## The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos - Brian Greene (2011)

### Chapter 4. Unifying Nature’s Laws

*On the Road to String Theory*

From the big bang to inflation, modern cosmology traces its roots to a single scientific nexus: Einstein’s general theory of relativity. With his new theory of gravity, Einstein upended the accepted conception of a rigid and immutable space and time; science now had to embrace a dynamic cosmos. Contributions of this magnitude are rare. Yet, Einstein dreamed of scaling even greater heights. With the mathematical arsenal and geometric intuition he’d amassed by the 1920s, he set out to develop a *unified field theory*.

By this, Einstein meant a framework that would stitch all of nature’s forces into a single, coherent mathematical tapestry. Rather than have one set of laws for these physical phenomena and a different set for those, Einstein wanted to fuse all the laws into a seamless whole. History has judged Einstein’s decades of intense work toward unification as having had little lasting impact—the dream was noble, the timing was early—but others have taken up the mantle and made substantial strides, the most refined proposal being *string theory*.

My previous books *The Elegant Universe* and *The Fabric of the Cosmos* covered the history and essential features of string theory. In the years since they appeared, the theory’s general health and status have faced a spate of public questioning. Which is completely reasonable. For all its progress, string theory has yet to make definitive predictions whose experimental investigation could prove the theory right or wrong. As the next three multiverse varieties we will encounter (in __Chapters 5__ and __6__) emerge from a string theoretic perspective, it’s important to address the current state of the theory as well as the prospects for making contact with experimental and observational data. Such is the charge of this chapter.

**A Brief History of Unification**

At the time Einstein pursued the goal of unification, the known forces were gravity, described by his own general relativity, and electromagnetism, described by Maxwell’s equations. Einstein envisioned melding the two into a single mathematical sentence that would articulate the workings of all nature’s forces. Einstein had high hopes for this unified theory. He considered Maxwell’s nineteenth-century work on unification an archetypal contribution to human thought—and rightly so. Before Maxwell, the electricity flowing through a wire, the force generated by a child’s magnet, and the light streaming to earth from the sun were viewed as three separate, unrelated phenomena. Maxwell revealed that, in actuality, they formed an intertwined scientific trinity. Electric currents *produce* magnetic fields; magnets moving in the vicinity of a wire *produce* electric currents; and wavelike disturbances rippling through electric and magnetic fields *produce* light. Einstein anticipated that his own work would carry forward Maxwell’s program of consolidation by making the next and possibly final move toward a fully unified description of nature’s laws—a description that would unite electromagnetism and gravity.

This wasn’t a modest goal, and Einstein didn’t take it lightly. He had an unparalleled capacity for single-minded devotion to problems he’d set for himself, and during the last thirty years of his life the problem of unification became his prime obsession. His personal secretary and gatekeeper, Helen Dukas, was with Einstein at the Princeton Hospital during his penultimate day, April 17, 1955. She recounts how Einstein, bedridden but feeling a little stronger, asked for the pages of equations on which he had been endlessly manipulating mathematical symbols in the fading hope that the unified field theory would materialize. Einstein didn’t rise with the morning sun. His final scribblings shed no further light on unification.^{1}

Few of Einstein’s contemporaries shared his passion for unification. From the mid-1920s through the mid-1960s, physicists, guided by quantum mechanics, were unlocking the secrets of the atom and learning how to harness its hidden powers. The lure of prying apart matter’s constituents was immediate and powerful. While many agreed that unification was a laudable goal, it was of only passing interest in an age when theorists and experimenters were working hand in glove to reveal the laws of the microscopic realm. With Einstein’s passing, work on unification ground to a halt.

His failure was compounded when subsequent research showed that his quest for unity had been too narrowly focused. Not only had Einstein downplayed the role of quantum physics (he believed the unified theory would supersede quantum mechanics and so it needn’t be incorporated from the outset), but his work failed to take account of two additional forces revealed by experiments: the *strong nuclear force* and the *weak nuclear force*. The former provides a powerful glue that holds together the nuclei of atoms, while the latter is responsible for, among other things, radioactive decay. Unification would need to combine not two forces but four; Einstein’s dream seemed all the more remote.

During the late 1960s and 1970s, the tide turned. Physicists realized that the methods of quantum field theory, which had been successfully applied to the electromagnetic force, also provided descriptions of the weak and strong nuclear forces. All three of the nongravitational forces could thus be described using the same mathematical language. Moreover, detailed study of these quantum field theories—most notably in the Nobel Prize–winning work of Sheldon Glashow, Steven Weinberg, and Abdus Salam, as well as in the subsequent insights of Glashow and his Harvard colleague Howard Georgi—revealed relationships suggesting a potential unity among the electromagnetic, weak, and strong nuclear forces. Following Einstein’s nearly half-century-old lead, theoreticians argued that these three apparently distinct forces might actually be manifestations of a single monolithic force of nature.^{2}

These were impressive advances toward unification, but set against the encouraging backdrop was a pesky problem. When scientists applied the methods of quantum field theory to nature’s fourth force, gravity, the math just wouldn’t work. Calculations involving quantum mechanics and Einstein’s general relativistic description of the gravitational field yielded jarring results that amounted to mathematical gibberish. However successful general relativity and quantum mechanics had been in their native domains, the large and the small, the nonsensical output from the attempt to unite them spoke to a deep fissure in the understanding of nature’s laws. If the laws you have prove mutually incompatible, then—clearly—the laws you have are not the right laws. Unification had been an aesthetic goal; now it was transformed into a logical imperative.

The mid-1980s witnessed the next pivotal development. That’s when a new approach, *superstring theory*, captured the attention of the world’s physicists. It ameliorated the hostility between general relativity and quantum mechanics, and so provided hope that gravity could be brought within a unified quantum mechanical fold. The era of superstring unification was born. Research proceeded at an intense pace, and thousands of journal pages were quickly filled with calculations that fleshed out aspects of the approach and laid the groundwork for its systematic formulation. An impressive and intricate mathematical structure emerged, but much about superstring theory (*string theory*, for short) remained mysterious.^{3}

Then, beginning in the mid-1990s, theorists intent on unraveling those mysteries unexpectedly thrust string theory squarely into the multiverse narrative. Researchers had long known that the mathematical methods being used to analyze string theory invoked a variety of approximations and so were ripe for refinement. When some of those refinements were developed, researchers realized that the math suggested plainly that our universe might belong to a multiverse. In fact, the mathematics of string theory suggested not just one but a number of different kinds of multiverses of which we might be a part.

To fully grasp these compelling and contentious developments, and to assess their role in our ongoing search for the deep laws of the cosmos, we need to take a step back and first evaluate the state of string theory.

**Quantum Fields Redux**

Let’s begin by taking a closer look at the traditional, highly successful framework of quantum field theory. This will prepare us to string unification as well as highlight pivotal connections between these two approaches for formulating nature’s laws.

Classical physics, as we saw in __Chapter 3__, describes a field as a kind of mist that permeates a region of space and can carry disturbances in the form of ripples and waves. Were Maxwell to describe the light that’s now illuminating this text, for example, he’d wax enthusiastic about electromagnetic waves, produced by the sun or by a nearby lightbulb, undulating across space on their way to the printed page. He’d describe the waves’ movement mathematically, using numbers to delineate the field’s strength and direction at each point in space. An undulating field corresponds to undulating numbers: the field’s numerical value at any given location cycles down and up again.

When quantum mechanics is brought to bear on the concept of a field, the result is quantum field theory, which is characterized by two essential new features. We’ve already encountered both, but they’re worth a refresher. First, quantum uncertainty causes the value of a field at each point in space to jitter randomly—think of the fluctuating inflaton field from inflationary cosmology. Second, quantum mechanics establishes that, somewhat as water is composed of H_{2}O molecules, a field is composed of infinitesimally small particles known as the field’s *quanta*. For the electromagnetic field, the quanta are photons, and so a quantum theorist would modify Maxwell’s classical description of your lightbulb by saying that the bulb emits a steady stream comprising 100 billion billion photons each second.

Decades of research have established that these features of quantum mechanics as applied to fields are completely general. Every field is subject to quantum jitters. And every field is associated with a species of particle. Electrons are quanta of the electron field. Quarks are quanta of the quark field. For a (very) rough mental image, physicists sometimes think of particles as knots or dense nuggets of their associated field. This visualization notwithstanding, the mathematics of quantum field theory describes these particles as dots or points that have no spatial extent and no internal structure.^{4}

Our confidence in quantum field theory comes from one essential fact: there is not a single experimental result that counters its predictions. To the contrary, data confirm that the equations of quantum field theory describe the behavior of particles with astounding accuracy. The most impressive example comes from the quantum field theory of the electromagnetic force, *quantum electrodynamics*. Using it, physicists have undertaken detailed calculations of the electron’s magnetic properties. The calculations are not easy, and the most refined versions have taken decades to complete. But they’ve been worth the effort. The results match actual measurements to a precision of *ten* decimal places, an almost unimaginable agreement between theory and experiment.

With such success, you might anticipate that quantum field theory would provide the mathematical framework for understanding all of nature’s forces. An illustrious coterie of physicists shared this very expectation. By the late 1970s, the hard work of many of these visionaries had established that, indeed, the weak and strong nuclear forces fit squarely within the rubric of quantum field theory. Both forces are accurately described in terms of fields—the weak and the strong fields—that evolve and interact according to the mathematical rules of quantum field theory.

But, as I indicated in the historical overview, many of these same physicists quickly realized that the story for nature’s remaining force, gravity, was far subtler. Whenever the equations of general relativity commingled with those of quantum theory, the mathematics balked. Use the combined equations to calculate the quantum probability of some physical process—such as the chance of two electrons ricocheting off each other, given both their electromagnetic repulsion and their gravitational attraction—and you’d typically get the answer *infinity*. While some things in the universe can be infinite, such as the extent of space and the quantity of matter that may fill it, probabilities are not among them. By definition, the value of a probability must be between 0 and 1 (or, in terms of percentages, between 0 and 100). An infinite probability does not mean that something is very likely to happen, or is certain to happen; rather, it’s meaningless, like speaking of the thirteenth egg in an even dozen. An infinite probability sends a clear mathematical message: the combined equations are nonsense.

Physicists traced the failure to the jitters of quantum uncertainty. Mathematical techniques had been developed for analyzing the jitters of the strong, weak, and electromagnetic fields, but when the same methods were applied to the gravitational field—a field that governs the curvature of spacetime itself—they proved ineffective. This left the mathematics saturated with inconsistencies such as infinite probabilities.

To get a feel for why, imagine you’re the landlord of an old house in San Francisco. If you have tenants who throw raucous parties, it might take effort to deal with the situation, but you don’t worry that the festivities will compromise the building’s structural integrity. However, if there’s an earthquake, you’re facing something far more serious. The fluctuations of the three nongravitational forces—fields that are tenants within the house of spacetime—are like the building’s incessant partyers. It took a generation of theoretical physicists to grapple with their raucous jitters, but by the 1970s they’d developed mathematical methods capable of describing the quantum properties of the nongravitational forces. The fluctuations of the gravitational field, however, are qualitatively different. They’re more like an earthquake. Because the gravitational field is woven within the very fabric of spacetime, its quantum jitters shake the entire structure through and through. When used to analyze such pervasive quantum jitters, the mathematical methods collapsed.^{5}

For years, physicists turned a blind eye to this problem because it surfaces only under the most extreme conditions. Gravity makes its mark when things are very massive, quantum mechanics when things are very small. And rare is the realm that is both small and massive, so that to describe it you must invoke both quantum mechanics and general relativity. Yet, there are such realms. When gravity and quantum mechanics are together brought to bear on either the big bang or black holes, realms that *do* involve extremes of enormous mass squeezed to small size, the math falls apart at a critical point in the analyses, leaving us with unanswered questions regarding how the universe began and how, at the crushing center of a black hole, it might end.

Moreover—and this is the truly daunting part—beyond the specific examples of black holes and the big bang, you can calculate how massive and how small a physical system needs to be for both gravity and quantum mechanics to play a significant role. The result is about 10^{19} times the mass of a single proton, the so-called *Planck mass*, squeezed into a fantastically small volume of about 10^{–99} cubic centimeters (roughly a sphere with a radius of 10^{–33} centimeters, the so-called *Planck length* graphically illustrated in __Figure 4.1__).__ ^{6}__ The dominion of quantum gravity is thus more than a million billion times beyond the scales we can probe even with the world’s most powerful accelerators. This vast expanse of uncharted territory could easily be rife with new fields and their associated particles—and who knows what else. To unify gravity and quantum mechanics requires trekking from here to there, grasping the known and the unknown across an enormous expanse that, for the most part, is experimentally inaccessible. That’s a hugely ambitious task, and many scientists concluded that it was beyond reach.

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 = mc^{2}). 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 handles 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.*)

Kaluza revealed that in a universe with an additional dimension of space, gravity and electromagnetism can both be described in terms of spatial ripples. Gravity ripples through the familiar three spatial dimensions, while electromagnetism ripples through the fourth. An outstanding problem with Kaluza’s proposal was to explain why we don’t see this fourth spatial dimension. It was here that Klein made his mark by suggesting the resolution explained above: dimensions beyond those we directly experience can elude our senses and our equipment if they’re sufficiently small.

In 1919, after learning about the extra dimensional proposal for unification, Einstein vacillated. He was impressed by a framework that advanced his dream of unification but was hesitant about such an outlandish approach. After cogitating for a couple of years, in the process holding up publication of Kaluza’s paper, Einstein finally warmed to the idea and in time became one of the strongest champions of hidden spatial dimensions. In his own research toward a unified theory, he returned to this theme repeatedly.

Einstein’s blessing notwithstanding, subsequent research showed that the Kaluza-Klein program ran up against a number of hurdles, the most difficult being its inability to incorporate the detailed properties of matter particles, such as electrons, into its mathematical structure. Clever ways around this problem, as well as various generalizations and modifications of the original Kaluza-Klein proposal, were pursued on and off for a couple of decades, but as no pitfall-free framework emerged, by the mid-1940s the idea of unification through extra dimensions was largely dropped.

Thirty years later, along came string theory. Rather than allowing for a universe with more than three dimensions, the mathematics of string theory *required* it. And so string theory provided a new, ready-made setting for invoking the Kaluza-Klein program. In response to the question “If string theory is the long-sought unified theory, then why haven’t we seen the extra dimensions it needs?” Kaluza-Klein echoed across the decades, answering that the dimensions are all around us but are just too small to be seen. String theory resurrected the Kaluza-Klein program, and by the mid-1980s researchers worldwide were inspired to believe that it was only a matter of time—according to the most enthusiastic proponents, a short time—before string theory would provide a complete theory of all matter and all forces.

**Great Expectations**

During the early days of string theory, progress came at such a rapid clip that it was nearly impossible to keep up with all the developments. Many compared the atmosphere to that of the 1920s, when scientists stormed into the newly discovered realm of the quantum. With such excitement it’s understandable that some theoreticians spoke of a swift resolution to the major problems of fundamental physics: the merger of gravity and quantum mechanics; the unification of all of nature’s forces; an explanation of the properties of matter; a determination of the number of spatial dimensions; the elucidation of black hole singularities; and the unraveling of the origin of the universe. As more seasoned researchers anticipated, though, these expectations were premature. String theory is so rich, wide ranging, and mathematically difficult that research to date, nearly three decades after the initial euphoria, has taken us but partway down the road of exploration. And given that the realm of quantum gravity is some hundred billion billion times smaller than anything we can currently access experimentally, levelheaded assessments expect that the road will be long.

Where are we along it? In the rest of the chapter, I’ll survey the most advanced understanding in a number of key areas (saving those relevant to the theme of parallel universes for more detailed discussion in subsequent chapters), and I’ll appraise the achievements to date and the challenges still looming.

**String Theory and the Properties of Particles**

One of the deepest questions in all of physics is why nature’s particles have the properties they do. Why, for example, does the electron have its particular mass and the up-quark its particular electric charge? The question commands attention not only for its intrinsic interest but also because of a tantalizing fact we alluded to earlier. Had the particles’ properties been different—had, say, the electron been moderately heavier or lighter, or had the electric repulsion between electrons been stronger or weaker—the nuclear processes that power stars like our sun would have been disrupted. Without stars, the universe would be a very different place.__ ^{11}__ Most pointedly, without the sun’s heat and light, the complex chain of events that led to life on earth would have failed to happen.

This leads to a grand challenge: using pen, paper, possibly a computer, and one’s best understanding of the laws of physics, calculate the particle properties and find results in agreement with the measured values. If we could meet this challenge, we’d take one of the most profound steps ever toward understanding why the universe is as it is.

In quantum field theory, the challenge is insurmountable. Permanently. Quantum field theory requires the measured particle properties as input—these features are part of the theory’s definition—and so can happily accommodate a broad range of values for their masses and charges.__ ^{12}__ In an imaginary world where the electron’s mass or charge was larger or smaller than it is in ours, quantum field theory could cope without blinking an eye; it would simply be a matter of adjusting the value of a parameter within the theory’s equations.

Can string theory do better?

One of the most beautiful features of string theory (and the facet that most impressed me when I learned the subject) is that particle properties are *determined by the size and shape of the extra dimensions*. Because strings are so tiny, they don’t just vibrate within the three big dimensions of common experience; they also vibrate into the tiny, curled-up dimensions. And much as air streams flowing through a wind instrument have vibrational patterns dictated by the instrument’s geometrical form, the strings in string theory have vibrational patterns dictated by the geometrical form of the curled-up dimensions. Recalling that string vibrational patterns determine particle properties such as mass and electrical charge, we see that these properties are determined by the geometry of the extra dimensions.

So, if you knew exactly what the extra dimensions of string theory looked like, you’d be well on your way to predicting the detailed properties of vibrating strings, and hence the detailed properties of the elementary particles the strings vibrate into existence. The hurdle is, and has been for some time, that no one has been able to figure out the exact geometrical form of the extra dimensions. The equations of string theory place mathematical restrictions on the geometry of the extra dimensions, requiring them to belong to a particular class called *Calabi-Yau shapes* (or, in mathematical jargon, *Calabi-Yau manifolds*), named after the mathematicians Eugenio Calabi and Shing-Tung Yau, who investigated their properties well before their important role in string theory was discovered (__Figure 4.6__). The problem is that there’s not a single, unique Calabi-Yau shape. Instead, like musical instruments, the shapes come in a wide variety of sizes and contours. And just as different instruments generate different sounds, extra dimensions that differ in size and shape (as well as with respect to more detailed features we’ll come upon in the next chapter) generate different string vibrational patterns and hence different sets of particle properties. *The lack of a unique specification of the extra dimensions is the main stumbling block preventing string theorists from making definitive predictions*.

**Figure 4.6** *A close-up of the spatial fabric in string theory, showing an example of extra dimensions curled up into a Calabi-Yau shape. Like the pile and backing of a carpet, the Calabi-Yau shape would be attached to every point in the familiar three large spatial dimensions (represented by the two-dimensional grid), but for visual clarity the shapes are shown only on grid points*.

When I started working on string theory, back in the mid-1980s, there were only a handful of known Calabi-Yau shapes, so one could imagine studying each, looking for a match to known physics. My doctoral dissertation was one of the earliest steps in this direction. A few years later, when I was a postdoctoral fellow (working for the Yau of Calabi-Yau), the number of Calabi-Yau shapes had grown to a few thousand, which presented more of a challenge to exhaustive analysis—but that’s what graduate students are for. As time passed, however, the pages of the Calabi-Yau catalog continued to multiply; as we will see in __Chapter 5__, they have now grown more numerous than grains of sand on a beach. Every beach. Everywhere. By a long shot. To analyze mathematically each possibility for the extra dimensions is out of the question. String theorists have therefore continued the search for a mathematical directive from the theory that might single out a particular Calabi-Yau shape as “the one.” To date, no one has succeeded.

And so, when it comes to explaining the properties of the fundamental particles, string theory has yet to realize its promise. In this regard, it so far offers no improvement over quantum field theory.^{13}

Bear in mind, however, that string theory’s claim to fame is its ability to resolve *the* central dilemma of twentieth-century theoretical physics: the raging hostility between general relativity and quantum mechanics. Within string theory, general relativity and quantum mechanics finally join together harmoniously. *That’s* where string theory provides a vital advance, taking us beyond a critical obstacle that confounded the standard methods of quantum field theory. Should a better understanding of the mathematics of string theory enable us to pick out a unique form for the extra dimensions, one that furthermore allows us to explain all observed particle properties, that would be a phenomenal triumph. But there’s no guarantee that string theory can rise to the challenge. There’s also no necessity for it to do so. Quantum field theory has been rightly lauded as hugely successful, and yet it can’t explain the fundamental particle properties. If string theory also can’t explain the particle properties but goes beyond quantum field theory in one key measure, by embracing gravity, that alone would be a monumental achievement.

Indeed, in __Chapter 6__ we’ll see that in a cosmos replete with parallel worlds—as suggested by one modern reading of string theory—it may be plainly wrongheaded to hope that mathematics would pick out a unique form for the extra dimensions. Instead, much as the many different forms for DNA provide for the abundant variety of life on earth, so the many different forms for the extra dimensions may provide for the abundant variety of universes populating a string-based multiverse.

**String Theory and Experiment**

If a typical string is as small as __Figure 4.2__ suggests, to probe its extended structure—the very characteristic that distinguishes it from a point—you’d need an accelerator some million billion times more powerful than even the Large Hadron Collider. Using known technology, such an accelerator would need to be about as large as the galaxy, and would consume enough energy each second to power the entire world for a millennium. Barring a spectacular technological breakthrough, this ensures that at the comparatively low energies our accelerators can reach, strings will appear as though they are point particles. This is the experimental version of the theoretical fact I emphasized earlier: at low energy, the mathematics of string theory transforms into the mathematics of quantum field theory. And so, even if string theory is the true fundamental theory, it will impersonate quantum field theory in a wide range of accessible experiments.

That’s a good thing. Although quantum field theory is not equipped to combine general relativity and quantum mechanics, nor to predict the fundamental properties of nature’s particles, it can explain a great many other experimental results. It does this by taking the measured properties of particles as input (input that dictates the choice of fields and energy curves in the quantum field theory) and then uses the mathematics of quantum field theory to predict how such particles will behave in other experiments, generally accelerator-based. The results are extremely accurate, which is why generations of particle physicists have made quantum field theory their primary approach.

The choice of fields and energy curves in quantum field theory is tantamount to the choice of the extra dimensional shape in string theory. The particular challenge facing string theory, though, is that the mathematics linking particle properties (such as their masses and charges) to the shape of the extra dimensions is extraordinarily intricate. This makes it difficult to work backwards—to use experimental data to guide the choice of the extra dimensions, much as such data guide the choices of fields and energy curves in quantum field theory. One day we may have the theoretical dexterity to use experimental data to fix the form of string theory’s extra dimensions, but not yet.

For the foreseeable future, then, the most promising avenue for linking string theory with data are predictions that, while open to explanations using more traditional methods, are far more naturally and convincingly explained using string theory. Just as you might theorize that I’m typing these words with my toes, a far more natural and convincing hypothesis—and one I can attest to as correct—is that I’m using my fingers. Analogous considerations applied to the experiments summarized in __Table 4.1__ have the capacity to build a circumstantial case for string theory.

The undertakings range from particle physics experiments at the Large Hadron Collider (searching for supersymmetric particles and for evidence of extra dimensions), to tabletop experiments (measuring the gravitational strength of attraction on scales of a millionth of a meter and smaller), to astronomical observations (looking for particular kinds of gravitational waves and fine temperature variations in the cosmic microwave background radiation). The table explains the individual approaches, but the overall assessment is readily summarized. A positive signature from any of these experiments could be explained without invoking string theory. For example, although the mathematical framework of supersymmetry (see the first entry in __Table 4.1__) was initially discovered through theoretical studies of string theory, it has since been incorporated into non-string theoretic approaches. Discovering supersymmetric particles would thus confirm a piece of string theory, but would not constitute a smoking gun. Similarly, although extra spatial dimensions have a natural home within string theory, we’ve seen that they too can be part of non-string theoretic proposals—Kaluza, as a case in point, was not thinking about string theory when he proposed the idea. The most favorable outcome from the approaches in __Table 4.1__, therefore, would be a series of positive results that would show pieces of the string theory puzzle falling into place. Like touting touch-typing toes, non-string explanations would become overwrought when faced with such a collection of positive results.

**Table 4.1**. *Experiments and Observations with the Capacity to Link String Theory to Data*

**EXPERIMENT/OBSERVATION:** **Supersymmetry**

**EXPLANATION:** The “super” in superstring theory refers to *supersymmetry*, a mathematical feature with a straightforward implication: for every known particle species there should be a partner species that has the same electrical and nuclear force properties. Theorists surmise that these particles have so far evaded detection because they are heavier than their known counterparts, and so lie beyond the reach of well-worn accelerators. The Large Hadron Collider may have enough energy to produce them, so there’s broad anticipation that we may be on the threshold of revealing nature’s supersymmetric quality.

**EXPERIMENT/OBSERVATION:** **Extra Dimensions and Gravity**

**EXPLANATION:** Because space is the medium for gravity, more dimensions supply a larger domain within which gravity can spread. And just as a drop of ink grows more diluted when it spreads in a vat of water, the strength of gravity would become diluted as it spreads through the additional dimensions—offering an explanation for why gravity appears weak (when you pick up a coffee cup, your muscles beat out the gravitational pull of the entire earth). If we could measure gravity’s strength over distances smaller than the size of the extra dimensions, we’d catch it before it’s fully spread and so we should find its strength to be stronger. To date, measurements on scales as short as a micron (10^{–6} meters) have found no deviation from expectations based on a world with three spatial dimensions. Should a deviation be found as physicists push these experiments to ever-shorter distances, that would provide convincing evidence for additional dimensions.

**EXPERIMENT/OBSERVATION:** **Extra Dimensions and Missing Energy**

**EXPLANATION:** If the extra dimensions exist but are far smaller than a micron, they will be inaccessible to experiments that directly measure gravity’s strength. But the Large Hadron Collider provides another means of revealing their existence. Debris created by head-on collisions between fast-moving protons can be ejected from our familiar large dimensions and squeezed into the others (where, for reasons we’ll get to later, the debris would likely be particles of gravity, or *gravitons*). Were this to happen, the debris would carry away energy, and as a result our detectors would register a little less energy after the collision than was present before. Such missing energy signals could provide strong evidence for the existence of extra dimensions.

**EXPERIMENT/OBSERVATION:** **Extra Dimensions and Mini Black Holes**

**EXPLANATION:** Black holes are usually described as the remains of massive stars that have exhausted their nuclear fuel and collapsed under their own weight, but this is an unduly limited description. Anything would become a black hole if compressed sufficiently. Moreover, if there are extra dimensions that result in gravity being stronger when acting over short distances, it would be easier to form black holes, since a stronger gravitational force implies that it takes less compression to generate the same gravitational pull. Even just two protons, if slammed together at the velocities mustered by the Large Hadron Collider, may be able to cram enough energy into a sufficiently small volume to trigger the formation of a black hole. It would be a wisp of a black hole, but it would yield an unmistakable signature. Mathematical analysis, going back to the work of Stephen Hawking, shows that tiny black holes would quickly disintegrate into a spray of lighter particles whose tracks would be picked up by the collider’s detectors.

**EXPERIMENT/OBSERVATION:** **Gravitational Waves**

**EXPLANATION:** Although strings are tiny, if you could somehow grab hold of one, you could stretch it large. You’d need to apply a force in excess of 10^{20} tons, but stretching a string is merely a matter of exerting enough energy. Theorists have found exotic situations in which the energy for such stretching might be provided by astrophysical processes, generating long strings wafting through space. Even if they were very distant, these strings might be detectable. Calculations show that as a long string vibrates, it creates ripples in spacetime—known as *gravitational waves*—of a highly distinctive shape, and hence they offer a clear observational signature. Within the next few decades, if not sooner, highly sensitive detectors based on earth and, funding permitting, in space, may be able to measure these ripples.

**EXPERIMENT/OBSERVATION:** **Cosmic Microwave Background Radiation**

**EXPLANATION:** The cosmic microwave background radiation has already proved itself capable of probing quantum physics: the measured temperature differences in the radiation arise from quantum jitters stretched large by spatial expansion. (Recall the analogy of a tiny message scribbled on a shriveled balloon becoming visible once the balloon is inflated.) In inflation, the stretching of space is so enormous that even tinier imprints, perhaps laid down by strings, might also be stretched sufficiently to be detectable—perhaps by the European Space Agency’s Planck satellite. Success or failure turns on details of how strings would have behaved in the earliest moments of the universe—the nature of the message they would have imprinted on the deflated cosmic balloon. Various ideas have been developed and calculations made. Theorists are now waiting for the data to speak for themselves.

Negative experimental results would provide much less useful information. The failure to find supersymmetric particles might mean they don’t exist, but it also might mean they are too heavy for even the Large Hadron Collider to produce; the failure to find evidence for extra dimensions might mean they don’t exist, but it also might mean they are too small for our technologies to access; the failure to find microscopic black holes might mean that gravity does not get stronger on short scales, but it also might mean that our accelerators are too weak to burrow deeply enough into the microscopic terrain where the increase in strength is substantial; the failure to find stringy signatures in observations of gravitational waves or the cosmic microwave background radiation might mean string theory is wrong, but it might also mean that the signatures are too meager for current equipment to measure.

As of today, then, the most promising positive experimental results would most likely not be able to definitively prove string theory right, while negative results would most likely not be able to prove string theory wrong.__ ^{14}__ Yet, make no mistake. If we find evidence of extra dimensions, supersymmetry, mini black holes, or any of the other potential signatures, that will be a huge moment in the search for a unified theory. It would bolster confidence, and justifiably so, that the mathematical road we’ve been paving is headed in the right direction.

**String Theory, Singularities, and Black Holes**

In the vast majority of situations, quantum mechanics and gravity happily ignore each other, the former applying to small things like molecules and atoms and the latter to big things like stars and galaxies. But the two theories are forced to shed their isolation in the realms known as *singularities*. A singularity is any physical setting, real or hypothetical, that is so extreme (huge mass, small size, enormous spacetime curvature, punctures or rips in the spacetime fabric) that quantum mechanics and general relativity go haywire, generating results akin to the error message displayed on a calculator when you divide any number by zero.

A prize achievement of any purported quantum theory of gravity is to meld quantum mechanics and gravity in a manner that cures singularities. The resulting mathematics should never break down—even at the moment of the big bang or in the center of a black hole,__ ^{15}__ thus providing a sensible description of situations that have long baffled researchers. It is here that string theory has made its most impressive strides, taming a growing list of singularities.

In the mid-1980s, the team of Lance Dixon, Jeff Harvey, Cumrun Vafa, and Edward Witten realized that certain punctures in the spatial fabric (known as *orbifold singularities*), which leave Einstein’s mathematics in shambles, pose no problem for string theory. The key to this success is that whereas point particles can fall into punctures, strings can’t. Because strings are extended objects, they can bang into a puncture, they can wrap around it, or they can get stuck to it, but these mild interactions leave the equations of string theory perfectly sound. This is important not because such ruptures in space actually happen—they may or may not—but because string theory is delivering just what we want from a quantum theory of gravity: a means of making sense of a situation that lies beyond what general relativity and quantum mechanics can handle on their own.

In the 1990s, work I did with Paul Aspinwall and David Morrison, and independent results of Edward Witten, established that yet more intense singularities (known as *flop singularities*) in which a spherical portion of space is compressed to an infinitesimal size can also be handled by string theory. The intuitive reasoning here is that as a string moves it can sweep across the compressed chunk of space, like a hula hoop across a soap bubble, and thus act as an encircling protective barrier. The calculations showed that such a “string shield” nullifies any potentially disastrous consequences, ensuring that string theory’s equations suffer no ill effect—no “1 divided by 0” type errors—even though the equations of conventional general relativity would fall apart.

In the years since, researchers have shown that a variety of other more complicated singularities (with names like *conifolds, orientifolds, enhancons …*) are also under full control within string theory. So there’s a growing list of situations that would have left Einstein, Bohr, Heisenberg, Wheeler, and Feynman saying, “We just don’t know what’s going on,” and yet for which string theory gives a complete and consistent description.

This is great progress. But a remaining challenge for string theory is to cure the singularities of black holes and the big bang, which are more severe than those so far addressed. Theorists have expended much effort trying to reach this goal, and they’ve taken significant strides. But the executive summary is that there is still a way to go before these most puzzling and most relevant of singularities are fully understood.

Nevertheless, one major advance has illuminated a related aspect of black holes. As I will discuss in __Chapter 9__, the work of Jacob Bekenstein and Stephen Hawking in the 1970s established that black holes contain a very particular quantity of disorder, technically known as *entropy*. According to basic physics, much as the disorder within a sock drawer reflects the many possible haphazard rearrangements of its contents, the disorder of a black hole reflects the many possible haphazard rearrangements of the black hole’s innards. But try as they might, physicists were unable to understand black holes well enough to identify their innards, let alone analyze the possible ways they could be rearranged. The string theorists Andrew Strominger and Cumrun Vafa broke through the impasse. Using a mélange of string theory’s fundamental ingredients (some of which we will encounter in __Chapter 5__), they created a mathematical model for a black hole’s disorder, a model transparent enough to enable them to extract a numerical measure of the entropy. The result they found agreed spot-on with the Bekenstein-Hawking answer. While the work left open many deep issues (such as explicitly identifying a black hole’s microscopic constituents), it provided the first firm quantum mechanical accounting of a black hole’s disorder.^{16}

The remarkable advances in dealing with both singularities and black hole entropy give the community of physicists well-grounded confidence that in time the remaining challenges of black holes and the big bang will be conquered.

**String Theory and Mathematics**

Making contact with data, experimental or observational, is the only way to determine if string theory correctly describes nature. It’s a goal that’s proved elusive. String theory, for all its advances, is still a wholly mathematical undertaking. But string theory isn’t just a consumer of math. Some of its most important contributions have been *to* mathematics.

When he was developing the general theory of relativity in the early twentieth century, Einstein famously mined the mathematical archives in search of rigorous language for describing curved spacetime. The earlier geometrical insights of mathematicians such as Carl Friedrich Gauss, Bernhard Riemann, and Nikolai Lobachevsky provided an important foundation for his success. In a sense, string theory is now helping to repay Einstein’s intellectual debt by driving the development of new mathematics. There are numerous examples, but let me give one that captures the flavor of string theory’s mathematical achievements.

General relativity established a tight link between the geometry of spacetime and the physics we observe. Einstein’s equations, together with the distribution of matter and energy in a region, tell you the resulting shape of spacetime. Different physical environments (different configurations of mass and energy) yield differently shaped spacetimes; different spacetimes correspond to physically distinct environments. What would it feel like to fall into a black hole? Calculate with the spacetime geometry that Karl Schwarzschild discovered in his study of spherical solutions to Einstein’s equations. And if the black hole is rapidly spinning? Calculate with the spacetime geometry found in 1963 by the New Zealand mathematician Roy Kerr. In general relativity, geometry is the yin to physics’ yang.

String theory provides a twist to this conclusion by establishing that there can be *different* shapes for spacetime that nevertheless yield physically indistinguishable descriptions of reality.

Here’s one way to think about it. From antiquity to the modern mathematical era, we’ve modeled geometrical spaces as collections of points. A Ping-Pong ball, for example, is the collection of points that constitute its surface. Prior to string theory, the basic constituents making up matter were also modeled as points, point particles, and this commonality of basic ingredients spoke to an alignment between geometry and physics. But in string theory, the basic ingredient is not a point. It’s a string. This suggests that a new kind of geometry, based not on points but rather on loops, should be linked to string physics. The new geometry is called *stringy geometry*.

To get a feel for stringy geometry, picture a string moving through a geometrical space. Notice that the string can behave much like a point particle, innocently gliding from here to there, bumping into walls, navigating chutes and valleys, and so on. But in certain situations, a string can also do something novel. Imagine that space (or a piece of space) is shaped like a cylinder. A string can wrap itself around such a piece of space, much like a rubber band stretched around a can of soda, realizing a configuration that’s simply unavailable to a point particle. Such “wrapped” strings, and their “unwrapped” cousins, probe a geometrical space in different ways. Should a cylinder grow fatter, a string encircling it will respond by stretching, while an unwrapped string sliding on its surface won’t. In this way, wrapped and unwrapped strings are sensitive to different features of a shape through which they’re moving.

This observation is of great interest because it gives rise to a striking and thoroughly unexpected conclusion. String theorists have found special pairs of geometrical shapes for space that have completely different features when each is probed by unwrapped strings. They also have completely different features when each is probed by wrapped strings. But—and this is the punch line—when probed both ways, with wrapped and unwrapped strings, the shapes become indistinguishable. What the unwrapped strings see on one space, the wrapped strings see on the other, and vice versa, rendering identical the collective picture gleaned from the full physics of string theory.

Shapes that form such pairs provide a powerful mathematical tool. In general relativity, if you’re interested in one or another physical feature, you must complete a mathematical calculation using the unique geometrical space relevant to the situation being studied. But in string theory, the existence of pairs of physically equivalent geometrical shapes means that you have a newfound choice: you can choose to perform the necessary calculation using either shape. And the extraordinary thing is that while you’re guaranteed to get the same answer using either shape, the mathematical details en route to the answer can be vastly different. In a variety of situations, overwhelmingly difficult mathematical calculations on one geometrical shape translate into exceedingly easy calculations on the other. And any framework that makes hard mathematical calculations easy is, clearly, of great value.

Over the years, mathematicians and physicists have leveraged this hard-to-easy dictionary to make headway on a number of outstanding mathematical problems. One that I’m particularly fond of has to do with counting the number of spheres that can be packed (in a particular mathematical way) within a given Calabi-Yau shape. Mathematicians had been interested in this question for a long time but found the calculations in all but the simplest cases impenetrable. Take the Calabi-Yau shape of __Figure 4.6__. When a sphere is packed into this shape, it can wrap around a portion of the Calabi-Yau multiple times, much like a lasso can wrap multiple times around a beer barrel. So, how many ways can you pack a sphere into this shape if it wraps around, say, five times? When asked a question like this, mathematicians had to clear their throats, glance at their shoes, and quickly depart for pressing appointments. String theory flattened the hurdles. By translating such calculations into far easier ones on a paired Calabi-Yau shape, string theorists produced answers that knocked mathematicians back on their heels. The number of five-times-wrapped spheres packed into the Calabi-Yau in __Figure 4.6__? 229,305,888,887,625. And if the spheres wrap around themselves ten times? . Twenty times? . These numbers proved to be harbingers for a spectrum of results that have opened a whole new chapter in mathematical discovery.^{17}

So, whether or not string theory offers a correct approach to describing the physical universe, it has already established itself as a potent tool for investigating the mathematical one.

**The State of String Theory: An Evaluation**

Building on the last four sections, __Table 4.2__ provides a status report for string theory, including some additional observations that I didn’t explicitly call out in the text above. It paints a picture of a theory in progress, one that has produced stunning achievements but has not yet been tested on the most important scale: experimental confirmation. The theory will remain speculative until a convincing link to experiment or observation is forged. Establishing such a link is the great challenge. But it’s not a challenge that’s peculiar to string theory. Any attempt to unite gravity and quantum mechanics enters a domain that’s far beyond the cutting-edge of experimental research. It’s part and parcel of taking on such a supremely ambitious goal. Pushing the fundamental boundaries of knowledge, seeking answers to some of the deepest questions contemplated during the past few thousand years of human thought, is a formidable undertaking, one that won’t likely be completed overnight. Nor in a handful of decades.

In evaluating the state of the art, many string theorists argue that a crucial next step is to articulate the theory’s equations in their most exact, useful, and comprehensive form. Much of the research during the theory’s first couple of decades, through the mid-1990s, was carried out using approximate equations that many were convinced could reveal the theory’s gross features but were too coarse to yield refined predictions. Recent advances, to which we will now turn, have catapulted understanding far beyond what could be achieved by the approximate methods. While definitive predictions have remained elusive, a new perspective has emerged. It’s come from a series of breakthroughs that has opened grand new vistas on the theory’s potential implications, among which are new varieties of parallel worlds.

**Table 4.2**. *A summary status report for string theory*.

**GOAL:** **Unite gravity and quantum mechanics**

**IS GOAL REQUIRED?:** **Yes.**

The primary goal is to meld general relativity and quantum mechanics.

**STATUS:** **Excellent.**

A wealth of calculations and insights attest to string theory’s successful merger of general relativity and quantum mechanics.^{18}

**GOAL:** **Unify all** **forces**

**IS GOAL REQUIRED?:** **No.**

Unification of gravity and quantum mechanics does not require a further unification with the other forces of nature.

**STATUS:** **Excellent.**

While not required, a fully unified theory has long been a goal of physics research. String theory achieves this goal by describing all forces in the same manner—their quanta are strings executing particular vibrational patterns.

**GOAL:** **Incorporate key** **breakthroughs from past research**

**IS GOAL REQUIRED?:** **No.**

In principle, a successful theory need bear little resemblance to successful theories from the past.

**STATUS:** **Excellent.**

Though progress isn’t necessarily incremental, history shows that it usually is; successful new theories typically embrace past successes as limiting cases. String theory incorporates the essential key breakthroughs from previously successful physical frameworks.

**GOAL:** **Explain particle properties**

**IS GOAL REQUIRED?:** **No.**

This is a noble goal, and if achieved would provide a profound level of explanation—but it is not required of a successful theory of quantum gravity.

**STATUS:** **Indeterminate; no predictions.**

Going beyond quantum field theory, string theory offers a framework for explaining particle properties. But to date, this potential remains unrealized; the many different possible forms for the extra dimensions imply many different possible collections of particle properties. There is no currently available means to pick one shape from the many.

**GOAL:** **Experimental confirmation**

**IS GOAL REQUIRED?:** **Yes.**

This is the only way to determine whether a theory is a correct description of nature.

**STATUS:** **Indeterminate; no predictions.**

This is the most important criterion; to date, string theory has not been tested on it. Optimists hope that experiments at the Large Hadron Collider and observations by satellite-borne telescopes have the capacity to bring string theory much closer to data. But there’s no guarantee that current technology is sufficiently refined to reach this goal.

**GOAL:** **Cure singularities**

**IS GOAL REQUIRED?:** **Yes.**

A quantum theory of gravity should make sense of singularities arising in situations that are, even just in principle, physically realizable.

**STATUS:** **Excellent.**

Tremendous progress; many kinds of singularities have been resolved by string theory. The theory still needs to address black hole and big bang singularities.

**GOAL:** **Black hole entropy**

**IS GOAL REQUIRED?:** **Yes.**

A black hole’s entropy provides a hallmark context in which general relativity and quantum mechanics interface.

**STATUS:** **Excellent.**

String theory has succeeded in explicitly calculating, and confirming, the entropy formula proposed in the 1970s.

**GOAL:** **Mathematical contributions**

**IS GOAL REQUIRED?:** **No.**

There’s no requirement that correct theories of nature yield mathematical insights.

**STATUS:** **Excellent.**

Although mathematical insights aren’t necessary to validate string theory, significant ones have emerged from the theory, revealing the profound reach of its mathematical underpinnings.

__*__If you’d like to know how string theory surmounts the problems that blocked earlier attempts to join gravity and quantum mechanics, see *The Elegant Universe*, Chapter 6; for a sketch, see __note 8__. For an even briefer summary, note that whereas a point particle exists at a single location, a string, because it has length, is slightly spread out. This spreading, in turn, dilutes the raucous short-distance quantum jitters that stymied previous attempts. By the late 1980s, there was strong evidence that string theory successfully melds general relativity and quantum mechanics; more recent developments (__Chapter 9__) make the case overwhelming.