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The Elegant Universe

Brian Greene

Part III: The Cosmic Symphony

Part II: The Dilemma of Space, Time, and the Quanta

Part IV: String Theory and the Fabric of Spacetime

Chapter 6: Nothing but Music: The Essentials of Superstring Theory

The standard model, which describes the elementary particles of the universe as amorphous, zero-dimensional points, is not comprehensive because it ignores gravity. Superstring theory, on the other hand, describes the most basic ingredients of matter as Planck-length strings that vibrate perpetually, like tiny rubber bands.

Before explaining why only string theory can resolve the conflict between general relativity and quantum mechanics, Greene supplies a brief history of the origins of string theory. In 1968, theoretical physicist Gabriele Veneziano was trying to understand the strong nuclear force when he made a startling discovery. Veneziano found that a 200-year-old formula created by Swiss mathematician Leonhard Euler (the Euler beta function) perfectly matched modern data on the strong force. Veneziano applied the Euler beta function to the strong force, but no one could explain why it worked.

Two years later, Yochiro Nambu, Holger Nielsen, and Leonard Susskind unveiled the physics beneath Euler’s strictly theoretical formula. By representing nuclear forces as vibrating, one-dimensional strings, these physicists showed how Euler’s function accurately described those forces. But even after physicists understood the physical explanation for Veneziano’s insight, the string description of the strong force made many predictions that directly contradicted experimental findings. The scientific community soon lost interest in string theory, and the standard model, with its particles and fields, remained unthreatened.

Then, in 1974, John Schwarz and Joel Scherk studied the messenger-like patterns of string vibration and found that their properties exactly matched those of the gravitational force’s hypothetical messenger particle. Schwarz and Scherk argued that string theory had failed to catch on because physicists had underestimated its scope.

But the enthusiasm faded fast, for string theory’s conflicts with quantum mechanics remained unresolved. Then, in 1984, Schwarz and Michael Green declared that string theory was capable of explaining all four forces and all matter as well. String theory, they said, was not a strong force theory, but a quantum theory that also included gravity. Shwarz’s and Green’s reinterpretation marked the first-ever quantum mechanical theory of the gravitational force.

In 1984, Brian Greene started graduate school at Oxford University. The reaction to Schwarz’s and Green’s discovery was to influence the direction of his research for the next two decades. Greene came to believe that particle physics had no future: only string theory, he believed, could explain all properties of the microworld. String theory can make numeric predictions that the standard model, which is too flexible to explain the properties of elementary particles, can only assume.

From 1984 to 1986, the first superstring revolution swept the physics community, and more than a thousand research papers on the subject were published. The problem with string theory, then as now, was that the equations themselves were so difficult that physicists could only deduce approximations of both the equations and their solutions.

The second superstring revolution occurred in 1995, when Edward Witten delivered a groundbreaking lecture that introduced methods for dealing with the theory’s complexity.

Before detailing Witten’s insights, Greene returns to the basic question: what are strings made of? Well, like Greeks once supposed atoms to be, strings are indivisible (at least according to string theorists). They are nature’s most fundamental ingredient. Strings are like vibrating threads, and—like strings on a violin—can undergo an infinite number of vibrational patterns, which are called resonances. The different vibrational patterns yield all of the different force masses and force charges. This is the bottom line of string theory: elementary particles are determined by precise patterns of vibration.

Greater energy means greater mass (and vice versa). Therefore, the mass of an elementary particle is determined by the energy of its internal string’s vibrational pattern. Since all fundamental forces are influenced by mass and energy, the vibrational mode of any given string conveys electric charge, weak charge, and strong charge. Most exciting of all—and here is where string theory surpasses the standard model—is the fact that one vibrational mode promises to match the graviton exactly.

Because particle differences stem from different vibrational patterns, physicists believe that if they can work out the “notes” of string theory, they can explain the observed properties of the elementary particles. And with this claim we reach the radical proposition underlying the so-called Theory of Everything that is string theory: that the stuff of all matter and the forces of nature are . . . exactly the same thing!

The stiffer a string, the more energy it requires to set it in motion (think of plucking a very tight guitar string). Fundamental strings operate at a colossal tension called Planck tension, which equals a thousand billion billion billion billion tons.

These massive tensions make the strings contract to an extremely tiny size, meaning that the energy of a vibrating loop will be extremely high. This energy level is determined by two factors: the string’s vibrational pattern and its tension. The fundamental minimal energies are enormous because the strings are so stiff. This is called Planck energy. The corresponding mass, known as the Planck mass, is therefore enormous as well.

String theory, Greene says, cushions the violent quantum fluctuations that occur at Planck length by “smearing” space’s short-distance properties. Describing how this works is tricky. Essentially, the size of the probe particle sets a lower limit to the sensitivity of the scale, meaning that smaller probes can determine finer detail. Particle accelerators use protons or electrons as probes (or “pellets”) because their tiny size makes it easier for them to gauge subatomic features.

In 1988, David Gross and Paul Mende showed that increasing the energy of a string does not increase its ability to probe more delicate structures. (The opposite is true with point particles.) Quantum fluctuations—the source of so many frustrations for physicists—are responsible for this “smearing.”

The whole conflict between general relativity and quantum mechanics occurs only on the smallest scale of the universe, at sub-Planck-length scales. In the point-particle standard model, interactions happen at a precise location in time, but interactions among strings are more spread out; different observers in different states of motion can observe different contact times. Smearing, within this framework, evens out the quantum fluctuations that distort the fabric of space at sub-Planck scale distances.

Previously, physicists who attempted to combine the equations of general relativity with the equations of quantum mechanics would come up with one impossible answer: infinity. But when strings are taken into account, the calculations yield finite answers, which resolve the mathematical incompatibility between general relativity and quantum mechanics.

This insight was a revelation to string theorists, providing convincing theoretical evidence that point particles were not the true rudiments of the universe. But string theory does not only deal with strings. It also includes multidimensional building blocks: two-dimensional Frisbee-like structures, three-dimensional blobs, and perhaps even more elaborate shapes.

Chapter 7: The “Super” in Superstrings

Einstein believed that general relativity was “almost too beautiful” to be wrong; Greene believes exactly the same thing about string theory. Of course, he reminds us, we are only interested in theories insofar as they apply to the real world. But though theories cannot survive on aesthetics alone, symmetry is as crucial in science as it is in art. The word elegance describes the complexity of diverse phenomena arising from a simple set of laws. The laws governing the universe must be fixed, unchanging, all-applicable, and, at their core, elegant.

The term supersymmetry was coined to describe theories that unite the four forces of nature with the elementary constituents of the universe—the supreme elegance that is string theory. It was the discovery of supersymmetry that helped resolve the original glitches with the first incarnation of string theory in the early 1970s.

Here’s where something called spin becomes important. In 1925, Dutch physicists George Uhlenbeck and Samuel Goudsmit proved that, just as the earth spins on its axis, all electrons both revolve and rotate, spinning at one fixed, never-changing rate forever. This quantum mechanical property is intrinsic to the electron, meaning that if it’s not spinning, it’s not an electron. And because point-particles are zero-dimensional, they cannot undergo this rotational motion.

In the early 1970s, physicists analyzed the vibrational patterns of the first incarnation of string theory, which is called bosonic string theory. Bosonic string theory means that the string’s vibrational patterns must have whole-number spins. Unfortunately, one pattern of vibration had a negative mass called a tachyon. The existence of a tachyon pointed to some essential missing component in bosonic string theory.

In 1971, Pierre Ramond managed to modify the equations of bosonic string theory to take half-integer vibrational patterns (called fermionic patterns) into account as well. Physicists soon realized that bosonic and fermionic vibrational patterns seemed to come in pairs, and this discovery gave rise to supersymmetry, a term that describes the relationship between these integer and half-integer spin values. (Because it is so complicated, Greene makes no attempt to describe the mathematical underpinnings of supersymmetry with any more precision.) Bosonic string theory was soon replaced by supersymmetrical string theory, which reflected the symmetrical character of bosonic and fermionic vibrational pattern. The tachyon vibration of the bosonic string has no effect on the superstring.

According to supersymmetry, particles of nature come in pairs with respective spins differing by half a unit; these are called superpartners. (Scientists differentiate superpartners from one another by adding an s: the quark joins with the “squark,” the electron with the “selectron,” and so forth. Force-particle superpartners take the “-ino” suffix: the photino, the wino and the zino, and so forth.) Since all particles of elementary matter—quarks, electrons, and muons—have spin-1/2 and messenger particles have spin-1, supersymmetry produces a tidy pairing between matter and force particle. (As usual, the mass-less, still-undetected graviton is the exception. Scientists predict that graviton will have spin-2.)

The standard model demands extremely fine-tuned parameters for its particle interactions. With supersymmetry, on the other hand, the superpartners cancel one another out. The anomalies that once seemed so perilous to string theory cease to exist. The resulting cosmic system is far less sensitive than the one the standard model describes.

In 1974, Howard Georgi, Helen Quinn, and Weinberg studied the effect that quantum physics has on force strengths. At the level of quantum fluctuations, the eruptions amplify the strengths of both the strong and the weak force. The strengths get weaker when probed at shorter distances. Georgi, Quinn, and Weinberg concluded that the strengths of the three nongravitational forces are driven together at this scale. They found that the strengths of these three forces are almost—but not quite—identical at microscopic distance scales. But when you factor in supersymmetry, these tiny strength differences disappear altogether

Beyond these contributions, supersymmetrical string theory promises to unify gravity with the other three fundamental forces in one coherent framework. Schwarz and Scherk realized that one particular vibrational pattern of string corresponded exactly to the hypothetical properties of the graviton particle, which led them to believe that string theory alone could fuse quantum mechanics with gravity.

But in 1985, in the wake of the first superstring revolution, physicists found that supersymmetry could be incorporated into string theory in a grand total of five different ways. What Greene describes as a “super-embarrassment of riches” troubled string theorists who were searching for a single, inevitable theory. It was not until 1995 that Edward Witten showed that these five versions of string theory were really just five different ways of understanding the same theory.

Chapter 8: More Dimensions Than Meet the Eye

Einstein resolved the two biggest scientific conflicts of the past century with special and then general relativity. String theorists have set out to tackle the third large conflict.

In 1919, the all-but-unknown German mathematician Theodor Kaluza made the outlandish suggestion that the universe might have more than three spatial dimensions. To illustrate Kaluza’s claim, Greene asks readers to imagine an ant traversing a garden hose. From far away, the hose resembles a one-dimensional line. But the hose also has a circular dimension. The naked eye cannot perceive this extra dimension from afar, but that doesn’t mean that it doesn’t exist. This analogy shows that dimensions can come in two different varieties: those that are big and easy to spot, like the left/right dimension of the garden hose; and those that are smaller and more difficult to see, like the clockwise/counterclockwise dimension wrapping the surface of the hose.

In 1926, Swedish physicist Oskar Klein refined Kaluza’s hypothesis by proposing that this extra dimension might take the form of tiny circles as small or smaller than Planck length. Perhaps the three dimensions we recognize are simply like the left/right line of the garden hose. If the garden hose has another curled-up, hard-to-see dimension, maybe the fabric of the universe does as well.

Kaluza-Klein theory developed from a combination of the two men’s hypotheses about additional, ultramicroscopic dimensions in space. Applying quantum mechanical principles to Kaluza’s initial observations, Klein found that the radius of another circular dimension would be about Planck length—in other words, far too small for even the most advanced equipment to detect.

Adding another spatial dimension produced the unforeseen result of unifying Einstein’s theory of gravity with Maxwell’s theory of light. Before Kaluza, everyone assumed that gravity and electromagnetism were two completely unrelated forces. But although Einstein took a brief interest in Kaluza’s postulation, most physicists ignored it. Einstein dabbled with Kaluza-Klein theory through the early 1940s, but when it proved impossible to include the electron in the extra dimension, he dropped the idea altogether.

Then, in the mid-1970s, physicists applied their more advanced understanding of physics to Kaluza’s fifty-year-old suggestion. The problem, they found, was not that Kaluza had been too radical, but that he had been too conservative. Kaluza, and later Klein, had proposed adding only one dimension of space, but string theory’s early quantum mechanical equations necessitated adding even more. Physicists began feverishly researching the possibility of an extradimensional universe, and the term higher-dimensional supergravity was invented to describe theories that include gravity, additional dimensions, and supersymmetry.

When physicists posited the existence of nine spatial dimensions, probability calculations no longer yielded negative numbers. (These results were mathematically unfeasible, since all probabilities must fall between 0 and 1, or—when expressed as percentages—0 and 100 percent.) This meant that, according to string theory, the universe had ten dimensions: nine of space and one of time. (In the 1990s, Witten rocked the physics community by suggesting that string theory requires not nine but ten dimensions of space and one of time, for a grand total of eleven dimensions.)

The shape and size of the extra six dimensions has a huge impact on the vibrational patterns of the tiny, curled-up strings, so it’s crucial to understand the geometry. The more dimensions that exist, the more directions that strings can vibrate. Extradimensional geometry determine the basic physical characteristics of elementary particles, like particle masses and charges, all of which can influence the physical features of our universe—even though we can only observe our universe in three dimensions.

Figuring out what these extra dimensions look like isn’t easy, mostly because they are so tiny—far too small for even the most advanced scientific equipment to pick up. The likeliest configuration seems to be a six-dimensional geometrical shape called a Calabi-Yau space, named after the mathematicians Eugenio Calabi and Shing-Tung Yau, who discovered these shapes mathematically long before they had any bearing on string theory. Greene suggests that the basic structure of the cosmos could be found in the geometry of a Calabi-Yau space. But which one? Herein lies the difficulty. Calabi-Yau spaces come in thousands of varieties, all of which require extremely precise computations to verify.

Chapter 9: The Smoking Gun: Experimental Signatures

Now, back to the usual problem: theories have no value unless they can be confirmed experimentally and applied to the real world. String theory could well be the most predictive cosmic theory scientists have ever studied, but the experimental data is not yet precise enough to allow any predictions. The “instruction model,” as Greene calls it, is not yet written.

Since its earliest incarnation, string theory has attracted a great many doubters and detractors, physicists who question the utility of a theory that cannot be verified experimentally. Prominent among these naysayers is Harvard physicist Sheldon Glashow, who wonders if the elegance of a proposition has any bearing on its accuracy.

Because a particle accelerator capable of detecting Planck-length-scaled strings would require a tremendous amount of energy, string theorists must seek to confirm their theories indirectly, through mathematical proofs.

Witten and fellow string theorists believe that a family of particles exists that corresponds with every hole in the Calabi-Yau space. The problem is that no one knows which Calabi-Yau space correctly describes the additional spatial dimensions. The math is still so complicated that physicists must rely on a formal practice called perturbation theory, which allows them to make convoluted calculations involving multiple variables. Perturbation theory is a mathematics of approximation that physicists hope will lead them to the correct Calabi-Yau shape.

Progress in the field is slow but constant. In 1999, when The Elegant Universe was first published, Greene and his string theorist colleagues were focused on reducing the number of possible Calabi-Yau spaces by finding shapes (like that of a three-holed donut) that can be distorted in many ways without losing their essential shape.

At CERN in Geneva, a mammoth accelerator called the Large Hadron Collider is under construction and will be completed in 2010. The Large Hadron Collider is designed to prove the theoretical existence of superparticles, which would provide experimental proof of supersymmetry. String theory predicts that every known particle has a superpartner, and while physicists have determined these particles’ force changes, they cannot predict their masses. Physicists also hope to find fractionally charged particles. As it is, the elementary particles of the standard model have extremely limited electric charges. String theory predicts that resonant vibrational patterns can correspond to particles with a much wider range of charges.

Other string theorists hope to connect their theories to direct experimental observation using a variety of long-shot methods. These include: finding strings much larger than the Planck length; determining if neutrinos are extremely light or massless; locating new, tiny, long-range force fields; and finally, proving (or disproving) astronomers’ evidence that the entire universe is submerged in dark matter. For the moment, however, the terrain of applied superstring theory remains mostly uncharted. Physicists, Greene warns, can expect to labor for several more generations without making another sustaining breakthrough. Without experimental results to guide them, string theorists must simply brace themselves and continue plugging in numbers.

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