by Brian Greene
You can thus imagine the surprise and skepticism when, in the mid-1980s, rumors started racing through the physics community that there had been a major theoretical breakthrough toward unification with an approach called string theory.
Figure 4.1 The Planck length, where gravity and quantum mechanics confront each other, is some 100 billion billion times smaller than any domain that’s been explored experimentally. Reading across the chart, each of the equally spaced tick marks represents a decrease in size by a factor of 1,000; this allows the chart to fit on a page but visually downplays the huge range of scales. For a better feel, note that if an atom were magnified to be as large as the observable universe, the same magnification would make the Planck length the size of an average tree.
String Theory
Although string theory has an intimidating reputation, its basic idea is easy to grasp. We’ve seen that the standard view, prior to string theory, envisions nature’s fundamental ingredients as point particles—dots with no internal structure—governed by the equations of quantum field theory. With each distinct species of particle is associated a distinct species of field. String theory challenges this picture by suggesting that the particles are not dots. Instead, the theory proposes that they’re tiny, stringlike, vibrating filaments, as in Figure 4.2. Look closely enough at any particle previously deemed elementary and the theory claims you’ll find a minuscule vibrating string. Look deep inside an electron, and you’d find a string; look deep inside a quark, and you’d find a string.
With even more precise observation, the theory argues, you’d notice that the strings within different kinds of particles are identical, the leitmotif of string unification, but vibrate in different patterns. An electron is less massive than a quark, which according to string theory means that the electron’s string vibrates less energetically than the quark’s string (reflecting again the equivalence of energy and mass embodied in E = mc2). The electron also has an electric charge whose magnitude exceeds that of a quark, and this difference translates into other, finer differences between the string vibrational patterns associated with each. Much as different vibrational patterns of strings on a guitar produce different musical notes, different vibrational patterns of the filaments in string theory produce different particle properties.
Figure 4.2 String theory’s proposal for the nature of physics at the Planck scale envisions that the fundamental constituents of matter are string-like filaments. Because of the limited resolving power of our equipment, the strings appear as dots.
In fact, the theory encourages us to think of a vibrating string not merely as dictating the properties of its host particle but rather as being the particle. Because of the string’s infinitesimal size, on the order of the Planck length—10–33 centimeters—even today’s most refined experiments cannot resolve the string’s extended structure. The Large Hadron Collider, which slams particles together with energies just beyond 10 trillion times that embodied by a single proton at rest, can probe to scales of about 10–19 centimeters; that’s a millionth of a billionth the width of a strand of hair, but still orders of magnitude too large to resolve phenomena at the Planck length. And so, just as earth would look dotlike if viewed from Pluto, strings would appear dotlike when studied even with the most advanced particle accelerator in the world. Nevertheless, according to string theory, particles are strings.
In a nutshell, that’s string theory.
Strings, Dots, and Quantum Gravity
String theory has many other essential features, and the developments it has undergone since it was first proposed have greatly enriched the bare-bones description I’ve so far given. In the rest of this chapter (as well as Chapters 5, 6, and 9), we will encounter some of the most pivotal advances, but I want to stress here three overarching points.
First, when a physicist proposes a model of nature using quantum field theory, he or she needs to choose the particular fields the theory will contain. The choice is guided by experimental constraints (each known particle species dictates the inclusion of an associated quantum field) as well as theoretical concerns (hypothetical particles and their associated fields, like the inflaton and Higgs fields, are invoked to address open problems or puzzling issues). The Standard Model is the prime example. Considered the crowning achievement of twentieth-century particle physics because of its capacity to accurately describe the wealth of data collected by particle accelerators worldwide, the Standard Model is a quantum field theory containing fifty-seven distinct quantum fields (the fields corresponding to the electron, the neutrino, the photon, and the various kinds of quarks—the up-quark, the down-quark, the charm-quark, and so on). Undeniably, the Standard Model is tremendously successful, but many physicists feel that a truly fundamental understanding would not require such an ungainly assortment of ingredients.
An exciting feature of string theory is that the particles emerge from the theory itself: a distinct species of particle arises from each distinct string vibrational pattern. And since the vibrational pattern determines the properties of the corresponding particle, if you understood the theory well enough to delineate all vibrational patterns, you’d be able to explain all properties of all particles. The potential and the promise, then, is that string theory will transcend quantum field theory by deriving all particle properties mathematically. Not only would this unify everything under the umbrella of vibrating strings, it would also establish that future “surprises”—such as the discovery of currently unknown particle species—are built into string theory from the outset and so would be accessible, in principle, to sufficiently industrious calculation. String theory doesn’t build piecemeal toward an ever more complete description of nature. It seeks a complete description from the get-go.
The second point is that among the string’s possible vibrations, there is one with just the right properties to be the quantum particle of the gravitational field. Even though pre–string theoretic attempts to merge gravity and quantum mechanics were unsuccessful, research did reveal the properties that any hypothesized particle associated with the quantum gravitational field—dubbed the graviton—would necessarily possess. The studies concluded that the graviton must be massless and chargeless, and must have the quantum mechanical property known as spin-2. (Very roughly, the graviton should spin like a top, twice as fast as the spin of a photon.)7 Wonderfully, early string theorists—John Schwarz, Joël Scherk, and, independently, Tamiaki Yoneya—found that right there on the list of the string’s vibrational patterns was one whose properties matched those of the graviton. Precisely. When convincing arguments were put forward in the mid-1980s that string theory was a mathematically consistent quantum mechanical theory (largely due to the work of Schwarz and his collaborator Michael Green), the presence of gravitons implied that string theory provided a long-sought quantum theory of gravity. This is the most important accomplishment on string theory’s résumé and the reason it quickly soared to worldwide scientific prominence.*8
Third, however radical a proposal string theory may be, it recapitulates a revered pattern in the history of physics. Successful new theories usually do not render their predecessors obsolete. Instead, successful theories typically embrace their predecessors, while greatly extending the range of physical phenomena that can be accurately described. Special relativity extends understanding into the realm of high speeds; general relativity extends understanding further still, to the realm of large masses (the domain of strong gravitational fields); quantum mechanics and quantum field theory extend understanding into the realm of short distances. The concepts these theories invoke and the features they reveal are unlike anything previously envisioned. Yet, apply these theories in the familiar domains of everyday speeds, sizes, and masses and they reduce to the descriptions developed prior to the twentieth century—Newton’s classical mechanics and the classical fields of Faraday, Maxwell, and others.
String theory is potentially the next and final step in this progression. In a single framework, it handle
s the domains claimed by relativity and the quantum. Moreover, and this is worth sitting up straight to hear, string theory does so in a manner that fully embraces all the discoveries that preceded it. A theory based on vibrating filaments might not seem to have much in common with general relativity’s curved spacetime picture of gravity. Nevertheless, apply string theory’s mathematics to a situation where gravity matters but quantum mechanics doesn’t (to a massive object, like the sun, whose size is large) and out pop Einstein’s equations. Vibrating filaments and point particles are also quite different. But apply string theory’s mathematics to a situation where quantum mechanics matters but gravity doesn’t (to small collections of strings that are not vibrating quickly, moving fast, or stretched long; they have low energy—equivalently, low mass—so gravity plays virtually no role) and the math of string theory morphs into the math of quantum field theory.
This is graphically summarized in Figure 4.3, which shows the logical connections between the major theories physicists have developed since the time of Newton. String theory could have required a sharp break from the past. It could have stepped clear off the chart provided in the figure. Remarkably, it doesn’t. String theory is sufficiently revolutionary to transcend the barriers that hemmed in twentieth-century physics. Yet, the theory is sufficiently conservative to allow the past three hundred years of discovery to snuggly fit within its mathematics.
Figure 4.3 A graphical representation of the relationships between the major theoretical developments in physics. Historically, successful new theories have extended the domain of understanding (to faster speeds, larger masses, shorter distances) while reducing to previous theories when applied in less extreme physical circumstances. String theory fits this pattern of progress: it extends the domain of understanding while, in suitable settings, reducing to general relativity and quantum field theory.
The Dimensions of Space
Now for something stranger. The passage from dots to filaments is only part of the new framework introduced by string theory. In the early days of string theory research, physicists encountered pernicious mathematical flaws (called quantum anomalies), entailing unacceptable processes like the spontaneous creation or destruction of energy. Typically, when problems like this afflict a proposed theory, physicists respond swiftly and sharply. They discard the theory. Indeed, many in the 1970s thought this the best course of action regarding strings. But the few researchers who stayed the course came upon an alternative way of proceeding.
In a dazzling development, they discovered that the problematic features were entwined with the number of dimensions of space. Their calculations revealed that were the universe to have more than the three dimensions of everyday experience—more than the familiar left/right, back/forth, and up/down—then string theory’s equations could be purged of their problematic features. Specifically, in a universe with nine dimensions of space and one of time, for a total of ten spacetime dimensions, the equations of string theory become trouble-free.
I’d love to explain in purely nontechnical terms how this comes about, but I can’t, and I’ve never encountered anyone who can. I made an attempt in The Elegant Universe, but that treatment only describes, in general terms, how the number of dimensions affects aspects of string vibrations, and doesn’t explain where the specific number ten comes from. So, in one slightly technical line, here’s the mathematical skinny. There’s an equation in string theory that has a contribution of the form (D — 10) times (Trouble), where D represents the number of spacetime dimensions and Trouble is a mathematical expression resulting in troublesome physical phenomena, such as the violation of energy conservation mentioned above. As to why the equation takes this precise form, I can’t offer any intuitive, nontechnical explanation. But if you do the calculation, that’s where the math leads. Now, the simple but key observation is that if the number of spacetime dimensions is ten, not the four we expect, the contribution becomes 0 times Trouble. And since 0 times anything is 0, in a universe with ten spacetime dimensions the trouble gets wiped away. That’s how the math plays out. Really. And that’s why string theorists argue for a universe with more than four spacetime dimensions.
Even so, no matter how open you may be to following the trail blazed by mathematics, if you’ve never encountered the idea of extra dimensions, the possibility may nevertheless sound nutty. Dimensions of space don’t go missing like car keys or one member of your favorite pair of socks. If there were more to the universe than length, width, and height, surely someone would’ve noticed. Well, not necessarily. Even as far back as the early decades of the twentieth century, a prescient series of papers by the German mathematician Theodor Kaluza and by the Swedish physicist Oskar Klein suggested that there might be dimensions that are proficient at evading detection. Their work envisioned that unlike the familiar spatial dimensions that extend over great, possibly infinite, distances, there might be additional dimensions that are tiny and curled up, making them difficult to see.
To picture this, think of a common drinking straw. But for the purpose at hand, make it decidedly uncommon by imagining it as thin as usual but as tall as the Empire State Building. The surface of the tall straw (like that of any straw) has two dimensions. The long vertical dimension is one; the short circular dimension, which curls around the straw, is the other. Now imagine viewing the tall straw from across the Hudson River, as in Figure 4.4a. Because the straw is so thin, it looks like a vertical line stretching from ground to sky. At this distance, you don’t have the visual acuity to see the straw’s tiny circular dimension, even though it exists at every point along the straw’s long vertical extent. This leads you to think, incorrectly, that the straw’s surface is one-dimensional, not two.9
For another visualization, think of a huge carpet blanketing Utah’s salt flats. From an airplane, the carpet looks like a flat surface with two dimensions that extend north/south and east/west. But after you parachute down and view the carpet up close, you realize that its surface is composed of a tight pile: tiny cotton loops attached to each point on the flat carpet backing. The carpet has two large, easy-to-see dimensions (north/south and east/west), but also one small dimension (the circular loops) that is harder to detect (Figure 4.4b).
The Kaluza-Klein proposal suggested that a similar distinction, between dimensions that are big and easily seen, and others that are tiny and thus more difficult to reveal, might apply to the fabric of space itself. The reason we are all aware of the familiar three dimensions of space would be that their extent, like the vertical dimension of the straw and the north/south and east/west dimensions of the carpet, is huge (possibly infinite). However, if an extra dimension of space were curled up like the circular part of the straw or carpet, but to an extraordinarily small size—millions or even billions of times smaller than a single atom—it could be as ubiquitous as the familiar unfurled dimensions and yet remain beyond our ability to detect even with today’s most powerful magnifying equipment. The dimension would indeed go missing. Such was the beginning of Kaluza-Klein theory, the proposition that our universe has spatial dimensions beyond the three of everyday experience (Figure 4.5).
This line of thought establishes that the suggestion of “extra” spatial dimensions, however unfamiliar, is not absurd. That’s a good start, but it invites an essential question: Why, back in the 1920s, would someone invoke such an exotic idea? Kaluza’s motivation came from an insight he had shortly after Einstein published the general theory of relativity. He found that with a single stroke of the pen—literally—he could modify Einstein’s equations to make them apply to a universe with one additional dimension of space. And when he analyzed those modified equations, the results were so thrilling that, as his son has recounted, Kaluza discarded his normally reserved demeanor, pounded his desk with both hands, shot to his feet, and erupted into an aria from The Marriage of Figaro.10 Within the modified equations, Kaluza found the ones Einstein had already used successfully to describe gravity in the familiar three dimensions of
space and one of time. But because his new formulation included an additional dimension of space, Kaluza found an additional equation. Lo and behold, when Kaluza derived this equation he recognized it as the very one Maxwell had discovered half a century earlier to describe the electromagnetic field.
Figure 4.4 (a) The surface of a tall straw has two dimensions; the vertical dimension is long and easy to see, while the circular dimension is small and harder to detect. (b) A gigantic carpet has three dimensions; the north/south and east/west dimensions are big and easy to see, while the circular part, the carpet’s pile, is small and therefore harder to detect.
Figure 4.5 Kaluza-Klein theory posits tiny extra spatial dimensions attached to every point in the familiar three large spatial dimensions. If we could magnify the spatial fabric sufficiently, the hypothesized extra dimensions would become visible. (For the sake of visual clarity, extra dimensions are attached only on grid points in the illustration.)