The differing behavior of the forces is reflected in the nature of the bosons that convey them. Gravitation and electromagnetism are infinite in range—which is why our galaxy “feels” the gravitational pull of the Virgo Cluster of galaxies, and why we can see starlight coming from billions of light-years away—because the bosons that carry these two forces, known respectively as gravitons and photons, have zero mass. The weak nuclear force has a very short range because the particles that convey it, called weak bosons, are massive. The strong force is carried by particles called gluons; they are massless, but have the curious and quite beautiful property of increasing, not decreasing, in strength when the quarks between which they are exchanged move apart: A quark that starts to stray from its two companions soon finds itself hauled back by a gluon lattice, like a finger trapped in a woven Chinese finger-cuff. Consequently quarks in the contemporary universe remain bound up inside their protons and neutrons; no free quark has yet been observed, though they have been searched for in everything from accelerator collisions to moon dust to oysters (which filter seawater and so might catch stray quarks).
The four fundamental forces known to operate in nature today are here depicted in terms of characteristic interactions. In a typical electromagnetic interaction, a pair of electrons (symbolized e−) exchange a photon. In the weak force interaction portrayed here, a neutron (n) decays into a proton (p) via the exchange of a weak boson; the event also converts a positron (e+) into a neutrino (v). In a strong interaction, quarks (q) exchange a gluon. Gravitation involves the exchange of a graviton between any two massive particles (m).
The fermions that constitute matter, though notoriously numerous and varied, can all be classified as either quarks, which respond to the strong force, or leptons, which do not. Leptons are light particles; their ranks include the electrons that orbit atomic nuclei. Quarks are the building blocks of protons and neutrons: Three quarks make a nucleon.* There are thought to be six varieties each of leptons and quarks. Neither quarks nor leptons show any sign of having an internal structure, though their anatomy has been probed on scales down to some 10−18 meter. This is to say that if a single atom were enlarged to the dimensions of the earth, any subcomponents of quarks and leptons would have to be smaller than a grapefruit to have escaped detection. So quarks and leptons are the bedrock particles of matter, so far as we know.
The Building Blocks of Matter
Particle Description Examples
Leptons “Dimensionless” (i.e., radius < 10−35 meter); do not participate in the strong force. Electron
Muon
Neutrino
Quarks Small (< 10−18 meter) but finite in size; do participate in the strong force. Hadrons (three quarks) Mesons (two quarks)
Trios of quarks are thought to compose the nucleons—protons and neutrons—that in turn constitute the nuclei of atoms. According to this model, a proton consists of two “up” quarks, each of which carries an electrical charge of +⅔, and one “down” quark, which has a charge of -⅓ the total charge of the proton therefore is 4/3 - 1/3 = + 1. A neutron consists of two down quarks and one up quark; consequently its charge equals 0.
Every fundamental—meaning simple—event in the universe can in principle be interpreted by means of the standard model. When a child looks at a star, photons of starlight strike electrons in the outer atoms of the receptors of the child’s retina, setting off further electron interactions that convey the image to the brain; all this is the work of electromagnetism. The nuclear processes that produced the starlight are generated by the strong and weak nuclear forces at work inside the star. And gravitation is the force that holds the star together and keeps the child’s feet (if only intermittently) on the ground.
Electromagnetic energy is generated by natural processes across a wide range of wavelengths, including gamma rays and X rays from gas falling into black holes, light from stars, microwaves from the cosmic background radiation, and radio from interstellar clouds.
The scientific accounts of how the various particles of matter behave under the influence of three of the four forces are known as relativistic quantum field theories. They are so called because they incorporate both the quantum precept and the special theory of relativity, in order to take into account such effects as increases in the mass of particles traveling at close to the velocity of light. Electromagnetism is described, with exquisite accuracy, by the theory of quantum electrodynamics, or QED. The strong force is described by quantum chromodynamics, or QCD. (The “chromo” comes from a quantum number, whimsically called “color,” that plays a role for quarks comparable to that played by electrical charge in the affairs of electrons.) The weak force, as we will see, has recently come under the purview of the “electroweak” unified theory.
Gravitation remains the odd man out. Its workings are still described by Einstein’s general theory of relativity, which is a classical theory, meaning that it does not incorporate the quantum principle. This does not cause problems under most conditions, but relativity breaks down when it comes to extremely intense gravitational fields, like those inside a black hole or in the universe at the very beginning of its expansion. There the curvature of space goes to infinity, at which point the theory tips its hat and makes a graceful exit. There was, by the late 1980s, still no quantum theory of gravitation with which to supplement general relativity. One reason for this is that gravity is weak. Individual subatomic particles normally are so little influenced by the gravitational force exerted by their colleagues that gravity can be ignored. Another reason is that gravitational interactions are interpreted, through Einstein’s general theory of relativity, as resulting from the geometry of space itself. The “gravitons” thought to convey gravitation must therefore dictate the very shape of space, and for a theory to elucidate how they manage that is no simple matter.
Particle physics today is a house divided, and though the standard model gets results, few imagine that it represents the last word on the subject. The model is a crazy quilt, not a mándala. To fire it up on all cylinders requires inputting some seventeen separate parameters, numbers the values of which have been determined experimentally but whose fundamental significance is not yet understood. We know, for instance, that the electrical charge carried by an electron is equal to 1.6021892 × 10−19 coulomb, and that the mass of the proton is 938.3 MeV, equal to 0.9986 the mass of the neutron, but nobody knows why these numbers are as they are and not otherwise. The roots of discontent with the standard model were described this way by Leon Lederman, the director of the Fermilab particle accelerator in Illinois:
The trouble we’re in now is that the standard model is very elegant, it’s very powerful, it explains so much—but it’s not complete. It has some flaws, and one of its greatest flaws is aesthetic. It’s too complicated. It has too many arbitrary parameters. We don’t really see the creator twiddling seventeen knobs to set seventeen parameters to create the universe as we know it. The picture is not beautiful, and that drive for beauty and simplicity and symmetry has been an unfailing guidepost to how to go in physics.9
So it was that physicists late in the century were still searching for a simpler and more efficient account of the fundamental interactions. The object of their quest went by the name “unified” theory, by which they usually meant a single theory that would account for two or more of the forces currently handled by separate theories. They were guided, to be sure, by experimental data and by the challenges immediately at hand—the theorist resembles, as Einstein said, an “unscrupulous opportunist,” more often trying to find a specific solution to an immediate problem than to write a grand explication of everything. But they were guided as well, as Lederman mentions, by the hope that their accounts of nature could more nearly approach the elegant simplicity and superlative creativity of nature herself.
*The “spin” referred to here is a familiar, mechanical spin, though it is quantized, and is measured in terms of h, the quantum of action.
*The medicin
e balls are purely repulsive, as in the interaction between two electrons or other fermions of like charge. For attractive forces (as between a proton and an electron), imagine that the bosons are elastic bands that stretch when the skaters move apart, drawing them together. For the exclusion principle, let each skater wear a hoopskirt that forbids their colliding…. And that is quite enough of that.
*The name “quark” was conferred by Murray Gell-Mann, the Caltech physicist who came up with the idea. It comes from a line in James Joyce’s Finnegans Wake, “Three quarks for Muster Mark!” George Zweig, a physicist at Caltech who arrived at the same idea independently, called the entities “aces,” a term that lost out to Gell-Mann’s, perhaps because there are four aces, not three, in a deck of cards.
16
RUMORS OF PERFECTION
Spirit of BEAUTY, that dost consecrate
With thine own hues all thou dost shine
upon
Of human thought or form, where art
thou gone?
Why dost thou pass away, and leave our
state,
This dim vast vale of tears, vacant and
desolate?
—Shelley, “Hymn to Intellectual
Beauty”
The Universe is built on a plan the profound symmetry of which is somehow present in the inner structure of our intellect.
—Paul Valéry
Theoretical physicists, like artists (one is tempted to say, like other artists) are guided in their work by aesthetic as well as rational concerns. “To make any science, something else than pure logic is necessary,” wrote Poincaré, who identified this additional element as intuition, involving “the feeling of mathematical beauty, of the harmony of numbers and forms and of geometric elegance.”1 Heisenberg spoke of “the simplicity and beauty of the mathematical schemes which nature presents us. You must have felt this too,” he told Einstein, “the almost frightening simplicity and wholeness of the relationship which nature suddenly spreads out before us.”2 Paul Dirac, the English theoretical physicist whose relativistic, quantum mechanical description of the electron ranks with the masterpieces of Einstein and Bohr, went so far as to maintain that “it is more important to have beauty in one’s equations than to have them fit experiment.”3*
Aesthetics are notoriously subjective, and the statement that physicists seek beauty in their theories is meaningful only if we can define beauty. Fortunately this can be done, to some extent, for scientific aesthetics are illuminated by the central sun of symmetry.
Symmetry is a venerable and all but bottomless concept, with many implications in both science and art; long after the Chinese-American physicist Chen Ning Yang had won a Nobel Prize for his work in developing a symmetry theory of fields, he was still pointing out that “we do not yet comprehend the full scope of the concept of symmetry” (Yang’s italics).4 In Greek, the word means “the same measure” (sym, meaning “together,” as in symphony, a bringing together of sounds, and metron, for “measurement”); its etymology thus informs us that symmetry involves the repetition of a measurable quantity. But by symmetry the Greeks also meant “due proportion,” suggesting that the repetition involved ought to be harmonious and pleasing; this suggests that a symmetrical relationship is to be judged by a higher aesthetic criterion, an idea to which I will return at the end of this chapter. In twentieth-century science, however, the former aspect of the old definition is emphasized; symmetry is said to exist when a measurable quantity remains invariant (meaning unchanging) under a transformation (meaning an alteration). Because this definition is the most relevant to the subject at hand, I will employ it in discussing all aspects of symmetry, including those that were in general use before there was such a thing as science.
Most of us were first introduced to symmetry through its visual manifestations in geometry and art. When we say, for instance, that a sphere is rotationally symmetrical, we indicate that it possesses a characteristic—in this case, its circular silhouette—that remains invariant throughout the transformation introduced by rotating it. The sphere can be rotated on any axis and to any degree without changing its silhouette, which makes it more symmetrical than, say, a cylinder, which enjoys a similar symmetry only when rotated on its long axis; if rotated on its short axis, the cylinder shrinks to a circle. Translational symmetries, like those found in palm fronds and building facades like that of the Doge’s Palace in Venice, occur when a shape remains invariant when moved (“translated”) a given distance along one axis. (See previous page.)
Symmetries are commonplace in sculpture, beginning with the human nude, which is (approximately) bilaterally symmetrical when viewed from the front or back, and in architecture, as in the cross-shaped floor plans of medieval cathedrals, and they turn up elsewhere in everything from weaving to square dancing. There are many symmetries in music. Bach, in the following passage from the Toccata and Fugue in E Minor, moves little tentlike trios of notes up and down the staff. Except for the occasional difference in a note here and there, the construct is translationally symmetrical: If we were to peel off any one trio and lay it over another, it would fit perfectly:
The first two bars of Claude Debussy’s Deux Arabesques are bilaterally symmetrical both within themselves and relative to each other: The sheet music can be folded vertically at the bar, or midway within each bar, and the notes will still fit atop one another:
Beneath these visible and audible manifestations of symmetry lie deeper mathematical invariances. The spiral patterns found inside the chambered nautilus and on the faces of sunflowers, for instance, are approximated by the Fibonacci series, an arithmetic operation in which each succeeding unit is equal to the total of the preceding two (1, 1, 2, 3, 5, 8 …). The ratio created by dividing any number in such a series by the number that follows it approaches the value 0.618* (see page 306). This, not incidentally, is the formula of the “golden section,” a geometrical proportion that shows up in the Parthenon, the Mona Lisa, and Botticelli’s The Birth of Venus, and is the basis of the octave employed in Western music since the time of Bach. All the fecund diversity of this particular symmetry, expressed in myriad ways from seashells and pine cones to the Well-Tempered Clavier, therefore derives from a single invariance, that of the Fibonacci series. The realization that one abstract symmetry could have such diverse and fruitful manifestations occasioned delight among Renaissance scholars, who cited it as evidence of the efficacy of mathematics and of the subtlety of God’s design. Yet it was only the beginning. Many other abstract symmetries have since been identified in nature—some intact and some “broken,” or flawed—and their effects appear to extend to the very bedrock foundations of matter and energy.
The Fibonacci series, represented in the abstract (above) is embodied in the architecture of the chambered nautilus (below).
Which brings us back to science. When mathematicians in the early twentieth century began to look more closely at the concept of symmetry, they realized that the laws that science finds in nature are expressions of invariances—and may, therefore, be based on symmetries. This first became evident with regard to the conservation laws: The laws of thermodynamics, for instance, identify a quantity (energy) that remains invariant under a transformation (work). The German mathematician Emmy Noether demonstrated in 1918 that every conservation law implies the existence of a symmetry. The same evidently is true of the other laws as well. As the Hungarian physicist Eugene Wigner put it, “Laws of nature could not exist without principles of invariance,”6 and invariance, keep in mind, is the signature of symmetry.
If natural laws express symmetries, then one ought to be able to search for previously unknown laws by looking for symmetrical relationships (invariances) in nature. Einstein absorbed this lesson in his bones, and employed symmetry as a lamp to guide his way in the creation of new theories. In special relativity (which, we recall, he originally called “invariance theory”), he employed the Lorentz transformations to maintain the invariance of Maxwell’s field equati
ons for observers in motion; in the general theory of relativity he did much the same thing for observers in strong gravitational fields. As his friend Wigner remarked in 1949, speaking at a Princeton celebration in Einstein’s honor, “It is now natural for us to try to derive the laws of nature and to test their validity by means of the laws of invariance, rather than to derive the laws of invariance from what we believe to be the laws of nature.”7
If the symmetry concept was powerful in relativity, it proved to be even more so when applied to the quantum physics of particles and fields. We can appreciate why this would be the case if we consider that subatomic particles of a given variety are indistinguishable from one another. All protons are identical. So are all neutrons and electrons. The same is true of fields; if we have two electromagnetic or gravitational fields with the same quantum numbers, and a trickster switches them while our backs are turned, we can never tell that we have been tricked, for the fields are identical. Inasmuch as identity is a form of invariance, we can, therefore, choose to regard the individual electron or photon as representative of a symmetry group that embraces all its fellow particles of the same species. Moreover, we can look for larger symmetries that might link the various groups involved—revealing, say, a previously undiscerned invariance linking photons and electrons.
Coming of Age in the Milky Way Page 31