10

The Origin of Mass

Knowing how to calculate something is not the same as understanding it. Having a computer calculate the origin of mass for us may be convincing, but it is not satisfying. Fortunately, we can understand it too.

 

 

HAVING A COMPUTER SPIT OUT ANSWERS after gigantic and totally opaque calculation does not satisfy our hunger for understanding. What would?

Paul Dirac was famously taciturn, but when he spoke, what he said was often profound. He once said, “I feel I understand an equation, when I can anticipate the behavior of its solutions without actually solving it.”

What’s the value of such understanding?

“Solving” equations is just one tool—an imperfect one—for working with them. The calculations we discussed in the preceding chapter are an instructive example. They show conclusively that the equations for quark and gluon Grid accurately account for the masses of protons, neutrons, and other hadrons. They also show that those equations keep quarks and gluons hidden. (You can interpret the nonappearance of isolated quarks or gluons as a calculation of their mass, when you include their virtual particle clouds—the answer is infinity!)

Those are glorious results, won after heroic efforts by human and machine. But the need for heroic efforts is one of the biggest drawbacks of “solving” equations. We don’t want to tie up expensive computer resources and wait a long time for answers every time we ask a slightly different question. Even more important, we don’t want to tie up expensive computer resources and wait a very long time when we ask more complicated questions. For example, we’d like to be able to predict the masses not only of single protons and neutrons but also of systems containing several protons and neutrons—atomic nuclei. In principle we have the equations to do it, but solving them is impractical. For that matter, we have equations adequate to answer any question of chemistry, in principle. But that hasn’t put chemists out of business or replaced them with computers, because in practice the calculations are too hard.

In both nuclear physics and chemistry, we are happy to sacrifice extreme precision for ease of use and flexibility. Rather than brutally “solving” the equations by crunching numbers, we make simplified models and find rules of thumb that can give us practical guidance in complicated situations. These models and rules of thumb can grow out of experience in solving the equations, and they can be checked by solving the equations when that’s practical, but they have a life of their own. This reminds me of the distinction between graduate students and professors: a graduate student knows everything about nothing, a professor knows nothing about everything. Solving the equations is what graduate students do, understanding them is what professors do.

We are as far as we can be from understanding, when solving the equations reveals behavior that’s totally unexpected and appears miraculous. The computers have given us mass—and not just any mass, but our mass, the mass of the protons and neutrons we’re made from—from quarks and gluons that are themselves massless (or nearly so). The equations of QCD output Mass Without Mass. It sounds suspiciously like something for nothing. How did it happen?

Fortunately, it’s possible get a rough, professor-like understanding of that apparent miracle. We just have to put together three ideas we’ve already discussed separately. Let’s briefly recall and assemble them.

First Idea: Blossoming Storms

The color charge of a quark creates a disturbance in the Grid—specifically, in the gluon fields—that grows with distance. It’s like a strange storm cloud that blossoms from a wispy center into an ominous thunderhead. Disturbing the fields means putting them into a state of higher energy. If you keep disturbing the fields over an infinite volume, the energy cost will be infinite. Even Exxon Mobil wouldn’t claim that Nature has the resources to pay that price.³⁹ So isolated quarks can’t exist.

Second Idea: Costly Cancellations

The blossoming storm can be short-circuited by bringing an antiquark, with the opposite color charge, close to the quark. Then the two sources of disturbance cancel, and calm is restored.

If the antiquark were located accurately right on top of the quark, the cancellation would be complete. That would produce the minimal possible disturbance in the gluon fields—namely, none. But there’s another price to be paid for that accurate cancellation. It comes from the quantum-mechanical nature of quarks and antiquarks.

According the Heisenberg uncertainty principle, in order to have accurate knowledge of the position of a particle, you must let that particle have a wide spread in momentum. In particular, you must allow that the particle may have large momentum. But large momentum means large energy. And the more accurately you fix the position of the particle (“localize” it, in the jargon), the more energy it costs.

(It’s also possible to cancel the color charge of a quark using the complementary color charges of two other quarks. This is what happens for baryons, including the proton and neutron—as opposed to mesons, based on quark-antiquark. The principle is the same.)

Third Idea: Einstein’s Second Law

So there are two competing effects, which pull in opposite directions. To cancel the field disturbance accurately and minimize that energy cost, Nature wants to localize the antiquark on the quark. But to minimize the quantum-mechanical cost of localizing a position, Nature wants to let the antiquark wander a bit.

Nature compromises. She finds ways of striking a balance between the demands of the gluon fields that don’t want to be disturbed, and those of the quarks and antiquarks that want to roam free. (You might think of a family gathering where the gluon fields are the old curmudgeons, the quarks and antiquarks are the rambunctious kids, and Nature is the responsible adult.)

As with any compromise, the result is—well, a compromise. Nature can’t make both energies zero simultaneously. So the total energy won’t be zero.

Actually there can be different accommodations that are more-or-less stable. Each will have its own nonzero energy E. And thus, according to Einstein’s second law, each will have its own mass, m = E/c².

And that is the origin of mass. (Or at least of 95% of the mass of ordinary matter.)

Scholium

Such a climax deserves some commentary. Indeed, it deserves a scholium, which is simply Latin for “commentary” but sounds much more impressive.

1. Nothing in this account of the origin of mass referred to, or depended on, the quarks and gluons having any mass. We really do get Mass Without Mass.

2. It wouldn’t work without quantum mechanics. You cannot understand where your mass comes from if you don’t take quantum mechanics into account. In other words, without quantum mechanics you’re doomed to be a lightweight.

3. A similar mechanism, though much simpler, works in atoms. Negatively charged electrons feel an attractive electric force from the positively charged nucleus. From that point of view, they’d like to snuggle right on top of it. Electrons are wavicles, though, and that inhibits them. The result, again, is a series of possible compromise solutions. These are what we observe as the energy levels of the atom.

4. The title of Einstein’s original paper was a question, and a challenge:

 

Does the Inertia of a Body Depend on Its Energy Content?

 

If the body is a human body, whose mass overwhelmingly arises from the protons and neutrons it contains, the answer is now clear and decisive. The inertia of that body, with 95% accuracy, is its energy content.