# Basic Concepts: Fields

In the initial “Basic Concepts” post, I discussed the concept of Force in physics. This time out, I’ll be talking about fields, which is a much dicier proposition. Not only are fields considerably more abstract than forces, but I’ve never lectured on fields in general (specific instances of fields, yes, but not fields as abstract objects). For that matter, I’ve never taken a real field theory class. The chances of my saying something stupid about fields are exponentially greater than the chances of my saying something stupid about forces.

In a certain sense, though, “Fields” is a good topic to follow “Force,” because both are concerned with interactions between objects, or between an object and its environment. In fact, I’ll make that the one-sentence blockquote summary:

Fields are a way of understanding the interaction between an object and its environment.

Fields and forces are intimately linked, and the typical introduction to fields is through force, specifically in the context of electrostatic forces between charged object. In introductory physics classes, we define the electric field of an object in terms of the force on a small test charge placed at a particular point. We define an “electric field vector” at each point in space, which tells you how big the force is, and what direction it’s in. We define “electric field lines” by connecting the vectors together. An electric field line essentially shows you the path that a test charge would follow, if it were released from rest at a point along the line.

We can extend this basic idea to other interactions. Magnetic forces are really only understood through the concept of fields and field lines, and light is understood as a combination of electric and magnetic fields. You’ll sometimes hear gravity discussed in terms of a field, but not that often. The strong and weak nuclear forces can in principle be described in terms of a field, though it’s not that common, because people talking about those forces usually go directly to the quantum picture (about which more later). In all of these cases, the idea is the same– the field is a mathematical object that tells you how to determine the effect of some interaction on an object at some point in space.

So, why do we do this?

The field description is useful for a number of reasons, starting with the fact that it immediately conveys the idea of an interaction filling a region of space. If you bring two charged objects close together, they will interact with one another, no matter how you arrange them. The force between two charges is not a special property of any particular arrangement of those objects, it’s a consequence of the interaction between them, and there will be some interaction no matter where the two are placed. The field gets you that idea.

Fields are also a useful tool for simplifying complicated problems, particularly when it comes to describing interactions between an object and its environment. If you want to talk about a charged particle near a metal surface, you don’t need to actually calculate the electromagnetic force between the particle of interest and each of the atoms making up the surface. Instead, you can calculate a field from the surface, and use that to determine what happens to the particle.

The biggest mathematical advantages of the field picture is that fields are generally governed by simple and elegant mathematical rules. This is most apparent in electromagnetism, where everything you want to know about the interactions between charged particles and their environment can be expressed in four short statements about the properties and behavior of electric and magnetic fields, known as “Maxwell’s Equations.” They fit very nicely on a T-shirt, and are a testament to the mathematical power of the idea.

So are fields just a calculational trick? Yes and no. While fields are fundamentally about describing the way a particular interaction propagates through space, the fields themselves do have a certain degree of physical reality, independent of particular objects. You can see this from, well, from the fact that you can see– the light we use to see by is a combination of oscillating electric and magnetic fields, propagating through space. The fields aren’t really tied to any particular object, and you can talk sensibly about light in terms of fields without worrying at all about what it was that produced the light in the first place. There’s also energy contained in fields, even in a region of space that doesn’t contain any particles.

The other tremendously confusing thing about the concept of fields is that physicists will talk about them as both a continuous mathematical object extending through all of space, and a collection of particles. High-energy theorists are particularly bad about this, talking about a “Higgs field” permeating all of space in one sentence, and then a “Higgs particle” that can be detected in an accelerator in the very next sentence. It can get kind of baffling.

The key idea here is called field quantization, and it’s best explained in terms of light, mostly because I only understand the process in terms of light… I’ll attempt to give a (highly idiosyncratic) explanation of what’s going on when people switch back and forth between particle and field descriptions, but it might be heavy going. I fully expect to have to come back to this idea again, and this first attempt may get ripped to shreds in the comments, but I’ll give it a try anyway.

The theory of electric and magnetic fields was developed in a classical context, and runs into two problems when you try to bring it into line with modern physics. The first of these is a problem with relativity– in principle, the classical description has the fields changing instantaneously. An electron sitting on my desk creates an electric field that extends out to the orbit of Jupiter and beyond. If I move it a centimeter to the right, the whole field changes, including the bit out at the orbit of Jupiter. In a purely classical world, you might expect this to happen instantaneously– that is, the very second I move the electron in Niskayuna, the alien monolith in orbit around Jupiter detects the change in the field. That would involve transmission of information faster than the speed of light, though, which isn’t allowed in relativity.

The solution to the problem is contained within Maxwell’s Equations. Among other things, you can use them to construct oscillating electric and magnetic fields that support one another and propagate through space– light waves. These waves move at the speed of light (duh), and can be used to carry energy and information.

To bring the electric field idea into line with relativity, you need to look in detail and how the field propagates through space, which you can do by describing the field everywhere in terms of these oscillating solutions. From math, we know that any arbitrary pattern can be made up by adding together sufficient numbers of sine and cosine functions with different frequencies and different amplitudes (it’s called the Fourier Theorem), and the same thing works for electromagnetic fields. We can describe the field due to an electron sitting on my desk as the sum of lots of different oscillating electric and magnetic fields, oscillating with different frequencies and traveling at the speed of light. It takes an infinite number of these “modes” of the field to describe any particular pattern (there are an infinite number of possible frequencies, and an infinite number of possible directions, so we’re talking one of the big inifinities, here), but that’s what integral calculus is for. You can think of the electron, in some sense, as constantly emitting all sorts of electromagnetic waves at different frequencies, heading in different directions.

Shifting the position of the electron, then, amounts to changing the distribution of these modes– we have a little more field at one frequency, a little less at another, and so on through the whole infinite number. And when you change the distribution, it takes time for that change to propagate. The alien monolith doesn’t know that there’s more field at one frequency until the electromagnetic wave carrying that frequency reaches Jupiter, and the wave travels at the speed of light. So Maxwell is happy, Einstein is happy, and the rest of us have to wait a few minutes before the aliens turn Jupiter into a star.

(What I describe is mathematically pretty cumbersome, which is why nobody actually describes fields this way, but it’s what’s going on in a conceptual sense. It’s rare to find a situation in which you really need to worry about the propagation delay of electric and magnetic fields, but they do come up, and there are simpler ways to calculate the effect.)

The second problem you hit is with quantum mechanics. Those electric and magnetic fields contain energy (a future Basic Concept), and quantum theory tells us that energy has to come in discrete quantities. If a really weak field contains one unit of energy, we can expect a stronger field to contain two or three times as much energy, but not two and a half times, or π times as much energy.

Mathematically, we deal with this problem by introducing the concept of “photons.” We can describe any arbitrary electromagnetic field as the sum of an infinite number of oscillating modes of the electromagnetic field, each with a particular frequency, a particular direction, and its own amplitude (which determines the amount of energy carried in that mode). If we write this description in quantum-mechanical terms, we find that each mode is quantized– I can have zero, one, or two (or any integer number) units of energy in that field, but not two and a half or π units. This applies for every individual mode of the field, and we can describe the field in terms of the number of units of energy in each mode.

If we take a single mode of the field, and put a single unit of energy in it, and look at its behavior, it turns out to behave exactly like a particle– it carries a small amount of energy, it carries a small amount of momentum, it can be detected at a particular position, and all the other things that we associate with particles. We call these units of energy “photons,” and describe them as particles of light.

(Again, describing a real macroscopic field this way would be incredibly cumbersome. When we actually work with the photon description of electromagnetic fields, we generally assume that the number of available modes is restricted in some way– we have a laser that we describe as a large number of photons in a single mode, for example, and we only worry about what happens to that one mode. In principle, though, the field of an electron sitting on my desk could be described in terms of a constant stream of photons pouring out of the electron all the time. You’d probably go crazy doing it, but that’s the microscopic picture.)

There are two steps from classical fields to quantum particles, then: First, we take the classical description (“The field at this point is so many volts per meter in this direction”) and split it into a sum of oscillating modes (“The field at this position is the sum of these waves headed in these directions with these amplitudes”), and then we treat each mode as a collection of photons (“The field at this position is the sum of these numbers of photons in these modes”). The two descriptions are mathematically perfectly equivalent, so physicists feel free to switch back and forth between them, talking about fields in some contexts, and forces in others.

Why the switching back and forth? Because describing a macroscopic electromagnetic field in terms of photon numbers is an absolute nightmare, and generally not necessary. For big-picture stuff, the classical description works just fine, and we talk about atoms sitting in electric or magnetic fields. When you’re dealing with very weak fields, or particular types of interactions, though, you need to move to the quantum picture, and talk about things in terms of numbers of photons. In either case, though, the fundamental process is the same: you have an object that interacts with its environment via the electromagnetic interaction.

The same thing can be done for the other forces, as well, and for anything else you can describe as a field. The gluons that carry the strong force, and the W and Z bosons that carry the weak force can be thought of as particles of the strong and weak “fields,” though you rarely hear them spoken of that way. Which description you use follows the same basic pattern, too– when talking about the universe in general, physicists will speak of a “Higgs field” as a continuous thing that permeates all of space, just as we talk about electric and magnetic fields filling some region of space. When talking about specific experiments, they’ll talk about a “Higgs boson” as a specific particle that can be created in accelerator experiments, just as we talk about the absorption and emission of single photons in quantum optics experiments. In either case, they’re talking about the same thing– the postulated “Higgs interaction” that gives rise to the observed masses of fundamental particles.

And that brings us back around to the initial point, after some heavy going: “Fields” are a way of dealing with different types of interaction between an object and its environment.

(And yes, I skipped the whole issue of zero-point energy and vacuum fluctuations. That’s a completely fascinating topic, but I’m trying to keep the number of brain-exploding steps per post to a minimum…)