The Hitchhiker’s Guide to Quantum Field Theory
The mathematical language of particle physics
Apr 13, 2016
25 Min read time
Photo: Ineke Huizing
Editors' Note: This is the second piece in a series on the search for new physics at CERN.
This series explores an anomaly CERN scientists announced last December at the Large Hadron Collider (LHC), where protons are smashed together very close to the speed of light. My first installment explained how two detectors observed results at odds with predictions of the Standard Model. In the jargon of the field, they found a “diphoton excess at 750 GeV.” (My first piece explains what that means.)
This might be a very big deal. The Standard Model, which has withstood all experimental challenges for forty years, is our best theory of the fundamental particles that make up the matter and forces we know about. If the anomaly holds up, we will have come face to face with the Standard Model’s limitations.
But that’s a big “if.” The results are too preliminary for us to say anything for sure right now. Fortunately, CERN restarted the LHC experiments this month and is expected to make another announcement this summer. The new data may show that the anomaly was just statistical noise, but whatever happens, there is much to be learned from these efforts to probe the edges of our understanding. We may learn something about Nature, or we may learn that the existing theory has survived yet another test. In either case, by following how science gets done you can see why it is so exciting—the process as well as the results.
In the lead up to this summer’s announcement, I will take you through our present understanding of particle physics: the Standard Model, the Higgs boson, and why we suspect there is something beyond the Standard Model for the LHC to find. To do that, I need to give you a way to picture how the Universe works at these incredibly small scales. This second installment lays the foundation by exploring the basic language of particle physics. That language is called quantum field theory, but it is not so much a specific theory as the framework for all our fundamental theories of Nature, both the well tested (quantum electrodynamics and quantum chromodynamics, which are parts of the Standard Model) and the more speculative (supersymmetry and quantum gravity).
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The quantum world is strange. On small scales, particles can act like waves, objects can tunnel straight through solid walls, and the more you know about where something is, the less you know how fast it is going. Thanks to these bizarre phenomena, the word “quantum” is often synonymous with “magic” in films, books, and popular culture. (This attitude sadly abused by many charlatans who wish to wrap themselves in the trappings of science.) It is not magic, of course, but it is genuinely hard to understand. It was the great Danish physicist and Nobel Laureate Niels Bohr who said, “Anyone who is not shocked by quantum theory has not understood it.” There are indeed fundamental questions about the behavior of physics at the smallest scales that we do not have satisfying answers to—yet.
Quantum field theory is the basic language of the most accurate physical theory yet devised.
However, our understanding of the quantum world has grown enormously since Bohr’s time. Indeed, there are important differences between the quantum mechanics developed in the early twentieth century and the quantum field theory I will talk about here. We still use the former in many situations, not only in physics but also engineering: your computer relies on the “magic” of quantum mechanics for its existence. But the latter is a deeper, more fundamental theory, what you get when you require quantum mechanics to abide by the rules of Einstein’s theory of special relativity—the description of what happens when things are moving near the speed of light. Special relativity is the theory that gives us the most famous equation in history, E = mc2, which tells us that mass can be converted to energy and vice versa. As a result, any theory that includes special relativity must be able to deal with the creation or destruction of particles. Quantum mechanics could not accommodate this possibility; quantum field theory can.
When you use the rules of quantum field theory to make calculations, it is truly incredible. It has enabled us to formulate the most precise scientific theory ever devised, capable of making predictions to one part in a trillion. I will try to explain some of those rules, to give a picture of how I think about things and their interactions at these incredibly small scales (and very high speeds). For example, how should you picture what’s going on when a physicist says “at the LHC, we collided two protons traveling close to the speed of light and created a Higgs boson; it lasted for roughly 10−22 seconds before decaying into some photons”?
I will evade the philosophical questions that lurk at the heart of the quantum world: questions about the reality of the wavefunction, hidden variables, or multiverses. In my day-to-day working life, I take the attitude that is sometimes summed up as “shut up and calculate.” The philosophical questions are interesting and important, but I do not need to answer them to carry out my research. From this vantage point, I will give you a sense of how I view the quantum world: how particles move and how they interact with each other. Of course this won’t give you a deep working knowledge of quantum field theory, but my goal is to convey some useful analogies for making sense of what goes on at the LHC.
Fields and Quantum Fields
The fundamental problem we face when we try to visualize quantum behavior is how to describe what things are. Consider electrons, photons, and quarks. Are they particles or waves? We think of particles as pointlike things that have a definite location—small billiard balls, say. Waves, by contrast, ripple out through space and exhibit behavior such as diffraction and interference.
But asking whether something is a particle or a wave is like asking whether Hobbes from the comic strip Calvin and Hobbes was an anthropomorphic tiger (as Calvin saw him) or a stuffed tiger toy (as everyone else saw him). Clearly, in the world of the comic strip, Hobbes was simply Hobbes: he had properties like anthropomorphic tigers and properties like stuffed animals, but he was his own thing, and fitting him into either category would miss some of Hobbes’s important properties, his intrinsic Hobbes-ness. Likewise, we should not ask whether an electron is a particle or a wave; it is its own thing. In particular, electrons are excitations of a particular quantum field—the electron field. In some situations an excitation can look like a particle, and in others it can act like a wave, but to force a quantum field into either of these categories would miss some of its intrinsic properties.
So what is a quantum field? There are a lot of quantum fields, including one for every fundamental particle: an electron field, a photon field, many quark fields, a Higgs field, and so on. The next couple of articles will explore the fields that we know exist around us—the ones that make up everything you see and touch. For now, I will talk in more generality: How should you picture these fields? Well, a quantum field is, first of all, a field. Not all fields are quantum fields, so let us first think about fields in general.
A field is a set of values assigned to every location in space and time. Consider some everyday, non-quantum fields. Right now you are sitting in a room with a temperature field. At each location in the room and at every moment in time, the value of the field is given by a single number, the temperature. You are also in a wind field. At each location and every moment in time, the value of the wind field is given by a series of numbers, which give both the speed of the wind and its direction. The simplest examples of quantum fields are like the temperature field: they have a single value at each point. More complicated quantum fields are like the wind field: they have a set of values at each point, which allow them to encode information such as directionality.
You might think that a field is just a clever trick to keep track of something more fundamental. After all, fields like the temperature field and the wind field are just mathematical devices for representing some more fundamental physical quality of the matter at every point in space and time. So is the world really fields all the way down? As far as we can tell right now, quantum fields can’t be explained away by something more fundamental. They’re what everything is made of: distributions of values everywhere in space and time. (If there is something more fundamental, we don’t know what it is, and all we know is that when you stand back a bit from that deeper theory, the quantum fields appear as the basic building blocks.) The fields are what they are, rather than mathematical stand-ins for something else. If you find that a bit unsatisfying, and want to know what these fields “really are,” I can only say, join the club. It is part of the nature of this kind of scientific inquiry that the answers we find sometimes just raise more questions.
One important feature of fields is that they can change over time. The temperature field changes when temperature at some location changes from one moment in time to the next. Similarly, a quantum field can vary. Normally, the field sits in an unperturbed state, like the surface of a lake on a calm day. For most quantum fields we know about, that resting value is “zero.” (The one exception we know about is the Higgs field, and the resulting weirdness of having a non-zero resting value everywhere will be the focus of another article.)
There is an important difference between water fields and quantum fields that I must mention. When you look for changes on the surface of a lake, you are picking out one particular frame of reference to work in—the frame in which you are stationary relative to the lake’s surface. Your description of the lake could change drastically if you picked a different reference frame. Quantum field theory is different: as I mentioned earlier, it is a relativistic theory, which means that it works no matter how you are moving through the Universe. Unfortunately, perhaps the only thing we humans are worse at visualizing than quantum behavior is relativity, so I will gloss over that aspect here. But I do want to point out that, in making this simplification, I’ve made quantum fields sound a little like the old (wrong) theory called the aether. You’ll have to trust me that the two are not equivalent.
Returning to the example of the water field: if you throw a stone into a calm lake, you’ll cause motion of the water. By pushing the water level down in one place, it will rebound upwards, starting an oscillation. That vertical motion of water will spread out horizontally across the lake, causing the water elsewhere to oscillate up and down. Put simply, the water “field” will develop a wave. The same is true for a quantum field: if you perturb it, you’ll cause the field in that location to start oscillating back and forth around its resting value. (Note that when I talk about a quantum field oscillating, I don’t mean that the field is moving spatially—up/down, left/right, forward/back, unlike the motion of the water on the lake’s surface. Instead, the field values at each point are changing. A wave in the field means a certain pattern in the values of the field at different spacetime locations. The field itself does not move through space: it is already everywhere.)
How do you perturb a quantum field? I will explain that in a bit. For now, just imagine that you’ve made the field oscillate at a certain point, like the water at the point where a stone is thrown in. That oscillation will cause the field to oscillate nearby, which will lead to more oscillations elsewhere, spreading out as a wave. These oscillatory states are called “excitations” of the field, and they have more energy than the resting state. It is these excitations that we might call particles: they are things that carry energy around.
Particles as Excitations of Quantum Fields
Perhaps you are puzzled. We started with a field, an assignment of values to every location in space and time. Then we caused some oscillations, which added energy to the field. The values of the field are changing, and the changes are spreading—but where are particles in all this?
This is where the quantum nature of these fields comes into play. Quantum waves—oscillations in a quantum field—are different from the waves on a lake or from oscillations in the wind field or the temperature field. The essential difference is that you can always imagine a wave on a lake, or the changes in temperature, or in wind speed or direction, as smaller and smaller. But the quantum excitations are quantized, which means that they come in discrete lumps. There is no way, for example, to generate a wave in the electron field that corresponds to half an electron, with half a unit of electric charge. You get the whole thing, or you get nothing.
Nevertheless, there are ways to generate a wave in the electron field that correspond to a particle having more energy or less energy or more or less momentum. In the non-quantum world, the energy and momentum of a particle must combine to what we call the “mass” of the particle. In other words, for a non-quantum object, if you tell me how much energy an object has and how fast it is moving, I can tell you what the object’s mass is. The weird thing is that, in general, the excitations of the quantum field don’t need to satisfy this relation. This would seem to imply that you can get an excitation of the electron field that doesn’t have the mass of an electron. But if you try to do that, something very interesting happens.
Think of a child on a swing. If you pull the swing back and let it go once, the child will start swinging back and forth with a specific period. If you want the child to swing higher, you have to push them. You instinctively know how to push them in order to make them swing higher: you push in time with the motion of the swing. If you don’t, you’ll be pushing this way while the child is swinging that way: you’ll damp the motion, and the swinging will quickly stop. We say the swing-set has a “natural frequency,” and if you want the oscillation to continue, you have to work with it. Physicists call this phenomenon “resonance.”
A quantum field has a property similar to the swing’s natural frequency. Call that property the “mass.” If you kick the field so that the excitation has an energy and momentum in just the right combination to add up to this “mass,” the wave you’ve set up in the field will continue on its merry way (This is why we can call this property “mass,” since it corresponds to the minimum energy you can give the field to set up a self-sustaining excitation, and from Einstein, we know that E = mc2). But if the relation between the energy and momentum doesn’t add up to the mass, the excitation will not last very long: like pushing out of time on the swing set, the wave will die away quickly.
This property of quantum fields explains a lot. Why, for example, does every electron in the universe have the same mass of 510,998,910 electron volts? Well, any experiment takes some time to perform, so the electrons we can measure are electrons that have stuck around long enough to be measured—they are the excitations that didn’t die away quickly. This means that these particular waves in the quantum field are set up just right so that the energy and momentum sum up correctly. Thus, when we measure their properties, they’ll have the “mass” that is set by the field’s intrinsic properties. These properties are the same everywhere, as the field permeates the entire Universe. So every electron we can find and measure is an excitation of the same field, and thus has the same mass. There can be electron field excitations that don’t have the right mass, but these don’t last long enough for us to weigh them. If you have access to a particle collider though, you might be able to set up some of these short-lived excitations and demonstrate that particles can indeed exist with the “wrong” mass, albeit briefly.
Quantum fields also help to explain Heisenberg’s uncertainty principle, one of the basic facts about quantum mechanics, which says that the more precisely you know a particle’s location at any one time, the less you are able to know about its momentum. Think of a quantum excitation whose location you know exactly, like an upward spike of water the moment after you throw a rock into a lake. But then think of what that spike of water will do, immediately after the rock sinks: it will spread out in all directions. Which way then is the “particle” moving? You don’t know—it is moving out in all directions. Alternatively, think of a wave whose direction of motion you know exactly. On the surface of the water, this would be an infinite series of waves, marching steadily in one direction, like ocean waves before they crash onto the beach. But an infinite series of waves is not localized in one place. So you can’t predict where the “particle” is. When you look for it, you might find it anywhere these waves exist.
Understanding quantum fields on their own is an achievement, but the real action is in understanding how particles and fields interact. You can picture particles as waves in their fields—electrons in the electron fields, photons in the photon field, and so on. But these fields must be able to talk to each other. After all, moving electrons around can generate excitations in the photon field—the basis of radio, light bulbs, and lasers. How do these conversations happen?
If you pluck a string on a guitar, it vibrates—you’ve created a wave on the string. But the strings are nailed to the wood, which can transfer the vibrations from one to the other strings. You can increase or decrease the mechanical connection between the two strings, making the transfer of energy easier or harder. In addition, the frequency of waves that can be sustained on each string are different, so a wave that will last a long time on one string may die out quickly on another. Plucking a string to produce an A will not work well on a string tuned to produce a B. (As with the swing-set, the “pushes” from the A string will be out of sync with the natural frequency of the B string.)
This is how you can picture the interaction of particles. Different strings on a guitar are like different types of fields—though fields all coexist at every point in space and time, which makes them harder to picture than the nicely separated guitar strings. Just as different strings are mechanically linked, different fields are sometimes linked: we say such fields are “coupled.” An excitation of one field can cause excitations in other fields coupled to it, just as a wave on one guitar string can cause nearby strings to vibrate. And just as the different tunings of guitar strings can prevent waves on one string from causing vibrations on another, the properties of each field (akin to the “mass” parameter I spoke about above) can prevent an excitation in one particular field from jumping to another. The fact that the fields are quantized means that an excitation in one field must be of just the right size to produce a sustained excitation in another field. The rules for how these fields speak to each other appear complicated, but deeper examination reveals a stunning simplicity, falling under an umbrella we call “symmetry”—an idea I will talk about in the next installment of this series.
For example, an electron moving around on the electron field deforms the coupled photon field. If the electron is not accelerating, the deformation in the photon field is what we call the “electric and magnetic field” of an electron. But if you accelerate the electron, you start to shake the photon field, and the deformation can turn into excitations we call photons. The result is what we call light.
We have talked about how fields can excite other fields, but it is also possible for fields to excite themselves. The photon field, however, is relatively weakly coupled to the quantum fields of the charged particles (such as electrons) and doesn’t couple to itself at all. As a consequence of this latter fact, as I sit here looking at the light coming from the window, the waves of the photons coming toward my eye from the sun are not affected by the photon waves coming down from the lights in the ceiling. This is how water waves act: if you watch the waves coming from the wake of a boat pass through the waves on a lake, where the waves overlap the water will be choppy and confused, but after the waves pass through each other, they continue on as if nothing had happened. Water waves, like light, don’t couple to each other.
Fields that are weakly coupled to each other and to themselves are a great boon for particle physicists. The mathematics of uncoupled (“free”) quantum fields is easy to work out, but the equations describing the behavior of coupled fields can’t be solved exactly, so we must approximate the answer. Fields that don’t couple strongly allow us to say our approximations are good—the error is small. For example, we can calculate how the electron field affects the photon field without worrying too much about how the photon field, in turn, affects the electron field—and then how the photon field then responds to that response, ad infinitum. If we carry out these calculations, we know we can safely ignore this feedback loop at some point, because at each iteration the ability of one wave to affect another is very small. It is in the analysis of these sorts of “weakly coupled” fields where quantum field theory truly shines, allowing us to make incredibly precise predictions about how particles move and interact.
Indeed, the most accurate physical theory yet devised is quantum electrodynamics, the quantum field theory account of electricity, magnetism, and light, and it owes its great success to weak coupling. Experimental physicists can measure one particular quantity, the magnetic moment of the electron, to twelve decimal places. Theoretical physicists can calculate the same quantity to the same precision using four iterations of the idea that the electron field talks to the photon field, which talks back to the electron field, and so on. The remarkable thing is that the theoretical value agrees with the experimental value. Theorists could go further, calculating this quantity to five iterations, or six, and so on, but since the two fields are weakly coupled, each iteration will be progressively less important to the final answer, and since the experimental value isn’t known beyond twelve decimal places, there’s little point to performing these more precise theoretical calculations.
However, other fields are tightly coupled to each other: unlike waves on a lake or photons, their waves do not pass through each other freely. Instead, when they interact, the waves throw each field into a chaotic mess; the motion of one field depends powerfully on the motion of the other field, and vice versa. If water waves were strongly coupled, then when two waves intersected, after moving “past” each other, the shape of each wave would have changed. Quarks, for example, are strongly coupled to a field called the gluon field. Unlike the weak coupling between photons and electrons (or between photons and quarks), any perturbation in the quark field causes the gluon field to move in lockstep; but as soon as the gluon field moves, the quark field in turn reacts to this motion, and so on, back and forth. At first glance, this strong coupling makes it hard to see that there are two fields present. After all, if we can’t move the quark field without moving the gluon field, how can we know that they are distinct?
This is why we speak of protons and neutrons even though they are really bags of quarks and gluons. When you wiggle the quark or the gluon field, you force all the other fields to move with it. This still causes excitations and waves, which still look like particles. But since we can’t see the distinct fields, we see the whole conglomeration moving as one, so we talk about the “proton field” or “neutron field.” (There are many more such fields, all just complicated combinations of quark and gluon fields.) Only by looking at very short distances and over very short times can we pry apart this motion of all the fields and manage to see only one quark or one gluon acting independently.
Creating the Higgs Boson
We have seen that the most fundamental things in the universe are quantum fields. They are quantized, and they oscillate, and they sometimes transfer their oscillations to other fields. Let us put this all together and try to visualize a Higgs boson being created at the LHC.
When I picture what is going on at the LHC when a Higgs boson is born, I picture the waves of two protons, moving toward each other from opposite directions. The protons are a bundle of oscillating fields—quark and gluon fields—all moving together to form the thing we see as a single particle. They are guided along the ring of the LHC by deformations of the photon field (that is, by magnetic fields generated by powerful magnets), which act like barriers, pushing these waves along a curved path. Eventually, these two proton waves are guided toward a crossing point and move past each other with incredible energies and at incredible speeds (99.9999991 percent of the speed of light). At this moment, occasionally one of the oscillating gluons in one of the bundles that make up a proton will combine with an oscillating gluon in the other proton to create a wave in the previously calm Higgs field. This is called “gluon fusion.” A bit of the energy from the protons leaks over, and a new excitation—a new particle—is born in the Higgs field. The Higgs field itself always exists, everywhere in spacetime, but an excitation of the field was only created when oscillations in coupled fields were transferred over. Since the gluon fields are only tenuously and indirectly connected to the Higgs field, this is a very rare event.
As the excitation we call the Higgs boson moves off, the remnants of the combined quark and gluon fields which made up the colliding protons find themselves momentarily out of balance; after losing energy to the Higgs field, the waves no longer have the right properties to move as one in the form we call protons. For example, their energy and momentum will no longer add up to the correct mass for a sustained excitation of the proton field. Eventually, after a jumble of confusing rearrangement of energy among all the quark and gluon fields, the fields sort themselves out, forming new stable waves corresponding to new particles. But that takes a bit of time, and by then the oscillations that once were protons will have moved far away from the newly created Higgs particle.
As this oscillation in the Higgs fields evolves in time, its vibration might leak over to one of the other fields that the Higgs field is coupled to: the photon field, or the quark fields, for example. It is impossible to predict exactly when this will happen, or into which field the transfer will take place, but eventually, the energy carried in the Higgs field will transfer over, starting up oscillations in one of those other fields. When this happens, we say that the Higgs boson decayed: the Higgs field is quiet again, and in its place are new excitations in other fields (a pair of photons, a quark and an antiquark, or something else). Those particles will then themselves travel off, scattering against other excitations of other fields, pulling and pushing other fields, a wave in their own quantum sea.
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The analogies I have used are only analogies. You can certainly bend them in the wrong direction, and the picture I have painted is only a poor reflection of the mathematics of quantum field theory. To make predictions and analyze experiments, we of course turn this picture into something much more precise—something we can use to do calculations. For example, you cannot say exactly where the energy in the Higgs field will end up, but you can say very precise things about the probability of its ending up in different fields; there is a 0.23 percent probability that the Higgs will decay into two photons.
But you don’t need to know the mathematics to appreciate what physicists are doing at the LHC. Armed with this picture of the quantum world, we will turn in the next article to all the fields we know about, as described by the Standard Model of particle physics. After introducing the zoo of particles around us, I will explain what is going on with the Higgs field and the Higgs boson, in particular—and why they reveal how deeply strange our Universe is.
April 13, 2016
25 Min read time