Physics Has Demoted Mass

You’re sitting here, reading this article. Maybe it’s a hard copy, or an e-book on a tablet computer or e-reader. It doesn’t matter. Whatever you’re reading it on, we can be reasonably sure it’s made of some kind of stuff: paper, card, plastic, perhaps containing tiny metal electronic things on printed circuit boards. Whatever it is, we call it matter or material substance. It has a characteristic property that we call solidity. It has mass.

But what is matter, exactly? Imagine a cube of ice, measuring a little over one inch (or 2.7 centimeters) in length. Imagine holding this cube of ice in the palm of your hand. It is cold, and a little slippery. It weighs hardly anything at all, yet we know it weighs something.

Let’s make our question a little more focused. What is this cube of ice made of? And, an important secondary question: What is responsible for its mass?

To understand what a cube of ice is made of, we need to draw on the learning acquired by the chemists. Building on a long tradition established by the alchemists, these scientists distinguished between different chemical elements, such as hydrogen, carbon, and oxygen. Research on the relative weights of these elements and the combining volumes of gases led John Dalton and Louis Gay-Lussac to the conclusion that different chemical elements consist of atoms with different weights which combine according to a set of rules involving whole numbers of atoms.

The mystery of the combining volumes of hydrogen and oxygen gas to produce water was resolved when it was realized that hydrogen and oxygen are both diatomic gases, H₂ and O₂. Water is then a compound consisting of two hydrogen atoms and one oxygen atom, H₂O.

This partly answers our first question. Our cube of ice consists of molecules of H₂O organized in a regular array. We can also make a start on our second question. Avogadro’s law states that a mole of chemical substance will contain about 6 × 10²³ discrete “particles.” Now, we can interpret a mole of substance simply as its molecular weight scaled up to gram quantities. Hydrogen (in the form of H₂) has a relative molecular weight of 2, implying that each hydrogen atom has a relative atomic weight of 1. Oxygen (O₂) has a relative molecular weight of 32, implying that each oxygen atom has a relative atomic weight of 16. Water (H₂O) therefore has a relative molecular weight of 2 × 1 + 16 = 18.

About 99 percent of the masses of the proton and neutron seem to be unaccounted for.

It so happens that our cube of ice weighs about 18 grams, which means that it represents a mole of water, more or less. According to Avogadro’s law it must therefore contain about 6 × 10²³ molecules of H₂O. This would appear to provide a definitive answer to our second question. The mass of the cube of ice derives from the mass of the hydrogen and oxygen atoms present in 6 × 10²³ molecules of H₂O.

But, of course, we can go further. We learned from J.J. Thomson, Ernest Rutherford, and Niels Bohr and many other physicists in the early 20th century that all atoms consist of a heavy, central nucleus surrounded by light, orbiting electrons. We subsequently learned that the central nucleus consists of protons and neutrons. The number of protons in the nucleus determines the chemical identity of the element: A hydrogen atom has one proton, an oxygen atom has eight (this is called the atomic number). But the total mass or weight of the nucleus is determined by the total number of protons and neutrons in the nucleus.

Hydrogen still has only one (its nucleus consists of a single proton—no neutrons). The most common isotope of oxygen has—guess what?—16 (eight protons and eight neutrons). It’s obviously no coincidence that these proton and neutron counts are the same as the relative atomic weights I quoted above.

If we ignore the light electrons, then we would be tempted to claim that the mass of the cube of ice resides in all the protons and neutrons in the nuclei of its hydrogen and oxygen atoms. Each molecule of H₂O contributes 10 protons and eight neutrons, so if there are 6 × 10²³ molecules in the cube and we ignore the small difference in mass between a proton and a neutron, we conclude that the cube contains in total about 18 times this figure, or 108 × 10²³ protons and neutrons.

So far, so good. But we’re not quite done yet. We now know that protons and neutrons are not elementary particles. They consist of quarks. A proton contains two up quarks and a down quark, a neutron two down quarks and an up quark. And the color force binding the quarks together inside these larger particles is carried by massless gluons.

Okay, so surely we just keep going. If once again we approximate the masses of the up and down quarks as the same we just multiply by three and turn 108 × 10²³ protons and neutrons into 324 × 10²³ up and down quarks. We conclude that this is where all the mass resides. Yes?

The naked quark is acutely embarrassed, and it quickly dresses itself with a covering of gluons.

No. This is where our naïve atomic preconceptions unravel. We can look up the masses of the up and down quarks on the Particle Data Group website. The up and down quarks are so light that their masses can’t be measured precisely and only ranges are quoted. The following are all reported in units of MeV/c². In these units the mass of the up quark is given as 2.3 with a range from 1.8 to 3.0. The down quark is a little heavier, 4.8, with a range from 4.5 to 5.3. Compare these with the mass of the electron, about 0.51 measured in the same units.

Now comes the shock. In the same units of MeV/c² the proton mass is 938.3, the neutron 939.6. The combination of two up quarks and a down quark gives us only 9.4, or just 1 percent of the mass of the proton. The combination of two down quarks and an up quark gives us only 11.9, or just 1.3 percent of the mass of the neutron. About 99 percent of the masses of the proton and neutron seem to be unaccounted for. What’s gone wrong?

To answer this question, we need to recognize what we’re dealing with. Quarks are not self-contained “particles” of the kind that the Greeks or the mechanical philosophers might have imagined. They are quantum wave-particles; fundamental vibrations or fluctuations of elementary quantum fields. The up and down quarks are only a few times heavier than the electron, and we’ve demonstrated the electron’s wave-particle nature in countless laboratory experiments. We need to prepare ourselves for some odd, if not downright bizarre behavior.

And let’s not forget the massless gluons. Or special relativity, and E = mc². Or the difference between “bare” and “dressed” mass. And, last but not least, let’s not forget the role of the Higgs field in the “origin” of the mass of all elementary particles. To try to understand what’s going on inside a proton or neutron we need to reach for quantum chromodynamics, the quantum field theory of the color force between quarks.

Quarks and gluons possess color “charge.” Just what is this, exactly? We have no way of really knowing. We do know that color is a property of quarks and gluons and there are three types, which physicists have chosen to call red, green, and blue. But, just as nobody has ever “seen” an isolated quark or gluon, so more or less by definition nobody has ever seen a naked color charge. In fact, quantum chromodynamics (QCD) suggests that if a color charge could be exposed like this it would have a near-infinite energy. Aristotle’s maxim was that “nature abhors a vacuum.” Today we might say: “nature abhors a naked color charge.”

So, what would happen if we could somehow create an isolated quark with a naked color charge? Its energy would go up through the roof, more than enough to conjure virtual gluons out of “empty” space. Just as the electron moving through its own self-generated electromagnetic field gathers a covering of virtual photons, so the exposed quark gathers a covering of virtual gluons. Unlike photons, the gluons themselves carry color charge and they are able to reduce the energy by, in part, masking the exposed color charge. Think of it this way: The naked quark is acutely embarrassed, and it quickly dresses itself with a covering of gluons.

This isn’t enough, however. The energy is high enough to produce not only virtual particles (like a kind of background noise or hiss), but elementary particles, too. In the scramble to cover the exposed color charge, an anti-quark is produced which pairs with the naked quark to form a meson. A quark is never—but never—seen without a chaperone.

But this still doesn’t do it. To cover the color charge completely we would need to put the anti-quark in precisely the same place at precisely the same time as the quark. Heisenberg’s uncertainty principle won’t let nature pin down the quark and anti-quark in this way. Remember that a precise position implies an infinite momentum, and a precise rate of change of energy with time implies an infinite energy. Nature has no choice but to settle for a compromise. It can’t cover the color charge completely but it can mask it with the anti-quark and the virtual gluons. The energy is at least reduced to a manageable level.

As we worked our way ever inward we lost sight of matter completely. Matter lost its tangibility.

This kind of thing also goes on inside the proton and neutron. Within the confines of their host particles, the three quarks rattle around relatively freely. But, once again, their color charges must be covered, or at least the energy of the exposed charges must be reduced. Each quark produces a blizzard of virtual gluons that pass back and forth between them, together with quark–anti-quark pairs. Physicists sometimes call the three quarks that make up a proton or a neutron “valence” quarks, as there’s enough energy inside these particles for a further sea of quark–anti-quark pairs to form. The valence quarks are not the only quarks inside these particles.

What this means is that the mass of the proton and neutron can be traced largely to the energy of the gluons and the sea of quark–anti-quark pairs that are conjured from the color field.

How do we know? Well, it must be admitted that it is actually really rather difficult to perform calculations using QCD. The color force is extremely strong, and the corresponding energies of color-force interactions are therefore very high. Remember that the gluons also carry color charge, so everything interacts with everything else. Virtually anything can happen, and keeping track of all the possible virtual and elementary-particle permutations is very demanding.

This means that although the equations of QCD can be written down in a relatively straightforward manner, they cannot be solved analytically, on paper. Also, the mathematical sleight-of-hand used so successfully in QED no longer applies—because the energies of the interactions are so high we can’t apply the techniques of renormalization. Physicists have had no choice but to solve the equations on a computer instead.

Considerable progress was made with a version of QCD called “QCD-lite.” This version considered only massless gluons and up and down quarks, and further assumed that the quarks themselves are also massless (so, literally, “lite”). Calculations based on these approximations yielded a proton mass that was found to be just 10 percent lighter than the measured value.

Let’s stop to think about that for a bit. A simplified version of QCD in which we assume that no particles have mass to start with nevertheless predicts a mass for the proton that is 90 percent right. The conclusion is quite startling. Most of the mass of the proton comes from the energy of the interactions of its constituent quarks and gluons.

John Wheeler used the phrase “mass without mass” to describe the effects of superpositions of gravitational waves which could concentrate and localize energy such that a black hole is created. If this were to happen, it would mean that a black hole—the ultimate manifestation of super-high-density matter—had been created not from the matter in a collapsing star but from fluctuations in spacetime. What Wheeler really meant was that this would be a case of creating a black hole (mass) from gravitational energy.

But Wheeler’s phrase is more than appropriate here. Frank Wilczek, one of the architects of QCD, used it in connection with his discussion of the results of the QCD-lite calculations. If much of the mass of a proton and neutron comes from the energy of interactions taking place inside these particles, then this is indeed “mass without mass,” meaning that we get the behavior we tend to ascribe to mass without the need for mass as a property.

Does this sound familiar? Recall that in Einstein’s seminal addendum to his 1905 paper on special relativity the equation he derived is actually m = E/c². This is the great insight (not E = mc²). And Einstein was surely prescient when he wrote: “the mass of a body is a measure of its energy content.”¹ Indeed, it is. In his book The Lightness of Being, Wilczek wrote:²

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 percent accuracy, is its energy content.

In the fission of a U-235 nucleus, some of the energy of the color fields inside its protons and neutrons is released, with potentially explosive consequences. In the proton–proton chain involving the fusion of four protons, the conversion of two up quarks into two down quarks, forming two neutrons in the process, results in the release of a little excess energy from its color fields. Mass does not convert to energy. Energy is instead passed from one kind of quantum field to another.

Where does this leave us? We’ve certainly come a long way since the ancient Greek atomists speculated about the nature of material substance, 2,500 years ago. But for much of this time we’ve held to the conviction that matter is a fundamental part of our physical universe. We’ve been convinced that it is matter that has energy. And, although matter may be reducible to microscopic constituents, for a long time we believed that these would still be recognizable as matter—they would still possess the primary quality of mass.

Modern physics teaches us something rather different, and deeply counter-intuitive. As we worked our way ever inward—matter into atoms, atoms into sub-atomic particles, sub-atomic particles into quantum fields and forces—we lost sight of matter completely. Matter lost its tangibility. It lost its primacy as mass became a secondary quality, the result of interactions between intangible quantum fields. What we recognize as mass is a behavior of these quantum fields; it is not a property that belongs or is necessarily intrinsic to them.

Despite the fact that our physical world is filled with hard and heavy things, it is instead the energy of quantum fields that reigns supreme. Mass becomes simply a physical manifestation of that energy, rather than the other way around.

This is conceptually quite shocking, but at the same time extraordinarily appealing. The great unifying feature of the universe is the energy of quantum fields, not hard, impenetrable atoms. Perhaps this is not quite the dream that philosophers might have held fast to, but a dream nevertheless.

Jim Baggott is a freelance science writer. He was a lecturer in chemistry at the University of Reading but left to work with Shell International Petroleum Company and then as an independent business consultant and trainer. His many books include Origins: The Scientific Story of Creation, Higgs: The Invention and Discovery of the ‘God Particle’, A Quantum Story: A History in 40 Moments, and A Beginner’s Guide to Reality.

References

1. Einstein, A. Does the inertia of a body depend upon its energy-content? Annalen der Physik 18 (1905).

2. Wilczek, F. The Lightness of Being Basic Books, New York, NY (2008).

Photocollage credits: Physicsworld.com; Thatree Thitivongvaroon / Getty Images

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