The Elegant Universe

by Brian Greene

Crossfire

And, if you're either relatively conservative or fond of playing devil's advocate, you might even say that physicists have no business wasting time on a theory that postulates a new feature of nature some hundred million billion times smaller than anything we can directly probe experimentally.

Yeah! That's me!

Richard Feynman, shortly before he died, made it clear that he did not believe that string theory was the unique cure for the problems—the pernicious infinities, in particular—besetting a harmonious merger of gravity and quantum mechanics: My feeling has been—and I could be wrong—that there is more than one way to skin a cat. I don't think that there's only one way to get rid of the infinities. The fact that a theory gets rid of infinities is to me not a sufficient reason to believe its uniqueness.

The Road to Experiment

If we are going to test string theory experimentally, it will have to be in an indirect manner. We will have to determine physical implications of string theory that can be observed on length scales that are far larger than the size of a string itself.

Why families and why three?

There is a family of lowest-energy string vibrations associated with each hole in the Calabi-Yau portion of space. Because the familiar elementary particles should correspond to the lowest-energy oscillatory patterns, the existence of multiple holes—somewhat like those in the multidoughnut—means that the patterns of string vibrations will fall into multiple families. If the curled-up Calabi-Yau has three holes, then we will find three families of elementary particles.

The problem is that at present no one knows how to deduce from the equations of string theory which of the Calabi-Yau shapes constitutes the extra spatial dimensions.

It's hard to visualize, but the idea is that as strings vibrate through the extra curled-up dimensions, the precise arrangement of the various holes and the way in which the Calabi-Yau shape folds around them has a direct impact on the possible resonant patterns of vibration. Although the details get difficult to follow and are really not all that essential, what is important is that, as with the number of families, string theory can provide us with a framework for answering questions—such as why the electron and other particles have the masses they do—on which previous theories are completely silent.

And so, once again, we come to realize that string theory provides a framework for explaining the observed messenger-particle content of our universe, that is, for explaining the properties of the fundamental forces, but that without knowing exactly which Calabi-Yau shape the extra dimensions are curled into, we cannot make any definitive predictions or postdictions (beyond Witten's remark regarding the postdiction of gravity).

Excuses.

In this approximation scheme, each possible Calabi-Yau shape appears to be on equal footing with every other; none is fundamentally singled out by the equations. And since the physical consequences of string theory depend sensitively on the precise form of the curled-up dimensions, without the ability to select one Calabi-Yau space from the many, no definitive experimentally testable conclusions can be drawn.

Exhausting Possibilities

A sensible start is to focus only on those Calabi-Yau shapes that yield three families.

But precise and definitive physical conclusions, such as the mass of the electron or the strength of the weak force, require equations that are far more exact than the present approximate framework.

Explaining the elementary matter and force particle properties would be among the greatest—if not the greatest—of scientific achievements. Nevertheless, you might ask whether there are any string theoretic pre- dictions—as opposed to postdictions—that experimental physicists could attempt to confirm, either now or in the foreseeable future. There are.

Superparticles

Even so, the prediction that superpartners exist is a generic feature of string theory; it is a property of string theory that is true, independent of those aspects of the theory we haven't yet figured out.

Where the hell are they, Brian?

No superpartners of the known elementary particles have ever been observed. This might mean that they do not exist and that string theory is wrong.

As Schwarz has said, "Supersymmetry ought to be discovered before too long. And when that happens, it's going to be dramatic."

Wrong!

Fractionally Charged Particles

Some Calabi-Yau shapes have this geometrical property while others do not, and for this reason the possibility of unusual electric-charge fractions is not as generic as the existence of superpartner particles.

As with the situation with superpartners, no such exotically charged particles have ever been observed, and our understanding of string theory does not allow for a definitive prediction of their masses should the extra dimensions have the correct properties to generate them.

Some Longer Shots

The energy of the big bang, in fact, would have been high enough to produce a few macroscopically large strings that, through cosmic expansion, might have grown to astronomical scales.

A challenge to string theory is to provide a compelling explanation of present and future neutrino data, especially if experiments ultimately show that neutrinos do have a tiny but nonzero mass.

Observations show that the cosmological constant is either zero (as Einstein ultimately suggested) or quite small; calculations indicate that quantum-mechanical fluctuations in the vacuum of empty space tend to generate a nonzero cosmological constant whose value is some 120 orders of magnitude (a 1 followed by 120 zeros) larger than experiment allows!

An Appraisal

The history of physics is filled with ideas that when first presented seemed completely untestable but, through various unforeseen developments, were ultimately brought within the realm of experimental verifiability.

And a plethora of them that have not.

The motivation for introducing string theory is at least as compelling as any of these three ideas—in fact, string theory has been hailed as the most important and exciting development in theoretical physics since the discovery of quantum mechanics.

By overly optimistic physicists who ignore lack of experimental evidence.

Unlike those who worked out quantum theory, today's string theorists do not have the shining light of nature—through detailed experimental results—to guide them from one step to the next.

The blind leading the blind.

Part IV

String Theory and the Fabric of Spacetime

Chapter 10

Quantum Geometry

Whereas general relativity asserts that the curved properties of the universe are described by Riemannian geometry, string theory asserts that this is true only if we examine the fabric of the universe on large enough scales. On scales as small as the Planck length a new kind of geometry must emerge, one that aligns with the new physics of string theory. This new geometrical framework is called quantum geometry.

The Heart of Riemannian Geometry

Mathematically, the curvature of spacetime—like the curvature of the trampoline—reflects the distorted distance relations between its points. Physically, the gravitational force felt by an object is a direct reflection of this distortion.

But string theory limits how precisely Riemann's geometrical formalism can be realized by the physics of gravity, because there is a limit to how small we can make any object. Once you get down to strings, you can't go any further. The traditional notion of a point particle does not exist in string theory—an essential element in its ability to give us a quantum theory of gravity.

A Cosmological Playground

If the average matter density exceeds a so-called critical density of about a hundredth of a billionth of a billionth of a billionth (10-29) of a gram per cubic centimeter—about five hydrogen atoms for every cubic meter of the universe—then a large enough gravitational force will permeate the cosmos to halt and reverse the expansion. If the average matter density is less than the critical value, the gravitational attraction will be too weak to stop the expansion, which will continue forever.

No one has figured out the identity of the dark matter, let alone the precise amount that exists.

The fate of our presently expanding universe, therefore, is as yet unclear.

The expansion of the universe is accelerating. If it continues to do so, the universe will expand without limit. So sad!

But when the distance scales involved are around the Planck length or less, quantum mechanics invalidates the equations of general relativity, as we are now well aware. We must instead make use of string theory.

Or some experimental a verifiable theory.

The Essential New Feature

The Physics of Wound Strings

The essential difference is that a wrapped string has a minimum mass, determined by the size of the circular dimension and the number of times it wraps around. The string's oscillatory motion determines a contribution in excess of this minimum.

A wound string has a minimum length determined by the circumference of the circular dimension and the number of times the string encircles it. The minimum length of a string determines its minimum mass: The longer this length, the greater the mass, since there is more of it.

(Unwrapped strings also have a tiny minimum length since if they didn't, we would be back in the realm of point particles. The same reasoning might lead to the conclusion that even unwrapped strings have a minuscule yet nonzero minimum mass. In a sense this is true, but the quantum-mechanical effects encountered in Chapter 6—remember The Price Is Right, again—are able to exactly cancel this contribution to the mass. This is how, we recall, unwrapped strings can yield the zero-mass photon, graviton, and the other massless or near-massless particles, for example. Wrapped strings are different in this regard.)

The circular dimension can shrink to the Planck-length. But because of the winding modes, attempts to shrink further actually result in expansion.

The Spectrum of String States*