© Matt Strassler [September 12, 2012]
This is article 7 in the sequence entitled Fields and Particles: with Math. Here is the previous article.
We’ve reached our goal finally: to understand what the things we call “particles”, such as electrons, photons, quarks, gluons and neutrinos, actually are [... as far as we understand it today. One must remember that in science, there is no guarantee that current understanding will not deepen further.]
The previous article described what fields are — objects that have a value at each point in space and each moment in time [i.e., ``functions'' of space and time] that satisfy an equation of motion, and are physically meaningful in that they can carry energy from place to place and actually affect physical processes in the universe.
We learned that most fields that we are familiar with describe a property of a medium, such as the height of a rope or the pressure in a gas. But we also learned that in Einstein’s theory of relativity, there are special classes of fields, relativistic fields, that do not require a medium [--- or at least, if they have a medium, it's quite an extraordinary one. Nothing in the equations for the field tells you there has to be a medium... or what property of that medium the relativistic fields would describe.]
So at least for now, we’re going to think of certain relativistic fields as elementary physical objects in the universe, not as particular properties of some as-yet unknown medium. Whether that’s the viewpoint that physicists adopt over the long term is a question for… well… the long term. See here for an ongoing discussion about the issue.
There were two classes of relativistic fields that we considered, and we’re going to look at them more closely now. They satisfy (read this article if you want a review of these equations) either a Class 0 equation of motion with cw=c (where “c” is the universal speed limit, often called “the speed of light”)
- d2Z/dt2 – c2 d2Z/dx2 = 0
or a Class 1 equation of motion with cw=c
- d2Z/dt2 – c2 d2Z/dx2 = – (2 π μ)2 (Z-Z0)
As we saw in a previous article, it turns out that μ is the minimum frequency for waves in this field, and we’re going to rename it νmin for the remainder of this article.
Note the universal speed limit is often called “the speed of light” for the following reason: Waves with a Class 0 equation all travel at speed cw; and light (the general term meaning electromagnetic waves of all possible frequencies, not just visible light) traveling through empty space satisfies the relativistic Class 0 equation, so light waves (and the waves of any relativistic field satisfying the relativistic Class 0 equation) move at the speed c.
Furthermore, we saw in the same previous article that if a Class 1 field has a wave of amplitude A, frequency ν, wavelength λ and equilibrium value Z0, the equation of motion requires that the frequency and wavelength be related to the quantity μ = νmin that appears in the equation by the formula
- ν2 = (c/λ)2+ μ2 = (c/λ)2+ νmin2
This is a sort of Pythagorean formula; if you like, you can represent it as a triangle, as in Figure 1. The minimum frequency for any wave is just νmin, and setting ν = νmin (and thus λ → ∞) corresponds to squashing the triangle to a vertical line, see Figure 1 at bottom. Meanwhile, you can get the similar Class 0 relation by just setting μ = νmin to zero; then you can take the square root if you like to get
- ν = c/λ
That’s a triangle that’s gotten so squashed that it’s just a horizontal line, see Figure 1 at right. In this case the minimum frequency is zero; you can make the field oscillate as slowly as you like.
Notice there is no constraint on A. But that’s because we’re ignoring quantum mechanics. It’s time to study relativistic quantum fields.
Relativistic Quantum Fields
The real world is quantum mechanical, so in fact (as described in this article) the amplitude A cannot be just anything; it takes discrete values, and those values are proportional to the square root of n, a positive integer (or zero), which is the number of quanta of oscillation in the wave. And the energy stored in the wave is
- E = (n+1/2) h ν
where h is Planck’s constant, which always appears when quantum mechanics is important. In other words, the energy associated with each quantum of oscillation depends only on the frequency of oscillation of the wave, and equals
- E = h ν (for each additional quantum of oscillation)
This relation was first suggested, for light waves specifically, by Einstein, in 1905, in his proposed explanation of the photo-electric effect.
But let’s remember our Pythagorean relation for frequency and wavelength, the purple equation above for Class 1 waves. If we multiply it by h2, we find that for a quantum of a Class 1 field,
- E2 = (hν)2 = (hc/λ)2 + (hνmin)2
This formula looks familiar. We already know (see this article) that any object in Einstein’s theory of relativity must satisfy an equation relating its energy, momentum and mass of the form
- E2 = (pc)2 + (mc2)2
another Pythagorean relation. The minimum energy that object can have is just mc2, which resembles the statement that the minimum frequency a Class 1 wave can have is νmin. And so we may well be tempted to suggest that perhaps, for a quantum of a relativistic field,
- p c = h c / λ
- m c2 = h νmin
(The first equation first appeared in Louis de Broglie’s 1924 paper; notice the nearly 20 years between this formula and Einstein’s energy/frequency relation. Why did it take so long? I’m not sure.)
Does this idea make sense? Well, as we noted, Class 0 relativistic fields include electric fields, and their waves are electromagnetic waves, in short, light. The version of the violet formula above that we get for Class 0 quanta is the same as for Class 1 fields with μ = νmin set equal to zero — in other words, with m = 0. Taking the square root we get
- E = p c
which is Einstein’s relation for massless particles. The quanta of electromagnetic waves (including all forms of light: visible light, ultraviolet light, infrared light, radio waves, gamma rays, etc., which differ only in their frequency and thus in their energy per quantum) are indeed, once we apply the two red equations above, massless particles. These are photons.
From the second red equation above, we can finally see what the mass of a particle is. Each particle that has a mass is a quantum of a Class 1 field whose waves have a minimum frequency νmin; the minimum energy of a single quantum of such a wave is h times its frequency; and the mass of the particle is simply that minimum energy divided by c2.
- m = h νmin /c2
If we want to know where the particle’s mass comes from, we need to learn what determines νmin, and why there’s a minimum frequency in the first place. For particles such as electrons and quarks, the full story isn’t known, but the Higgs field plays an important role, as will be discussed in a later series of articles.
And so we conclude: The particles of nature are quanta of relativistic quantum fields. The massless ones are quanta of waves in fields that satisfy a Class 0 equation. The ones with mass correspond to fields with a Class 1 equation. There are many more details to investigate. But this fact is among the most fundamental properties of our world.
Do These Quanta Really Behave Like Particles?
When we think of particles, we think of specks of dust, or grains of sand. Quanta are not particles in this sense; they are waves that, for a given frequency, have minimum energy and amplitude. But they behave so much like particles that we could be forgiven (well, almost forgiven) for using the word “particle” in describing them. Let’s see why.
If you make a wave in water, and you allow this wave to pass over some rocks that lie just below the water’s surface, some of the wave will cross over the line and some of it will reflect backward, as in Figure 3. Exactly how much crosses over will depend on the shape of the rocks, how close they are to the surface, etc. But the point is that some of the wave is transmitted across the rocks, and some is reflected. Some of the energy of the wave keeps going in the same direction that it was going initially; some goes back the other way.
But if you send a single photon at a piece of glass that is somewhat reflective, that photon either will be transmitted through the glass or it will be reflected (Figure 4.). [More precisely, if you measure what the photon does, then you will find it was either reflected or transmitted; if you don't measure it, you can't say what happened. Welcome to the murkiness of quantum mechanics. We will move right along now.] A photon is a quantum; its energy cannot be divided up into a part that went through the glass and a part that reflected back, because then there would be less than a quantum on either side, which isn’t allowed. [Fine point: Glass cannot change the frequency of the photon, and that's why the energy can't be shared among two or more quanta with lower frequencies.] So the photon, though it is a wave, behaves quite like a particle in this case; it either bounces off the glass, or it doesn’t. Whether it bounces or not is something that quantum mechanics equations do not predict; they give only the probability for the bounce. But they do predict that no matter what happens, the photon travels as a unit, and retains its identity.
What is it like to have two photons? Well, that depends. For example, if the photons were emitted at widely different times and places one will see two individual quanta widely separated in space, and perhaps moving in different directions (Figure 5). They may also have different frequencies (not shown).
Alternatively, in the very special case where two photons were emitted together in perfect synchrony (as is done in lasers), then two photons look as shown in Figure 6, which shows that if we send this combination of two photons at a piece of glass, not two but three things can happen: both photons can pass through the glass together; both can be reflected from the glass; or one of them can be reflected and the other transmitted. Either zero, one or two photons bounce off the glass; there is no other option. In this respect, these quanta of light waves again behave like particles, a little like tiny baseballs, because if I threw two balls at a fence with some gaps between the fenceposts, I’d expect zero, one or two of the balls would pass through the gaps, and two, one or zero would bounce back. The option of 1.538 balls bouncing back just doesn’t exist.
That’s for photons, which, being massless, must always travel at light speed, with E = p c. What about massive particles like electrons? Electrons are quanta in the electron field, and like photons they can only be emitted, absorbed, reflected or transmitted as a unit. They have definite energy and momentum, with E2 = (pc)2 + (mec2)2, where me is the mass of the electron. The one thing that’s new about electrons compared to photons is that they move slower than light, and they can therefore be stationary. A sketch (more a useful image for intuition than a precise one — in quantum mechanics, thanks to the uncertainty principle, nothing can be truly stationary and also in a known position) of an essentially stationary electron is given in Figure 7. It is a wave of minimum frequency, which is obtained by taking its wavelength very long, essentially infinite; that is why the spatial shape of the wave in the figure shows no wiggles; it just oscillates in time. (See also Figure 2 of this article.)
So yes, it is really true. Quanta behave enough like particles that calling electrons, quarks, neutrinos, photons, gluons, W particles and Higgs particles by the name “particles” isn’t a disastrously misleading thing to do. But the word “quanta” would be better — because that’s really what they are.
Fermions and Bosons
Quanta of fermion fields and boson fields are different from one another, and the key reason is explained here. More qualitative and less technical comments about the consequences of this difference are discussed here.
What I Haven’t Explained:
Here is a list of some of the most important things I haven’t explained, and will try to describe to you at some later time.
- How fields can interact with each other, and how those interactions allow particles to interact with other fields and particles.
- What spin is.
- What makes the Higgs field different from other known fields
- How the Higgs field can become non-zero
- Why a non-zero Higgs field can change the masses of particles
- Why the electron simply cannot have a mass unless there is a Higgs field