© Matt Strassler [October 9, 2012]
This is article 8 in the sequence entitled Fields and Particles: with Math. Here is the previous article.
In the previous article in this series, I explained that the particles of nature are quanta of relativistic fields satisfying Class 0 and Class 1 equations of motion. But what I didn’t say yet is that — fortunately — this statement is only approximately true. The real equations are always a bit more complicated, in such a way that retains this relation between particles and fields but allows for a much richer set of processes to occur, including the production of particles from the collision of other particles, the decays of some particles to other particles, and the scattering of particles off each other, as well as the formation of interesting objects such as protons and neutrons, atomic nuclei and atoms. I won’t be able to explain all of that in detail, but in this article I’ll start to give you a feel for how it works.
The key difference between the equations that I called “Class 0” and “Class 1”, and the equations that are really important in real-world physics, is that the real equations have additional terms that depend on two or more fields at a time, rather than just one. What I mean by this is the following: instead of a Class 0 equation for a relativistic field Z(x,t) that looks like
- d2Z/dt2 – c2 d2Z/dx2 = 0
real fields in the world have equations that look more like this
- d2Z/dt2 – c2 d2Z/dx2 = y’ Z(x,t)3 + y A(x,t)B(x,t)
where y and y’ are numbers (generally not larger than one), “Z” is shorthand for Z(x,t), and A(x,t) and B(x,t) are two other fields. Other possible terms that one might find appearing in these equations would be A(x,t)Z(x,t), or A(x,t)2 Z(x,t), or Z(x,t)2, or even A(x,t)d2Z/dt2, etc. Which terms are allowed and which are not depends on the details of the fields involved; the rules are straightforward but long-winded, so let me not go into that right now. Generally, we have learned from experiment (and understood from the theoretical point of view) that in nature
- any term which is not forbidden by a principle (such as conservation of electric charge or consistency with Einstein’s relativity) appears in the equations,
- but terms with many fields are usually very small and unimportant compared to terms with one, two and sometimes three fields (and terms with time or space derivatives are usually small too.)
Therefore in most interesting physical processes we can focus our attention on all the allowed terms with just one, two or three fields.
A little term-inology. The terms in equations that involve one power of a field are called “linear”; all the terms we saw in our Class 0 and Class 1 equations were linear. Terms that involve two or three fields are called “quadratic” or “cubic”; more generally they are called “non-linear.” All of the interesting phenomena in our world arise because of the non-linear terms — the interactions of fields with each other and with themselves — in the equations of motion. And we’re going to learn about one of them now.
Production of a New Wave from the Resonance of Two Others
To get a feel for how interesting things can become, let’s take three fields A(x,t), B(x,t), C(x,t), and to keep things simple, let’s imagine the A and B fields approximately satisfy Class 0 equations (and therefore have massless quanta) while the C field satisfies a Class 1 equation (and has waves whose minimum frequency is νmin and whose quanta therefore have mass m = hνmin / c2, where h is Planck’s constant). But we’ll also put non-linear terms in the their equations. Specifically, we’ll consider (with shorthand “A” for “A(x,t)”, et cetera, and with c the speed of light and y a number, generally not larger than 1)
- d2A/dt2 – c2 d2A/dx2 = y B C
- d2B/dt2 – c2 d2B/dx2 = y A C
- d2C/dt2 – c2 d2C/dx2 = (2 π νmin)2 C + y A B
[Fine Point: The reason I put nonlinear terms in all three equations is that it turns out it would be inconsistent to have non-linear terms in one equation without having them in the other two; energy wouldn't be conserved, for instance. For the process I'm about to describe, we'll only need to study the nonlinear terms in the C field's equations of motion.]
Let’s see what can happen, given this set of equations, if a wave in the A field meets a wave in the B field. We can actually guess, if we look carefully at the equations themselves. Where there is a wave in the A field, then A(x,t) is non-zero in most of that vicinity. When waves in the A and B field overlap each other, then the product A(x,t) times B(x,t) becomes non-zero. Now look at the C equation: it says that changes in the C field over time and over space (the two terms on the left-hand side of the equation) are related to A times B (one of the terms on the right-hand side of the equation.)
- d2C/dt2 – c2 d2C/dx2 = (2 π νmin)2 C + y A B
So even if the C field is zero to start with, once A(x,t)B(x,t) becomes non-zero, then soon enough C(x,t) will become non-zero in the same region. In short, some sort of small disturbance in the C field will result from the meeting of the A and B waves.
Fine point: You could then wonder, once C becomes non-zero, whether the A(x,t) C(x,t) term in the equation of motion for the B field would cause a further disturbance in the B field. The answer is yes, but it’s an even smaller effect. We’ll ignore it for now, and later will learn why that was a good idea.
In Figure 1 is shown a wave with frequency ν in the A field (green) meeting a field with frequency ν in the B field (blue). [I'm taking the frequencies equal to keep things simple and symmetric; later we'll see why all other cases in the end reduce to this one.] Click the figure to animate it. You can see how fast the frequency ν is by looking at the green wave and noting a vertical black bar that sits at one point in space and oscillates up and down with frequency ν as the green wave passes by.
Below them, in orange, is shown the product A(x,t)B(x,t); you will see that this product becomes non-zero where the two waves are overlapping. You can also see that it oscillates with time. If you look really closely, you’ll see (comparing it with the oscillating black bar that shows you how fast ν is) that A(x,t)B(x,t) oscillates with twice this frequency! This is important: keep it in mind. [More generally if a wave of frequency ν1 meets a wave of frequency ν2, their product will include oscillations of frequency ν1 + ν2.] One more thing: you notice that the oscillations of A times B aren’t moving right or left; they are standing in place. We’ll see why this is important in a moment.
Now how does the non-zero value of A times B affect the field C? The answer depends on the frequency ν, in a very big way! Let me tell you the answer, and then I’ll give you intuition for why it is the case. It is basically a phenomenon of resonance. Resonance is an essential phenomenon for all types of oscillations (i.e. vibrations) including waves. I have previously described how a ball on a spring oscillates with a natural frequency, and described how an oscillating force pushing on the ball can cause resonance if the frequency of the force’s oscillation matches that natural frequency of the ball on the spring.
Once you do understand resonance, you’ll recognize that the C field, when it oscillates with minimum frequency and with zero velocity to the right or left (as described at the end of this article in the section on Class 1 waves — see Figure 2), is like the ball on a spring, and A times B acts like an oscillating force trying to make that ball oscillate (as described in my article on resonance.) And so there is a resonance phenomenon, if the frequency with which A times B oscillates — namely, 2ν — happens to be at the minimum frequency for the field C — namely, νmin. More specifically
- If 2 ν does not equal νmin — if the force is off-resonance — then, in the region where A times B isn’t zero, C will begin oscillating irregularly and with a small amplitude.
- If 2 ν = νmin — if the force is on resonance — then C will begin oscillating smoothly, with a large amplitude, in the region where A times B isn’t zero, and will continue to oscillate even when A times B becomes zero again.
Figure 1 only shows the resonant situation 2 ν = νmin. You see there that when the wave in A passes the wave in B, they leave a stationary C wave, oscillating at νmin, behind. [Fine point: what is shown in the figure is a sketch, and not the exact solution to the equations. The exact solution would have lots of additional small complicated feature, which would obscure the main physics point I am making, so for clarity I have removed them.] We will look at the non-resonant situation, which is more complicated but also physically important, at a later time.
Production of a New Particle from the Annihilation of Two Others
I’ve just shown you that the non-linear A-times-B terms in the C equation can cause two overlapping waves in the A and B fields to generate oscillations in the C field, if the sum of the frequencies of the A and B fields is equal to the minimum frequency for the C field. But what if these waves have very small amplitude to start with? What may happen if a single quantum of the A field meets a single quantum of the B field? (You can remind yourself about quanta here.) The answer:
- If the frequencies of A times B are on-resonance for the C field, then a quantum of the C field — a real C particle — might be created (and the A and B quanta will disappear — will be “annihilated”).
- Alternatively the A and B quanta may just pass each other and no C particle will be created.
- The laws of quantum mechanics imply that the probability for a C particle to be created in this situation is proportional to the square of the number y that multiplies A times B in the equation of motion for C.
- If the frequencies are off-resonance, however, a real C particle will not be created. There may however be a temporary disturbance in the C field, an example of what is often called a C “virtual particle”, in which case the A and B quanta may again disappear. What happens as a result of this disturbance? I’ve put in a little preliminary discussion of its implications at the end of this article, in the “Loose Ends” section.
Ok; that’s the general sense of things. Let’s go into a few details.
What does being on- and off-resonance mean for `particles,’ i.e., for the quanta of the fields A, B and C? The thing to remember is that the energy of a quantum is related to its frequency by the equation E = h ν. So let’s do a little translating of our wave discussion above into a discussion for `particles’.
Suppose that the waves in the A and B field that we started off talking about consisted of just one quantum each. Those quanta are massless since A and B satisfy Class 0 equations. [More precisely, the linear terms in their equations of motion are those of Class 0 equations.] Since the A and B quanta have equal frequencies, they have equal energies E = h ν. And since, for a massless quantum, its momentum has magnitude p = E/c, the A and B quanta have momenta that are equal in magnitude to h ν/c but are obviously opposite in direction, since one moves to the left and the other to the right. Therefore,
- the total energy of the two quanta added together is 2hν
- the total momentum of the two quanta added together is zero
Since energy is conserved and momentum is conserved, the total energy after these two quanta collide will still be 2 hν, and the total momentum will remain zero.
Now what we saw for waves is that because the equation for C contains a term of the form A times B, there is a resonance when the frequency of A times B (which acts like an oscillating force) matches the minimum frequency of C (which acts like a ball on a spring). Let’s translate this statement into quanta.
The frequency of A times B is twice ν, and so the energy of the product of A times B, when A and B each have waves consisting of a single quantum, is the sum of the energies of A and B:
- EAB = 2 h ν = EA + EB
The minimum frequency of C is νmin, which means a stationary quantum of the C field has zero momentum and energy
- EC = h νmin = mc2,
where m is the mass of a C quantum.
On-resonance requires 2 ν = νmin, or in other words
- EA + EB = 2 h ν = h νmin = mc2 = EC
In short, resonance occurs when the sum of the (equal) energies for the A and B particles (which in this case have equal and opposite momentum) is just the right amount to make a stationary C particle! Along the way, the A and B particles are annihilated — their energy is entirely used up in creating the C particle. This is shown [somewhat schematically -- one can't really draw what happens in quantum mechanics] in Figure 2, which you should compare with Figure 1.
I’ve just shown you the process A + B → C. I used three different types of particles in my explanation to avoid any confusion about what was going on. But the same basic idea allows for processes such as A + A → C (for instance, gluon + gluon → Higgs particle, the main way that Higgs particles are produced at the Large Hadron Collider) and processes such as A + A* → C, where A* is the antiparticle of A (for instance, quark + antiquark → Z particle, the main way that Z particles are produced at the Large Hadron Collider.) Only little details have to be changed; the basic idea is the same.
Another process that is basically the same idea, just reversed in time, is particle decay. The decay of a Higgs particle to two photons, or of a Z particle to a quark and an anti-quark, is basically the same as in Figure 2 with the animation run in reverse.
Some Loose Ends
1. If the A-times-B term pushes the C field off-resonance — if 2 ν ≠ νmin — what physical processes can result? You may recall (see my article on resonance) that if you push a child’s swing with the wrong frequency, or apply a force to a ball on a spring that oscillates with a frequency different from the ball-and-spring’s natural frequency, you will get a jiggly motion that is a bit messy and has small amplitude. In the current context, that’s what happens to the C field also. The C field will do something, but it won’t form a nicely behaved C quantum. It will just rattle around a bit. This ill-behaved disturbance is one example of what is called a “virtual C particle” — but it isn’t a particle (a quantum of a wave in the C field) despite the name. Its mass is different from the mass of a C particle; it may be much larger or much smaller. Unlike a C particle, it won’t exist on its own for any length of time. And it does not satisfy the condition on its amplitude that a real quantum must satisfy. Instead, unlike the C particle in Figure 2, which endures for a while, the off-resonance disturbance lasts only as long as the A and B quanta are overlapping.
But that doesn’t mean it can’t have any effect. For one thing, it could cause the A and B particles to bounce off each other
- A (to the right) + B (to the left) → C field disturbance → A (to the left) + B (to the right)
More generally, in three dimensional space, the bounce, or “scattering”, could leave the A particle traveling in any direction, with B traveling in the opposite direction. Examples of this process and its variants include the scattering of an electron and a positron through the effects of a virtual photon, or the scattering of a quark and an anti-quark through the effects of a virtual gluon. (There are many other scattering processes which are common but require a more subtle discussion; perhaps I will cover these at a later time.)
Or if there are other fields, D and E, that interact with C and also appear in C’s equation of motion
- d2C/dt2 – c2 d2C/dx2 = (2 π νmin)2 C + y A B + y’ D E
then a much more interesting process can occur:
- A + B → C field disturbance → D + E
That’s right: A and B particles can annihilate (via the C virtual particle) and a D and an E particle can be created in their place. This is a second way (the first being the one in Figure 2) that we can create new particles! For instance, an electron and a positron can collide, annihilate via a virtual photon (remember to read that language appropriately, as “via the disturbance created by pushing the photon field with an off-resonance frequency”), and become a muon and an anti-muon, or a quark and an anti-quark. A down quark and a up anti-quark can collide, annihilate via a virtual W particle, and turn into an electron and an anti-neutrino. Or two gluons can collide, annihilate via a virtual gluon, and turn into a top quark and a top anti-quark (this is the most common way to make top quarks at the Large Hadron Collider.)
2. What if the two waves in the A and B field had had different frequencies, νA and νB? C particles can still be created if the frequencies are right, but the resonance condition is different, and the C particle that is created won’t be stationary. Let’s work that out.
If they have different frequencies, the two colliding (massless) quanta would have had
- different energies EA = h νA and EB = h νB
- different momenta pA = + h νA/ c and pB = – h νB/ c (here the plus sign means “to the right”, minus sign means “to the left”)
The total momentum pA + pB is now non-zero. But momentum is conserved. So if a C particle can be produced in the annihilation of the A and B particles, it will have momentum pC = pA + pB, and thus it will be moving right or left, rather than stationary (as in Figure 2). (If νA > νB it will be moving to the right, otherwise it will be moving to the left.)
Now, how much energy is required to make a C particle that is moving? It requires more than when it is stationary, of course: as for any massive particle, its energy and momentum have to satisfy
- EC2 = (pCc)2 + (mc2)2
which says EC = mc2 if pC = 0, or greater if the momentum is non-zero. Energy and momentum conservation tell us
- EC = EA + EB
- pC = pA + pB = EA /c – EB/c
Where did I get the last equation? Well, for a massless particle p = E/c in magnitude, and for our colliding A and B particles their momentum is in opposite directions, so their momenta differ in sign (see above where we first discussed their momenta.) Now we substitute this into the previous green equation, and find
- (EA + EB)2 = (EA – EB)2 + (mc2)2
The EA2 and EB2 terms cancel from both sides, and pulling the EA EB terms all to the left-hand side we find the result
- 4 EA EB = (mc2)2
Dividing by h2 and using the relation m = hνmin / c2 gives us the resonance condition
- (2νA)(2νB) = νmin2
(which correctly reduces, if νA = νB, to the resonance condition for a stationary C particle, 2 ν = νmin.) If this resonance condition isn’t met, a C particle can’t be created; if it is met, then a C particle might be created.
3. I noted that just as the equation of motion for the field C contains a term A-times-B, the equation for B contains a term A-times-C. Let’s recall those equations:
- d2B/dt2 – c2 d2B/dx2 = y A C
- d2C/dt2 – c2 d2C/dx2 = (2 π νmin)2 C + y A B
So if the overlapping of A and B waves produces a new wave in the C field, making it non-zero, as in Figure 1, shouldn’t I worry that the overlapping of that new C wave with the pre-existing A wave will cause a change in the B field? And my answer was “yes, but we can ignore this.” We can ignore it for a combination of two reasons.
First, there’s the fact that the number “y” which appears in front of both the A-times-B and B-times-C terms. The effect on the C field from A B is proportional to y times the size of A times B. The effect on the B field from A C is similarly proportional to y times the size of the A times C, but that in turn is equal to y2 times the size of A times A times B. So as long as y is less than 1, y2 is less than y, so the effect on B from A C is therefore small compared to the effect on C from A B — well, at least if the waves in A and B aren’t too big. But in particle physics applications, the waves are generally small: an A particle is a single quantum of a wave in the A field, and so corresponds to A having a wave with a small amplitude.
Which leads us to the second, weirder but a bit more convincing, reason: In fact, as we saw, if a single quantum of A meets a single quantum of B and turns into a single quantum of C, the waves in A and B completely disappear (i.e., are annihilated.) So after C is created there actually is no A wave left, and therefore A-times-C is zero after all, meaning there’s no effect on the B field.
A final caveat: although I can’t prove this to you without more work, even if A and B have waves of just one quantum each, as in Figure 2, the process depicted in Figure 2 would be much more complicated if y were very large compared to 1. So the simplicity of the story that I have told you does require y not be too large; otherwise it has to be reconsidered. It happens in nature that most of the non-linear terms that arise in particle physics have small “y”, and so what I’ve told you is relevant for most particle physics applications. The exceptions are interesting — they lead to complicated objects like protons and other hadrons.