© Matt Strassler [September 6, 2012]
This is article 6 in the sequence entitled Fields and Particles: with Math. Here is the previous article.
We’ve actually been dealing with fields for a while now, but I didn’t say so — because I wanted to focus on the waves in these fields. In describing what the waves have been doing, we’ve been expressing their shape and their dependence on time using a quantity Z(x,t). Well, Z(x,t) is a field; it is a function of space and time with an equation of motion that determines its behavior. (A suitable equation of motion will be one such that if Z increases or decreases at a particular location, it tends to make Z increase or decrease in nearby locations at a later time; this feature commonly allows waves to be among the solutions to the equation.)
In this article we’re going to look at a few examples of fields Z(x,t) whose equations of motion allow for waves. The physical interpretation of these fields will be very different; they describe different properties of different materials. But the equations that they satisfy, and the waves they exhibit, will satisfy the same (or very similar) mathematics, so they will behave very similarly despite their very different physical origins. This will turn out to be very important.
And then we’re going to do something radical. We’ll consider fields in the context of special relativity [completely avoiding the math of relativity, and just focusing on its consequences for the fields.] As Einstein showed, if you tweak space and time to make them behave differently from the way most of us would expect, you can have a new type of field — one whose physical interpretation is not that it is a property of something else, but rather that it is a fundamental physical object all on its own.
Some Ordinary Fields Describing Ordinary Stuff
A field Z(x,t) may represent many different physical quantities. Examples include
- the height of a rope extended horizontally across a room
- the height of water in a river
- the density of a crystal or of a gas
- the orientation of atoms in a magnet
- the speed of the wind
- the temperature, density or pressure of the air
In each of these cases there’s a field Z(x,t): a height field, a density field, an orientation field, a wind field, a temperature field. Its value as a function of space and time tells us what the height or density or orientation or wind speed or temperature of some material medium — rope, river, crystal, gas, magnet, air — is at all places and all times. Its equation of motion tells us what possible behaviors are allowed for Z(x,t); and it also tells us how to predict the future behavior of Z(x,t) if we know its present and recent past behavior.
In each example there is a field and a medium, and we must not confuse the field with the medium. The field merely describes and characterizes one of the many properties of the relevant medium. And very different media may have very similar-behaving fields, with very similar waves — as we’ll see.
Let me emphasize one last time a point that often causes confusion. In general the field may have nothing to do with a physical distance in space. Yes, in articles 3 and 4 I used the example of a wave on a rope to illustrate what Z(x,t) might be, because it’s nice and intuitive; and also I’ve made graphs of Z(x,t) for waves many times. Both of those things might tend to give you the erroneous impression that Z(x,t) always has to do with waves that cause a physical object (like the rope) to move a distance Z in a direction of space perpendicular to the x direction. But that’s not true, as three of our four examples will illustrate.
A Rope Height Field
First, let’s give consider the wiggling rope that was our initial example for waves (see Figure 2 of article 3): in this case the role of Z(x,t) is the height field, which we’ll call H(x,t), telling us how high the rope is at each point in space x along the rope, at every time t. If the rope just sits at its equilibrium height H0, then H(x,t) = H0; the height field is constant in space at all times. If it has a simple wave moving on it, the height field will be described by our famous wave formula that we saw in previous articles. [H(x,t) can behave in other ways too, if it is impinged on by external forces. But we won't consider such behaviors for now.]
If we know H(x,t), we know how high the rope is at all points in space and time. Alternatively, if we know what the rope is doing now and in the very recent past, we can predict, using the equation of motion, what it will do in the future. None of this tells us very much about the rope itself. The height field just tells us what its name implies: the height of the rope. The rope is the physical medium whose height is represented as the field H(x,t); this tells us nothing about the rope’s color, thickness, tension, material, etc.
In Figure 1 I’ve given you an animation that shows a wave in the height field, coming in from the left and moving off to the right. Naively it would seem I’ve plotted the same things twice, once in orange and once in green. But they’re not the same thing. The orange curve is supposed to be the rope itself, moving in physical space. The green curve is a graph, a simple mathematics plot that just represents what’s going on with H(x,t), without reference to what H(x,t) means (i.e. height) or what its medium is. In this case only, the green graph looks just like what’s happening in the physical world. But that won’t be the case for any of the other examples that we’re about to see.
A Lattice Displacement Field
Suppose we have a medium that consists of a crystal of equally spaced atoms. (I’ve drawn this with one space dimension in Figure 2; we could consider a similar situation with three space dimensions, but that would be an unnecessary complication for the moment. I’ve also labelled every 10th atom red so you can keep track of its motion more easily. And I’ve highly exaggerated the distance between the atoms; you might want to think about there being a few million atoms between each red atom rather than 10.)
Let’s look at the displacement field D(x,t) that tells us how far, at time t, the atom that is normally at equilibrium position x has moved away from its equilibrium position in the lattice. That means that in the static picture shown in Figure 2, the field is zero everywhere, D(x,t)=0, because all atoms are in their normal positions in the lattice. When you click to animate the figure, what you’ll see is (a) below, the individual atoms oscillate back and forth in a motion that, overall, propagates as a wave from left to right; (b) above, a graph of the lattice displacement field D(x,t) shows how it behaves as the wave moves past. Note D(x,t) wiggles back and forth in much the same way the height field did in Figure 1; the waves in the two fields behave similarly, even though the physical interpretation of the field is very different in the two cases.
A Magnet Orientation Field
What is a permanent magnet? It consists of a set of atoms, each of which is a tiny little magnet with a tiny magnetic field, all aligned so as to create cooperatively a large magnetic field. A magnet is shown in Figure 3; each atom points vertically. In this case, the orientation field Θ(x,t) tells us how far, at time t, the atom at position x points away from the vertical; Θ, in short, is the angle between each atom’s little magnet and the vertical direction. The animation in Figure 3 shows a wave in the magnet, where orientations of the atomic magnets oscillate left and right of vertical. Above the magnet, in green, is a graph of Θ(x,t); yet again, it looks the same as in the previous cases.
An Air Pressure Field
Let’s consider a gas of molecules placed in a long tube; the long direction of the tube we’ll call the direction x. The molecules in a gas move around almost randomly, banging into the walls of the tube and into each other. In equilibrium, the density (the number of molecules in a certain volume) and the pressure P(x,t) (the force per unit area that would be exerted by the molecules on the surface of a little ball inserted at the point x at time t) are constant. But sound waves passing through the gas will make the pressure (and density) oscillate, as shown in Figure 4 (click to animate). The density and pressure increase and decrease repeatedly; the molecules move back and forth, though on average they do not move at all, even though the wave and its energy move from left to right across the gas. The graph of P(x,t) looks yet again the same as before.
What do we learn from these four examples, all of which happen to exhibit Class 0 waves? (Recall that a Class 0 equation of motion has one quantity in it, called cw, and all waves in the corresponding field travel with speed cw. Different Class 0 fields have different values of cw.) We learn that the same field behavior can emerge from physically distinct fields in physically different media. Despite their different origins, the waves in a height field, in a lattice displacement field, in a magnet orientation field and in a gas pressure field may look identical as far as the field is concerned. They satisfy the same type of equation of motion and the same qualitative relationship between frequency and wavelength.
[Fine point: Strictly speaking, if you make the waves of sufficiently short wavelength, you will eventually be able to tell the difference between the different media. Once the wavelengths are as small as, say, the distance between the atoms in the rope, or crystal, or magnet, the wave equations that the waves satisfy will turn out to be more complicated than the ones we've written down, and the detailed difference will distinguish the media from one another. But often, in any practical experiment, we won't come even vaguely close to seeing these effects.]
A consequence of this is that studying the waves (and their quanta) associated with a field does not uniquely tell you what the underlying medium is or what the physical interpretation of the field is — what property of the medium it represents. Or even if for some reason you happen know the field is of a certain type — a pressure field, say — you will still not in general be able to tell, from its behavior, what it represents the pressure of. All you can do by studying the waves is learn whether the field is of Class 0 or Class 1, and what cw is for that field is; or perhaps it is of some other class altogether.
In some ways that’s too bad; the field only carries partial information about the medium. In other senses that’s kind of neat: the field is a more abstract and universal thing than the physical material that it characterizes.
The field thus does not determine the medium; and its behavior is often independent of the details of the medium. Which then raises a question.
Can one have a physical field — with waves made from quanta that move through space and carry energy — without having a medium to support it?
A Field With No Medium?
A song cannot be heard without a singer. Yet a song has some kind of independent existence; it sounds different in detail, depending on who is singing it, yet there is something essential about the song, some abstract quality that makes it always recognizable. That abstract entity is the tune of the song. We can discuss and study and learn the tune, and write it down in musical notation, without ever hearing it sung by a singer. Many of us can even hum the tune silently in our heads. Somehow the tune exists, even if no one sings the song.
If, as in all the examples I’ve given above, and in all the others I could give you that would make intuitive sense to you, a field describes a property of a medium, then how can one have a field without a medium? Yet somehow fields are also independent of their media, because many different fields can behave in the same way, despite describing many different properties of vastly differing media. So perhaps it is possible to abstract the field away from its medium.
Well, not only is it possible, it is apparently mandatory. At least, it is mandatory either to have no medium at all, or to have a medium which can’t be made from ordinary matter, and is vastly different from all the media we’ve considered up to now — in that it acts as though (for all experiments anyone has ever done) it isn’t there.
One of several radical elements in Einstein’s 1905 theory of special relativity was the notion that for light waves — which were known for decades to be waves in electric and magnetic fields (“electro-magnetic waves”), all of which travel at the same speed in empty space — there is no medium. There are only the fields. [The hypothetical medium had been called the ``aether''; Einstein argued there was no such thing, and wrote down a set of equations where indeed none was required. Note there is still debate (often more philosophy than physics) about whether one should or shouldn't think of there as being a weird sort of medium that's profoundly different from ordinary matter. At present there's no experimental evidence that requires it. Please take any comments on this point to this webpage, rather than using the comment space below, which I would like to reserve for more straightforward physics questions, rather than controversies and debates.]
The key elements of Einstein’s version of relativity (as opposed to Galileo and Newton’s) were that
- Space and time are not what you think; different observers, in steady motion relative to one another, disagree about the lengths of objects and the times between events (in a very precise and measurable fashion.)
- There is a universal speed limit, called “c”; any observer, measuring the speed relative to him or her of any passing object, will find it to be less than or equal to c.
- In such a world, certain fields — “relativistic fields” — can exist without a medium, and they satisfy special equations of motion. The simplest relativistic fields satisfy Class 0 or Class 1 equations of motion, with the wave speed cw that appears in the equation of motion taken equal to the universal speed limit c.
In short there are Class 0 relativistic fields that satisfy the equation
- Class 0: d2Z/dt2 – c2 d2Z/dx2 = 0 .
(Light, and indeed all electro-magnetic waves, furnishes the most famous but not unique example; that’s why “c” is often called “the speed of light”.) And there are Class 1 relativistic fields satisfying the equation
- Class 1: d2Z/dt2 – c2 d2Z/dx2 = – (2 π μ)2 (Z-Z0)
We’ll see examples in the next article; note there are no constraints from relativity on μ (except that μ2 be positive) or on Z0. [There are more complicated equations that are allowed for relativistic fields, but most of them, in simple physical processes, boil down to one of these two.]
Relativistic fields are physically real and physically meaningful in the universe, in that
- their waves carry energy and information from one place to another
- waves in one field can affect waves in another field and change physical processes that would happen in their absence
But unlike the fields in the examples given early in this article, relativistic fields do not describe a property of some ordinary physical medium that is made from anything resembling ordinary matter, and as far as is known experimentally, they do not describe the property of anything at all. These fields may be, as far as we know today, among the fundamental elements of the universe.
Click here for the next article, on how relativistic quantum fields give us the particles of nature.
If you want to learn more about why an ordinary medium (like the rope, the lattice, the gas and the magnet in the examples above) is unacceptable for light waves, click here for an article comparing ordinary and relativistic fields and looking at how media and waves behave as viewed by different observers. [under construction]
If you want to join an ongoing discussion about whether it makes sense for a field without a medium to be a physical object (rather than just an abstraction), and what properties a medium suitable for a relativistic field would have to have (hint: it would have to be essentially identical to having nothing), then click here .