**Table of Contents**

*.......The Elegant Universe*

**THE ELEGANT UNIVERSE,****Brian Greene,**1999, 2003

```(annotated and with added

**bold highlights by Epsilon=One**)

**Chapter 9 - The Smoking Gun: Experimental Signatures**

The Road to Experiment

Without monumental technological breakthroughs, we will never be able to focus on the tiny length scales necessary to see a string directly. Physicists can probe down to a billionth of a billionth of a meter with accelerators that are roughly a few miles in size. Probing smaller distances requires higher energies and this means larger machines capable of focusing that energy on a single particle. As the Planck length is some 17 orders of magnitude smaller than what we can currently access, using today's technology we would need an accelerator the size of the galaxy to see individual strings. In fact, Shmuel Nussinov of Tel Aviv University has shown that this rough estimate based on straightforward scaling is likely to be overly optimistic; his more careful study indicates that we would require an accelerator the size of the whole universe. (The energy required to probe matter at the Planck length is roughly equal to a thousand kilowatt-hours—the energy needed to run an average air conditioner for about one hundred hours—and so is not particularly outlandish. The seemingly insurmountable technological challenge is to focus all of this energy on a single particle, that is, on a single string.) As the U.S. Congress ultimately canceled funding for the Superconducting Supercollider—an accelerator a "mere" 54 miles in circumference—don't hold your breath while waiting for the money for a Planck-probing accelerator. 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.

In their groundbreaking paper, Candelas, Horowitz, Strominger, and Witten took the first steps toward this goal. They not only found that the extra dimensions in string theory must be curled up into a Calabi-Yau shape, but they also worked out some of the implications this has on the possible patterns of string vibrations. One central result they found highlights the amazingly unexpected solutions string theory offers to longstanding particle-physics problems.

Recall that the elementary particles that physicists have found fall into three families of identical organization, with the particles in each successive family being increasingly massive. The puzzling question for which there was no insight prior to string theory is, Why

There is a

You might think that the number of holes in the curled-up Planck-sized dimensions—mountaintop physics

The number of families is but one experimental consequence of the geometrical form of the extra dimensions. Through their effect on possible patterns of string vibrations, other consequences of the extra dimensions include the detailed properties of the force and matter particles. As one primary example, subsequent work by Strominger and Witten showed that the masses of the particles in each family depend upon—hang on, this is a bit tricky—the way in which the boundaries of the various multidimensional holes in the Calabi-Yau shape intersect and overlap with one another. 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. Once again, though, carrying through with such calculations requires that we know which Calabi-Yau space to take for the extra dimensions.

The preceding discussion gives some idea of how string theory may one day explain the properties of the matter particles recorded in Table 1.1. String theorists believe that a similar story will one day also explain the properties of the messenger particles of the fundamental forces, listed in Table 1.2. That is, as strings twist and vibrate while meandering through the extended and curled-up dimensions, a small subset of their vast oscillatory repertoire consists of vibrations with spin equal to 1 or 2. These are the candidate force-carrying string-vibrational states. Regardless of the shape of the Calabi-Yau space, there is always one vibrational pattern that is massless and has spin-2; we identify this pattern as the graviton. The precise list of spin-1 messenger particles—their number, the strength of the force they convey, the gauge symmetries they respect—though, does depend crucially on the precise geometrical form of the curled-up dimensions. 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).

Why can't we figure out which is the "right" Calabi-Yau shape? Most string theorists blame this on the inadequacy of the theoretical tools currently being used to analyze string theory. As we shall discuss in some detail in Chapter 12, the mathematical framework of string theory is so complicated that physicists have been able to perform only approximate calculations through a formalism known as

*15*In their groundbreaking paper, Candelas, Horowitz, Strominger, and Witten took the first steps toward this goal. They not only found that the extra dimensions in string theory must be curled up into a Calabi-Yau shape, but they also worked out some of the implications this has on the possible patterns of string vibrations. One central result they found highlights the amazingly unexpected solutions string theory offers to longstanding particle-physics problems.

Recall that the elementary particles that physicists have found fall into three families of identical organization, with the particles in each successive family being increasingly massive. The puzzling question for which there was no insight prior to string theory is, Why

*families*and why*three?*Here is string theory's proposal. A typical Calabi-Yau shape contains holes that are analogous to those found at the center of a phonograph record, or a doughnut, or a "multidoughnut", as shown in Figure 9.1. In the higher-dimensional Calabi-Yau context, there are actually a variety of different types of holes that can arise—holes which themselves can have a variety of dimensions ("multidimensional holes")—but Figure 9.1 conveys the basic idea. Candelas, Horowitz, Strominger, and Witten closely examined the effect that these holes have on the possible patterns of string vibration, and here is what they found.**Figure 9.1**A doughnut, or torus, and its multihandled cousins.

*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.**And so, string theory proclaims that the family organization observed experimentally, rather than being some unexplainable feature of either random or divine origin, is a reflection of the number of holes in the geometrical shape comprising the extra dimensions! This is the kind of result that makes a physicist's heart skip a beat.***16*You might think that the number of holes in the curled-up Planck-sized dimensions—mountaintop physics

*par excellence*—has now kicked an experimentally testable stone down to accessible energies. After all, experimentalists can establish—in fact, already have established—the number of particle families: 3. Unfortunately, the number of holes contained in each of the tens of thousands of known Calabi-Yau shapes spans a wide range. Some have 3. But others have 4, 5, 25, and so on—some even have as many as 480 holes.*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.*If we could find the principle that allows the selection of one Calabi-Yau shape from the numerous possibilities, then indeed a stone from the mountaintop would go tumbling down into the experimentalists' camp. If the particular Calabi-Yau shape singled out by the equations of the theory were to have three holes, we would have found an impressive postdiction from string theory explaining a known feature of the world that is otherwise completely mysterious. But finding the principle for choosing among Calabi-Yau shapes is a problem that as yet remains unsolved. Nevertheless—and this is the important point—we see that string theory provides the potential for answering this basic puzzle of particle physics, and this in itself is substantial progress.The number of families is but one experimental consequence of the geometrical form of the extra dimensions. Through their effect on possible patterns of string vibrations, other consequences of the extra dimensions include the detailed properties of the force and matter particles. As one primary example, subsequent work by Strominger and Witten showed that the masses of the particles in each family depend upon—hang on, this is a bit tricky—the way in which the boundaries of the various multidimensional holes in the Calabi-Yau shape intersect and overlap with one another. 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. Once again, though, carrying through with such calculations requires that we know which Calabi-Yau space to take for the extra dimensions.

The preceding discussion gives some idea of how string theory may one day explain the properties of the matter particles recorded in Table 1.1. String theorists believe that a similar story will one day also explain the properties of the messenger particles of the fundamental forces, listed in Table 1.2. That is, as strings twist and vibrate while meandering through the extended and curled-up dimensions, a small subset of their vast oscillatory repertoire consists of vibrations with spin equal to 1 or 2. These are the candidate force-carrying string-vibrational states. Regardless of the shape of the Calabi-Yau space, there is always one vibrational pattern that is massless and has spin-2; we identify this pattern as the graviton. The precise list of spin-1 messenger particles—their number, the strength of the force they convey, the gauge symmetries they respect—though, does depend crucially on the precise geometrical form of the curled-up dimensions. 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).

Why can't we figure out which is the "right" Calabi-Yau shape? Most string theorists blame this on the inadequacy of the theoretical tools currently being used to analyze string theory. As we shall discuss in some detail in Chapter 12, the mathematical framework of string theory is so complicated that physicists have been able to perform only approximate calculations through a formalism known as

*perturbation theory*. 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. A driving force behind present-day research is to develop theoretical methods that transcend the approximate approach in the hope that, among other benefits, we will be led to a unique Calabi-Yau shape for the extra dimensions. We will discuss progress along these lines in Chapter 13.