**Motivation**

There’s a fair amount of bruhaha in the popular press what with the announcement of the discovery of what is probably the Higgs Boson at CERN last summer. In describing why the Higgs Boson is so important, you will read that the Higgs Boson “gives other particles mass”. This is the so-called Higgs Mechanism, and is an esoteric thing arising from the mathematics of Quantum Field Theory (QFT) that’s very difficult to understand at a popular level. You can find a number of analogies out there, such as the description of the famous actor walking through the crowd at a Hollywood Party, and acquiring inertia (i.e., mass, i.e., resistance to being accelerated) as a result. (You can find that description here at the Exploratorium’s site, along with a brief mention of the W and Z bosons, a description of whose properties is what led to the Standard Model of Particle Physics predicting that the Higgs Boson must exist.)

I have to admit, this analogy has always left me cold. Part of the reason is that my familiarity with QFT (remember, that acronym stands for Quantum Field Theory) is quite shaky, and I didn’t *really* know how the Higgs mechanism worked. This analogy didn’t give me, as a physicist, any insight into the actual Higgs mechanism. It’s still on my list to become much more conversant with QFT. After all, QFT and GR (General Relativity) are the two most fundamental theories of reality that we’ve got, and while *most* physicists don’t actually work with them on a daily basis (they’re either working with more complicated and practical things like solids or fluids, or, as with this year’s Nobel Prize Winners, they’re working on fundamental physics in a manner that is adequately, and more easily, described with the non-relativistic version of quantum Mmechanics).

In this post, I’m going to try to describe, at a popular level, the view of reality that QFT gives us, and then, using only one equation (and it’s one you’ve seen before, many times), try to tell you how it is that the existence of the Higgs Boson leads to other particles having mass. here are the sections below:

- Fields in Physics
- The Magnetic Field
- The Photon, and the Electromagnetic Field
- Fields and Particles in General, and an Analogy to a Lake
- Energy in QFT
- Massive Fields
- The Higgs Field and Mass Terms in QFT
- Pithy Summary

**Fields in Physics**

One of the problems in physics (or nearly any other technical area of study, including, unfortunately, law) is that there is a *lot* more new vocabulary than there are new words. That is, there are words with everyday definitions that are *different* from the technical definition in physics. One of the most common ones you hear is “theory”, which is used in everyday parlance to mean “speculation”, but is something very different in science. In physics, we talk about “fields”, and it often sounds very scary and esoteric. However, the basic definition of a field is actually fairly straightforward:

field, n.: something that has a value everywhere in space

That’s it. I’ll give you a few examples of fields below, including everyday things you can understand, physics things you’ve heard of before, and esoteric physics things that might surprise you. Here’s a physics example: a *temperature field*. Look around the room you’re in. Imagine putting spatial coordinates on that room. That is, in one corner, draw three axes. Label the two on the ground the *x* and *y* axes, and label the one sticking up the *z* axis. Now, at any point in your room, in principle you could measure the temperature, with a thermometer or some other device. If, say, you’ve got a hot plate (or, equivalently, a computer) running in your room, just above it the temperature will be higher than it is somewhere else in the room. That is, the temperature has a different value everywhere in the room. You can just talk about the temperature at one point, but if you talk about all of the values of the temperature everywhere in the room, you could call that the “Temperature Field”, and even give it mathematical notation, *T*(*x,y,z*), and then deal with it conceptually and mathematically as if it were one thing— albeit one thing that’s got different values at different places.

**The Magnetic Field**

One physics field you’ve almost certainly heard mentioned is the magnetic field. Consider, for example, a bar magnet. There’s a magnetic field around it, and you may have seen it visualized similar to the following:

*A magnetic field. Image by Wimkimedia Commons users lscha1, Mirek2.*

A magnetic field is a little different from a temperature field. A temperature field’s value is just a single number, with temperature units. A magnetic field has both a strength and a direction. The arrows in the picture above tell you the direction of the field; the strength is weaker the farther away you get from the magnet.

This magnetic field can interact with other things. You could, for example, hold the magnet near a nail on a table, and pull the magnet; if you’re careful, the nail will be dragged along behind the magnet. (If you’re not careful, the nail will jump and stick to the magnet, or you’ll get the magnet too far away and the field won’t be strong enough to overcome friction from the table.) There are two jargon terms I want to introduce here: *interaction* and *coupling*. The magnetic field is clearly *interacting* with the nail. How strong the pull of the field on the nail is depends on how strong the iron in the nail *couples* to the magnetic field.

Indeed, there’s another coupling going on here that I want to draw your attention to. There’s also iron (or some other sort of atom) inside the bar magnet, which is itself coupling to the magnetic field. Wait! Did you notice what I just did there? The word “the” in “coupling to the magnetic field” is very important. Previously, I’d been saying “a” magnetic field, as if the magnetic field from the Earth and the magnetic field from the bar magnet were two different things, so that each one is just *a* magnetic field. However, really, each point in space has a magnetic field strength, whatever it was that gave rise to it. So, really, there is just *one* magnetic field, which is everywhere in space. If you go to a very empty part of space, the value of the field might be zero everywhere nearby, but you can still describe the field there, just by saying what the value (even if zero) is everywhere in space. In this way of looking at it, the iron atoms in the bar magnet have properties that couple them to the magnetic field in such a way that the strength of the magnetic field around the bar magnet takes on certain specific values.

**The Photon, and the Electromagnetic Field**

Two new concepts here. The first new concept is the *electromagnetic* field. Just as there is a non-zero magnetic field around a bar magnet, there is a non-zero electric field around a charged object, such as an electron, or a Van de Graaff machine that you’ve charged up.

*An electromagnet. Image: Gina Clifford.*

Way back in the 19th century, physicists figured out that the electric and magnetic fields are in fact *two aspects of the same thing*, which today we call the electromagnetic field. If you’ve ever built an electromagnet, say, by wrapping a wire around a nail many times and connecting the wire to a battery, you’ve been playing with the unification of the electric and the magnetic fields (albeit mediated by the moving charges in the wire). The different voltage on the two terminals of the battery create an electric field, which pushes around the charges in the wire, and the current (which is the moving charges in the wire) gives rise to a magnetic field.

In fact, you don’t need charges there at all. If you have a varying electric field, it automatically gives rise to a magnetic field, and vice versa. The classical physics of this is described by Maxwell’s Equations. Indeed, Maxwell’s Equations show that you can get *waves* in the electromagnetic field, and that these waves will freely propagate through space. We call such a wave an *electromagnetic wave*, although you probably more often call it by its more common name, *light*. Light that you see, the “heat” that you feel radiating off of a burner (which isn’t really *heat* in the physics sense), the ultraviolet radiation that gives you sunburns, the microwaves produced by a cellphone or a microwave oven, and radio waves are all the same physical phenomenon: electromagnetic waves. Another couple of vocabulary words here: these electromagnetic waves represent a *disturbance*, or an *excitation* of the electromagnetic field. If there’s a light wave passing by, the electromagnetic field doesn’t just have a zero (or even a constant non-zero) value, but it’s got some wiggly business going on.

Classical physics allows a disturbance of *any* size in the electromagnetic field. In particular, the energy in an electromagnetic wave can be arbitrary small. Have an electromagnetic wave with a certain amount of energy? Divide all the strengths of the electric and magnetic fields, and the energy in that wave goes down by a factor of four, but everything still solves the equation. This is what happens in classical field theory.

Here’s the next new concept. *Quantum* field theory (QFT), on the other hand, only admits *quantized* excitations of the field. With the electromagnetic field, a disturbance of a given frequency (which we would see as a given color, if it were a frequency in the range that our eyes can detect) can’t have any old energy. Rather, it must come in quantized *steps* of energy. There’s a minimum energy that you can have in a light wave of a given color, and the total energy you have must be an integer multiple of that minimum energy. When you have an excitation of the electromagnetic field that has this minimum energy, we call that excitation a *photon*.

You’ve probably heard the photon described as “the particle of light”, and this is an accurate description. However, when we say “particle” in quantum field theory, what we really mean is a *disturbance of a field*. So, a better way to describe a photon is to say that it is a *disturbance of the electromagnetic field*, or an *excitation of the electromagnetic field*. The field’s natural state, or *vacuum state*, is no electric or magnetic field anywhere. If there is any light propagating, it must come the form of these quantized excitations of the field, that correspond to some multiple of the energy that one photon represents.

**Fields and Particles in General, and an Analogy to a Lake**

So, great. We’ve got the electromagnetic field. It *may* be a newish concept, although almost certainly you’ve heard the word “electromagnetic” before; but, at the very least, you’ve heard about magnetic fields. And, we’ve got the idea that light is a wave propagating through this field, and that we might describe that as a “disturbance” or “excitation” of the field. And, we have the concept of the photon, as the minimal allowed disturbance (at a given frequency) of the electromagnetic field. We also will sometimes refer to this minimal disturbance as a particle, and thus call the photon the particle of light.

Let me give you another example of a field, that has a disturbance in it: the surface of the water on a lake.

*The surface of a lake represents a height field. Image: Kenneth Allen.*

Here, the field is a *height field*. Rather than having a value everywhere in three-dimensional space, it have a value everywhere in two-dimensional space. That is, you could draw *x* and *y* axes on the surface of the lake to represent a coordinate system to figure out where you are on the lake. For every value of (*x*,*y*)— that is, everywhere on the lake— the water level has a height above its average height. I chose this image because the field value is zero most places on the lake. You can see that it’s very still water, and the height field of the lake is undisturbed.

However, over on the left of the lake, you can see a localized disturbance. There are ripples propagating through the height field represented by the surface of the lake. In QFT terms, we’d say that this excitation of the field would represent the presence of a fair number of “lakeon particles” in that general area of the lake, just as a disturbed electromagnetic field represents the presence of photons. With a nod to Heisenberg’s Uncertainty Principle, you can’t figure out *exactly* where the lakeon particles are. Indeed, you could view the height field of the lake as being some sort of abstract thingy that you could use to figure out a *probability density* for there being a lakeon present. Where there’s some wiggly business going on, as is the case on the left side of the lake, there’s a non-zero probability that you’d find a lakeon at that spot were you hypothetically able to pinpoint the lakeon.

Here’s the fun thing: in QFT, *everything* is described by fields. The photon is the particle of electromagnetism, but rather than being what you’d think of as a particle (a little spec), in QFT it’s really a disturbance of the electromagnetic field, some wiggly business that propagates around like wave. Well, everything else is the same way. You might be used to thinking of particles like electrons as little specs the same way you might talk about particles of dust. That’s not how our most fundamental of theory describes them. Rather, there is an *electron field*, a field that is a weird and abstract thing that’s much harder to visualize than the height field of a lake or even the electromagnetic field. However, it is in fact a field in the physics sense of the word, in that it has a value everywhere in space. If you disturb this field, and get a wave moving through it, that excitation of the field is what an electron is. Or, to be more precise, the minimal allowed disturbance of that field would be an electron; a larger disturbance would represent a larger number of electrons.

**Energy in QFT**

Physicists are obsessed with energy. Isn’t everybody? It turns out, though, that when people talk about energy in popular parlance, they are talking about something vaguer, that incorporates aspects of both energy and entropy (and maybe perhaps your force of will and current capacity for focused cognitive activity). But that’s neither here nor there.

In physics, identifying the energy in a system is often a very useful thing to do, because energy is *conserved*. Saying that “energy is conserved” is very different from the popular parlance version of “energy conservation”, which is about keeping energy in a useful form. In physics, energy is neither created nor destroyed, so there’s no need to *try* to conserve energy; it just happens, always. That fact, plus a whole lot of math, allows us to make all kinds of predictions about how physical systems will behave.

In QFT terms, there are a few ways that energy can arise. There is the *self-energy* of a field. If you’ve got an electric field in space, there’s an energy density associated with that field. (I say “energy density” rather than just “energy”, because the field isn’t all at once place, but is distributed through space. So, within a given volume, there will be a certain amount of energy; energy within a volume divided by that volume gives you energy per volume, or energy density.) There is also the energy of interaction between fields. So, because a charged particle like an electron can interact with the electromagnetic field, there is an energy associated with the interaction of the electron field and the electromagnetic field. How much energy depends on the values of the two fields— that is, how probable it is that there are one or more electrons or photons at various points in space— and the *coupling strength* between those two fields. Another kind of particle, the neutrino, has no electric charge; it does not interact with the electromagnetic field at all, and so we would say that it does not *couple* with the field, or equivalently that the *coupling strength* the neutrino field and the electromagnetic field is zero.

For the electromagnetic field, that’s basically all you have to worry about: the self-energy of the field, and the energy that comes with coupling from other fields.

**Massive Fields**

Many of the fields in QFT, on the other hand, are *massive fields*. These are fields like the electron field, the quark field (quarks being the particles that make up protons and neutron), and several other of the fields we know about. A massive field is a field such that if you have an excitation of that field— that is, a particle— that particle has mass associated with it. The mass of the electron is not zero, nor are the masses of quarks zero. There *are* a couple of massless fields in QFT, such as the electromagnetic field— photons have zero mass.

It turns out that *mass is just another form of energy*. If you have something with mass *m*, it has energy *E* just as a result of its mass; the amount of mass-energy *E* that you’ve got when you have something of mass *m* is given by the most famous equation in all of physics:

E=mc^{2}

In this equation *c* is the speed of light. The equation is just the conversion factor between mass and energy; it tells you how much energy there is associated with a particle of mass *m*. Among other things, this means that it’s possible to create mass “out of nothing”, although you’re not really creating it out of nothing, you’re just converting other forms of energy into mass energy. If two photons with the enough energy come together and interact in just the right way, it’s possible that they’ll disappear and create two particles, a positron and an electron, the positron being a antimatter particle that’s sort of like the opposite of an electron. The mass of the positron is exactly the same as the mass of the electron.

*Pair production: two gamma ray photons come in, interact, and out comes a positron (e*

^{+}) and an electron (e^{–}).Notice that before, you had two photons, and zero mass; after you have mass. If you’ve taken a chemistry class, you may have learned about the law of “conservation of mass”. This law is in fact not strictly correct. For chemical reactions, the amount of mass that gets converted to energy and back is typically about a billionth the amount of mass present, so it’s correct to very good approximation. But, when we’re talking particle physics, it’s not true at all. You can convert mass energy to and from other forms of energy.

This means that when you’re writing down the energy expressions in QFT, you have to include not only the self-interaction of fields, and the coupling of fields to other fields, but also the mass energy associated with the particles of that field. Unfortunately, the way that you add this mass energy in QFT is rather ad-hoc. The coupling of the fields together come in a fairly elegant way (although the actual coupling strengths are arbitrary, and as of right now we have to take them as “just the way nature is” rather than determining them from fundamental principles). However, the mass terms show up in an ugly and tacked-on way.

**The Higgs Field and Mass Terms in QFT**

And so, finally, we come to the Higgs field. Now, if you’ve been paying attention, by introducing the “Higgs field”, I’m saying that there’s a new kind of particle, which we’d call the “Higgs particle”. In fact, you hear it called the “Higgs boson”, because physicists categorize things (for reasons that aren’t important here) as either fermions or bosons. Electrons, for instance, are fermions, while photons are bosons. The Higgs field is predicted by the standard model of particle physics in a fairly esoteric way. Suffice to say that the part of the standard model of particle physics that predicts the Higgs field *also* predicts other things that had previously been measured in physics experiments. That is, we have a theory that predicts various things, and some of the predictions of that theory had been validated. So, we had reason to take this theory seriously. The theory *also* predicted that there would be this Higgs field, and that it would be a massive field. In other words, there was a prediction for a new field, and excitations of that new field would show up as a massive particle. The mass of the particle is still tiny tiny tiny compared to everyday masses, but it’s huge compared to the masses of the other fundamental particles we know about. As such, it took accelerators that were able to accelerate other particles to very high energies before there was enough energy to create this new massive particle.

The Higgs field, however, has a key difference from the other fields. Above, I talked about the analogy to the surface of a lake. The “vacuum state”, that is, the natural, undisturbed, zero-energy state of the field was a field with a zero value everywhere— a level lake everywhere at its average height. Fields don’t necessarily have to be this way. For instance, it would be entirely possible to have an electric field that is doesn’t have any waves moving through it, but that is constant everywhere in space. The electric field inside some kinds of capacitors is very much like this. This electric field *would* have energy associated with it though (which is why capacitors work!), and so we wouldn’t call it the vacuum state of the field. The vacuum state of the electromagnetic field is in fact a field value of zero.

The Higgs field is different. It’s vacuum state is in fact *not* a field value of zero. This has consequences. One consequence is that when you figure out the interaction of the Higgs field with other fields, you get an additional energy term in the equations of QFT describing the energy of everything. The neat thing is that that extra energy term resulting from the non-zero vacuum value looks *exactly like the mass-energy term of the other field*. Consider, for example, the electron field. Where we used to have an ad-hoc mass term, we now just have another elegant field coupling term with the Higgs field, but that term looks, mathematically, just like the mass term. The energy we would have called the mass energy of the electron is in fact something that arises with the interaction of excitations of the electron field with the vacuum state of the Higgs field.

Notice that this doesn’t mean that the Higgs *boson* gives particles their mass. In the “star walking through a crowed in a party in Hollywood” analogy, you might be tempted to think that all the people in the room represent Higgs bosons. They do not. The sea of people, as it were, together represent the vacuum state of the Higgs field. Even though there aren’t any actual Higgs bosons tooling around, the interactions of other fields with the *field* of which the Higgs boson is an excitation is what gives rise to the mass energy terms in the QFT equations.

So what about photons and other massless particles? They don’t couple to the Higgs field; they ignore it, and so no mass-like terms show up in the equations for them. The different masses of all the other particles arises because of the different coupling strengths between those particles and the Higgs field.

**Pithy Summary**

The mass energy of particles in quantum field theory is in fact the result of interaction of the fields associated with these particles as they couple with the non-zero vacuum state of the Higgs field.

Right.