the "phase" of the field. The simplest picture of a phase is to imagine a clock face or a dial (or a gauge) with only one clock hand. If the hand points to 12 o'clock, then that is one possible phase, or if it points to half-past, then that would be a different phase. We have to imagine placing a tiny clock face at each and every point on our landscape, with each one telling us the phase of the field at that point. Of course, these are not real clocks (and they certainly do not measure time). The existence of the phase is something that was familiar to quantum physicists well before Glashow, Weinberg, and Salam came along. More than that, everyone knew that although the relative phase between different points of the field matters, the actual value does not. For example, you could wind all of the tiny clocks forward by ten minutes and nothing would change. The key is that you must wind every clock by the same amount. If you forget to wind one of them, then you will be describing a different electron field. So there appears to be some redundancy in the mathematical description of the world.

Back in 1954, several years before Glashow, Weinberg, and Salam constructed the Standard Model, two physicists sharing an office at the Brookhaven Laboratory, Chen Ning Yang and Robert Mills, pondered the possible significance associated with the redundancy in setting the phase. Physics often proceeds when people play around with ideas without any good reason, and Yang and Mills did just that. They wondered what would happen if nature actually did not care about the phase at all. In other words, they played around with the mathematical equations while messing up all the phases, and tried to work out what the consequences might be. This might sound weird, but if you sit a couple of physicists in an office and allow them some freedom, this is the sort of thing they get up to. Returning to the landscape analogy, you might imagine walking over the field, haphazardly changing the little dials by different amounts. What happens is at first sight simple—you are not allowed to do it. It is not a symmetry of nature.

To be more specific, let's go back and look at only the second line of the master equation. Now strike out all of the W, B, and G bits. What we have is then the simplest possible theory of particles that we could imagine: The particles just sit around and never interact with each other. That little portion of the master equation very definitely does not stay the same if we suddenly go and redial all the little clocks (that isn't something that you are supposed to be able to see by just looking at the equation). Yang and Mills knew this, but they were more persistent. They asked a great question: How can we change the equation so that it does stay the same? The answer is fantastic. We need to add back precisely the missing bits of the master equation that we just struck out, and nothing else will do. In so doing we conjure into existence the force mediators and suddenly we go from a world without any interactions to a theory that has the potential to describe our real world. The fact that the master equation does not care about the values on the clock faces (or gauges) is what we mean by gauge symmetry. The remarkable thing is that demanding gauge symmetry leaves us no choice in what to write down: Gauge symmetry leads inexorably to the master equation. To put it another way, the forces that make our world interesting exist as a consequence of the fact that gauge symmetry is a symmetry of nature. As a postscript, we should add that Yang and Mills set the ball rolling, but their work was primarily of mathematical interest and it came well before particle physicists even knew which particles the fundamental theory ought to describe. It was Glashow, Weinberg, and Salam who had the wit to take their ideas and apply them to a description of the real world.

So we have seen how the first two lines of the master equation that underpins the Standard Model of particle physics can be written, and we hope to have given some flavor as to its scope and content. Moreover, we have seen that it is not ad hoc; instead

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