The Hidden Reality Part 3

By this, Einstein meant a framework that would st.i.tch 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 n.o.ble, the timing was early-but others have taken up the mantle and made substantial strides, the most refined proposal being string theory string theory.

My previous books The Elegant Universe The Elegant Universe and and The Fabric of the Cosmos 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 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 Chapters 5 and and 6 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 produce magnetic fields; magnets moving in the vicinity of a wire magnetic fields; magnets moving in the vicinity of a wire produce produce electric currents; and wavelike disturbances rippling through electric and magnetic fields electric currents; and wavelike disturbances rippling through electric and magnetic fields produce produce light. Einstein antic.i.p.ated 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. light. Einstein antic.i.p.ated 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 pa.s.sion 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 const.i.tuents was immediate and powerful. While many agreed that unification was a laudable goal, it was of only pa.s.sing interest in an age when theorists and experimenters were working hand in glove to reveal the laws of the microscopic realm. With Einstein's pa.s.sing, 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 strong nuclear force and the and the weak nuclear force 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 n.o.bel Prizewinning 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 relations.h.i.+ps 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 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 a.n.a.lyze 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 a.s.sess 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.

Cla.s.sical physics, as we saw in Chapter 3 Chapter 3, describes a field as a kind of mist that permeates a region of s.p.a.ce 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 s.p.a.ce 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 s.p.a.ce. 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 s.p.a.ce to jitter randomly-think of the fluctuating inflaton field from inflationary cosmology. Second, quantum mechanics establishes that, somewhat as water is composed of H2O molecules, a field is composed of infinitesimally small particles known as the field's quanta quanta. For the electromagnetic field, the quanta are photons, and so a quantum theorist would modify Maxwell's cla.s.sical 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 a.s.sociated 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 a.s.sociated 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 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 ten decimal places, an almost unimaginable agreement between theory and experiment. decimal places, an almost unimaginable agreement between theory and experiment.

With such success, you might antic.i.p.ate that quantum field theory would provide the mathematical framework for understanding all of nature's forces. An ill.u.s.trious 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 infinity. While some things in the universe can be infinite, such as the extent of s.p.a.ce and the quant.i.ty 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 a.n.a.lyzing 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 s.p.a.cetime 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 s.p.a.cetime-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 s.p.a.cetime, its quantum jitters shake the entire structure through and through. When used to a.n.a.lyze 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 ma.s.sive, quantum mechanics when things are very small. And rare is the realm that is both small and ma.s.sive, 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 do involve extremes of enormous ma.s.s squeezed to small size, the math falls apart at a critical point in the a.n.a.lyses, leaving us with unanswered questions regarding how the universe began and how, at the crus.h.i.+ng center of a black hole, it might end. involve extremes of enormous ma.s.s squeezed to small size, the math falls apart at a critical point in the a.n.a.lyses, leaving us with unanswered questions regarding how the universe began and how, at the crus.h.i.+ng 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 ma.s.sive and how small a physical system needs to be for both gravity and quantum mechanics to play a significant role. The result is about 1019 times the ma.s.s of a single proton, the so-called times the ma.s.s of a single proton, the so-called Planck ma.s.s Planck ma.s.s, squeezed into a fantastically small volume of about 1099 cubic centimeters (roughly a sphere with a radius of 10 cubic centimeters (roughly a sphere with a radius of 1033 centimeters, the so-called centimeters, the so-called Planck length Planck length graphically ill.u.s.trated in graphically ill.u.s.trated in Figure 4.1 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 a.s.sociated 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. 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 a.s.sociated 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 s.p.a.ced 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 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 s.p.a.ced 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 a.s.sociated 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 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 ma.s.sive 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 ma.s.s 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 a.s.sociated 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 const.i.tuents of matter are string-like filaments. Because of the limited resolving power of our equipment, the strings appear as dots String theory's proposal for the nature of physics at the Planck scale envisions that the fundamental const.i.tuents 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 being the particle. Because of the string's infinitesimal size, on the order of the Planck length-10 the particle. Because of the string's infinitesimal size, on the order of the Planck length-1033 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 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 1019 centimeters; that's a millionth of a billionth the width of a strand of hair, but still orders of magnitude too centimeters; that's a millionth of a billionth the width of a strand of hair, but still orders of magnitude too large 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 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 are strings. strings.

In a nutsh.e.l.l, 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 Chapters 5, 6 6, and 9 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 a.s.sociated quantum field) as well as theoretical concerns (hypothetical particles and their a.s.sociated fields, like the inflaton and Higgs fields, are invoked to address open problems or puzzling issues). The Standard Model 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 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 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 a.s.sortment of ingredients. 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 a.s.sortment 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 all properties of properties of all 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. 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 prestring theoretic attempts to merge gravity and quantum mechanics were unsuccessful, research did reveal the properties that any hypothesized particle a.s.sociated with the quantum gravitational field-dubbed the graviton graviton-would necessarily possess. The studies concluded that the graviton must be ma.s.sless and chargeless, and must have the quantum mechanical property known as spin-2 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, Joel 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 Wonderfully, early string theorists-John Schwarz, Joel 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 string theory provided a long-sought quantum theory of gravity. This is the most important accomplishment on string theory's resume 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 ma.s.ses (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 ma.s.ses and they reduce to the descriptions developed prior to the twentieth century-Newton's cla.s.sical mechanics and the cla.s.sical fields of Faraday, Maxwell, and others.

String theory is potentially the next and final step in this progression. In a single 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 s.p.a.cetime picture of gravity. Nevertheless, apply string theory's mathematics to a situation where gravity matters but quantum mechanics doesn't (to a ma.s.sive 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 ma.s.s-so gravity plays virtually no role) and the math of string theory morphs into the math of quantum field theory. 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 s.p.a.cetime picture of gravity. Nevertheless, apply string theory's mathematics to a situation where gravity matters but quantum mechanics doesn't (to a ma.s.sive 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 ma.s.s-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 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 relations.h.i.+ps between the major theoretical developments in physics. Historically, successful new theories have extended the domain of understanding (to faster speeds, larger ma.s.ses, shorter distances) while reducing to previous theories when applied in less extreme physical circ.u.mstances. 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 A graphical representation of the relations.h.i.+ps between the major theoretical developments in physics. Historically, successful new theories have extended the domain of understanding (to faster speeds, larger ma.s.ses, shorter distances) while reducing to previous theories when applied in less extreme physical circ.u.mstances. 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 s.p.a.ce.

Now for something stranger. The pa.s.sage 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 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 s.p.a.ce. 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 s.p.a.ce and one of time, for a total of ten s.p.a.cetime 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 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 D represents the number of s.p.a.cetime dimensions and represents the number of s.p.a.cetime dimensions and Trouble 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 s.p.a.cetime dimensions is ten, not the four we expect, the contribution becomes 0 times 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 s.p.a.cetime dimensions is ten, not the four we expect, the contribution becomes 0 times Trouble Trouble. And since 0 times anything is 0, in a universe with ten s.p.a.cetime 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 s.p.a.cetime 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 s.p.a.ce 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 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 s.p.a.ce itself. The reason we are all aware of the familiar three dimensions of s.p.a.ce 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 s.p.a.ce 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 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 s.p.a.ce. And when he a.n.a.lyzed 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. 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 s.p.a.ce and one of time. But because his new formulation included an additional dimension of s.p.a.ce, Kaluza found an additional equation. Within the modified equations, Kaluza found the ones Einstein had already used successfully to describe gravity in the familiar three dimensions of s.p.a.ce and one of time. But because his new formulation included an additional dimension of s.p.a.ce, 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 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 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) (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 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 ill.u.s.tration. 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 ill.u.s.tration.) Kaluza revealed that in a universe with an additional dimension of s.p.a.ce, 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 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. 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 antic.i.p.ated, 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 a.s.sessments 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 ma.s.s 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. 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 ma.s.ses and charges.12 In an imaginary world where the electron's ma.s.s 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. In an imaginary world where the electron's ma.s.s 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 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 ma.s.s 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 cla.s.s called Calabi-Yau shapes Calabi-Yau shapes (or, in mathematical jargon, (or, in mathematical jargon, Calabi-Yau manifolds Calabi-Yau manifolds), named after the mathematicians Eugenio Calabi and s.h.i.+ng-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 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 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 a.n.a.lysis-but that's what graduate students are for. As time pa.s.sed, however, the pages of the Calabi-Yau catalog continued to multiply; as we will see in Chapter 5 Chapter 5, they have now grown more numerous than grains of sand on a beach. Every beach. Everywhere. By a long shot. To a.n.a.lyze 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 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. 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 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. 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 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. 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 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. 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 ma.s.ses 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. a.n.a.logous considerations applied to the experiments summarized in Table 4.1 Table 4.1 have the capacity to build a circ.u.mstantial case for string theory. have the capacity to build a circ.u.mstantial 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 a.s.sessment 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 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 const.i.tute 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 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 Experiments and Observations with the Capacity to Link String Theory to Data EXPERIMENT/OBSERVATION: Supersymmetry Supersymmetry.

EXPLANATION: The "super" in superstring theory refers to The "super" in superstring theory refers to supersymmetry 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 antic.i.p.ation that we may be on the threshold of revealing nature's supersymmetric quality.

EXPERIMENT/OBSERVATION: Extra Dimensions and Gravity Extra Dimensions and Gravity EXPLANATION: Because s.p.a.ce 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 Because s.p.a.ce 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