**Table of Contents**

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

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

```(annotated and with added

**bold highlights by Epsilon=One**)

**Chapter 8 - More Dimensions Than Meet the Eye**

More Dimensions and String Theory

By now you should be convinced that our universe

Such was the case until string theory. Here is a theory that resolves the central dilemma confronting contemporary physics—the incompatibility between quantum mechanics and general relativity—and that unifies our understanding of all of nature's fundamental material constituents and forces. But to accomplish these feats, it turns out that string theory

Here's why. One of the main insights of quantum mechanics is that our predictive power is fundamentally limited to asserting that such-and-such outcome will occur with such-and-such probability. Although Einstein felt that this was a distasteful feature of our modern understanding, and you may agree, it certainly appears to be fact. Let's accept it. Now, we all know that probabilities are always numbers between 0 and 1—equivalently, when expressed as percentages, probabilities are numbers between 0 and 100. Physicists have found that a key signal that a quantum-mechanical theory has gone haywire is that particular calculations yield "probabilities" that are not within this acceptable range. For instance, we mentioned earlier that a sign of the grinding incompatibility between general relativity and quantum mechanics in a point-particle framework is that calculations result in infinite probabilities. As we have discussed, string theory cures these infinities. But what we have not as yet mentioned is that a residual, somewhat more subtle problem still remains. In the early days of string theory physicists found that certain calculations yielded

With stubborn determination, physicists sought and found the cause of this unacceptable feature. The explanation begins with a simple observation. If a string is constrained to lie on a two-dimensional surface—such as the surface of a table or a garden hose—the number of independent directions in which it can vibrate is reduced to

We emphasize this fact of string vibrations because physicists found that the troublesome calculations were highly sensitive to the number of independent directions in which a string can vibrate. The negative probabilities arose from a

But are we? Taking a more than half-century-old lead, we see that Kaluza and Klein provide a loophole. Since strings are so small, not only can they vibrate in large, extended dimensions, they can also vibrate in ones that are tiny and curled up. And so we

*may*have additional curled-up spatial dimensions; certainly, so long as they are small enough, nothing rules them out. But extra dimensions may strike you as an artifice. Our inability to probe distances smaller than a billionth of a billionth of a meter permits not only extra tiny dimensions but all manner of whimsical possibilities as well—even a microscopic civilization populated by even tinier green people. While the former certainly seems more rationally motivated than the latter, the act of postulating either of these experimentally untested—and, at present, untestable—possibilities might seem equally arbitrary.Such was the case until string theory. Here is a theory that resolves the central dilemma confronting contemporary physics—the incompatibility between quantum mechanics and general relativity—and that unifies our understanding of all of nature's fundamental material constituents and forces. But to accomplish these feats, it turns out that string theory

*requires*that the universe have extra space dimensions.Here's why. One of the main insights of quantum mechanics is that our predictive power is fundamentally limited to asserting that such-and-such outcome will occur with such-and-such probability. Although Einstein felt that this was a distasteful feature of our modern understanding, and you may agree, it certainly appears to be fact. Let's accept it. Now, we all know that probabilities are always numbers between 0 and 1—equivalently, when expressed as percentages, probabilities are numbers between 0 and 100. Physicists have found that a key signal that a quantum-mechanical theory has gone haywire is that particular calculations yield "probabilities" that are not within this acceptable range. For instance, we mentioned earlier that a sign of the grinding incompatibility between general relativity and quantum mechanics in a point-particle framework is that calculations result in infinite probabilities. As we have discussed, string theory cures these infinities. But what we have not as yet mentioned is that a residual, somewhat more subtle problem still remains. In the early days of string theory physicists found that certain calculations yielded

*negative*probabilities, which are also outside of the acceptable range. So, at first sight, string theory appeared to be awash in its own quantum-mechanical hot water.With stubborn determination, physicists sought and found the cause of this unacceptable feature. The explanation begins with a simple observation. If a string is constrained to lie on a two-dimensional surface—such as the surface of a table or a garden hose—the number of independent directions in which it can vibrate is reduced to

*two:*the left-right and back-forth dimensions along the surface. Any vibrational pattern that remains on the surface involves some combination of vibrations in these two directions. Correspondingly, we see that this also means that a string in Flatland, the Garden-hose universe, or in any other two-dimensional universe, is also constrained to vibrate in a total of two independent spatial directions. If, however, the string is allowed to leave the surface, the number of independent vibrational directions increases to three, since the string then can also oscillate in the up-down direction. Equivalently, in a universe with three spatial dimensions, a string can vibrate in three independent directions. Although it gets harder to envision, the pattern continues: In a universe with ever more spatial dimensions, there are ever more independent directions in which it can vibrate.We emphasize this fact of string vibrations because physicists found that the troublesome calculations were highly sensitive to the number of independent directions in which a string can vibrate. The negative probabilities arose from a

*mismatch*between what the theory required and what reality seemed to impose: The calculations showed that if strings could vibrate in*nine*independent spatial directions, all of the negative probabilities would cancel out. Well, that's great in theory, but so what? If string theory is meant to describe our world with three spatial dimensions, we still seem to be in trouble.But are we? Taking a more than half-century-old lead, we see that Kaluza and Klein provide a loophole. Since strings are so small, not only can they vibrate in large, extended dimensions, they can also vibrate in ones that are tiny and curled up. And so we

*can*meet the nine-space-dimension requirement of string theory in*our*universe, by assuming—à la Kaluza and Klein—that in addition to our familiar three extended spatial dimensions there are six other curled-up spatial dimensions. In this manner, string theory, which appeared to be on the brink of elimination from the realm of physical relevance, is saved. Moreover, rather than just postulating the existence of extra dimensions, as had been done by Kaluza, Klein, and their followers, string theory*requires*them. For string theory to make sense, the universe should have nine space dimensions and one time dimension, for a total of ten dimensions. In this way, Kaluza's 1919 proposal finds its most convincing and powerful forum.