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Authors: Alex Vilenkin
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As I have just mentioned, string theory has no adjustable parameters. This is not an exaggeration: I mean
none
at all. Even the number of the dimensions of space is rigidly fixed by the theory. The problem is that it gives the wrong answer: it requires that space should have nine dimensions instead of three.
none
at all. Even the number of the dimensions of space is rigidly fixed by the theory. The problem is that it gives the wrong answer: it requires that space should have nine dimensions instead of three.
This sounds very embarrassing: Why should we even consider a theory that is in such blatant conflict with reality? The conflict, however, can be avoided if the extra six dimensions are curled up or, as physicists say,
compactified
. A soda straw is a simple example of compactification: it has one large dimension along the straw and another curled up in a small circle. When viewed from a distance, the straw looks like a one-dimensional line, but close by we can see that its surface is in fact a two-dimensional cylinder (see
Figure 15.3
). Quite similarly, the compact extra dimensions will not be visible if they are sufficiently small. In string theory, they are not expected to be much larger than the Planck length.
5
compactified
. A soda straw is a simple example of compactification: it has one large dimension along the straw and another curled up in a small circle. When viewed from a distance, the straw looks like a one-dimensional line, but close by we can see that its surface is in fact a two-dimensional cylinder (see
Figure 15.3
). Quite similarly, the compact extra dimensions will not be visible if they are sufficiently small. In string theory, they are not expected to be much larger than the Planck length.
5
The main problem with extra dimensions is that it is not clear exactly how they are to be compactified. If there were a single extra dimension, there would be only one way to compactify it: to curl it up into a circle. Two extra dimensions can be compactified as a sphere, as a doughnut, or as a more complicated surface with a large number of “handles” (
Figure 15.4
). As you go to higher dimensions, the number of possibilities multiplies. The vibrational states of the strings depend on the size and shape of extra dimensions, so each new compactification corresponds to a new vacuum with different types of particles, having different masses and different interactions.
Figure 15.4
). As you go to higher dimensions, the number of possibilities multiplies. The vibrational states of the strings depend on the size and shape of extra dimensions, so each new compactification corresponds to a new vacuum with different types of particles, having different masses and different interactions.
Figure 15.3
.
A soda straw has a two-dimensional cylindrical surface. It has a large dimension along the straw and a small dimension curled up in a circle.
.
A soda straw has a two-dimensional cylindrical surface. It has a large dimension along the straw and a small dimension curled up in a circle.
The hope of string theorists was that in the end the theory would yield a unique compactification describing our world and we would finally have an explanation for the observed values of all the particle physics parameters.
6
On the wave of excitement that followed some mathematical breakthroughs in the 1980s, it seemed that this goal might be just around the corner, and string theory was heralded as the future “theory of everything”—a tall order for a theory that was yet to make its first observational prediction! But gradually, a very different picture was beginning to emerge: the theory appeared to allow thousands of different compactifications.
6
On the wave of excitement that followed some mathematical breakthroughs in the 1980s, it seemed that this goal might be just around the corner, and string theory was heralded as the future “theory of everything”—a tall order for a theory that was yet to make its first observational prediction! But gradually, a very different picture was beginning to emerge: the theory appeared to allow thousands of different compactifications.
Figure 15.4
. Different ways to compactify two extra dimensions. The large, noncompact dimensions are not shown.
. Different ways to compactify two extra dimensions. The large, noncompact dimensions are not shown.
If this was not bad enough, things got considerably worse in the mid-1990s as a result of some unexpected new developments. As the mathematics of string theory was better understood, it became clear that in addition to one-dimensional strings, the theory must include two-dimensional membranes, as well as their higher-dimensional analogues. All these new arrivals
are collectively called
branes
.
be
Vibrating little branes would look like particles, but they are too massive to be produced in particle accelerators.
7
are collectively called
branes
.
be
Vibrating little branes would look like particles, but they are too massive to be produced in particle accelerators.
7
The branes have one unpleasant side effect: they dramatically increase the number of ways in which new vacua can be constructed. A brane can be wrapped, like a rubber band, around some of the compact dimensions. Every new stable brane configuration gives a new vacuum. You can wrap one, two, or more branes on each of the handles of the compact space, and with a large number of handles, the number of possibilities is enormous. The equations of the theory have no adjustable constants, but their solutions, describing different vacuum states, are characterized by several hundred parameters—the sizes of compact dimensions, the locations of the branes, and so on.
If we had just one parameter, it would be very similar to a scalar field in the usual particle theory. As we discussed in earlier chapters, it would then behave as a little ball in the energy landscape and would roll to the nearest minimum of the energy density. With two parameters, the landscape would be two-dimensional, as illustrated in
Figure 15.5
. It would have maxima (peaks) and minima (valleys), with minima representing the vacuum states. The altitude at each minimum gives the corresponding vacuum energy density (the cosmological constant).
Figure 15.5
. It would have maxima (peaks) and minima (valleys), with minima representing the vacuum states. The altitude at each minimum gives the corresponding vacuum energy density (the cosmological constant).
The actual energy landscape of string theory is much more complicated, since it includes many more parameters. This landscape cannot be drawn on a sheet of paper: to account for all the parameters, we would need a space of several hundred dimensions. But the landscape can still be mathematically analyzed. A rough estimate indicates that it contains about 10
500
(google to the fifth power!) different vacua. Some of these vacua are similar to ours; others have very different values for the constants of nature. Still others differ more drastically and have totally different kinds of particles and interactions, or more than three large dimensions.
500
(google to the fifth power!) different vacua. Some of these vacua are similar to ours; others have very different values for the constants of nature. Still others differ more drastically and have totally different kinds of particles and interactions, or more than three large dimensions.
As the outlines of the landscape were emerging, the hope of deriving a unique vacuum from string theory was rapidly slipping away. But string theorists were in denial and not ready to accept defeat.
Figure 15.5
.
Energy landscape in two dimensions. Each horizontal dimension (not to be confused with the dimensions of ordinary space) represents one of the parameters characterizing string theory vacua. The height represents the energy density.
.
Energy landscape in two dimensions. Each horizontal dimension (not to be confused with the dimensions of ordinary space) represents one of the parameters characterizing string theory vacua. The height represents the energy density.
The first physicists to break from the pack were Raphael Bousso, now at the University of California at Berkeley, and Joseph Polchinski of the Kavli Institute for Theoretical Physics at Santa Barbara. Remember Polchinski? He is the string theorist who could not stand the anthropic principle and pledged to quit physics if the cosmological constant was discovered.
bf
Luckily, he changed his mind—both about quitting physics and about the anthropic principle.
bf
Luckily, he changed his mind—both about quitting physics and about the anthropic principle.
Bousso and Polchinski combined the picture of the string theory landscape with the ideas of inflationary cosmology and argued that regions of all possible vacua will be created in the course of eternal inflation. The highest-energy vacuum will inflate the fastest. Bubbles of lower-energy vacua will nucleate and expand in this inflating background (as in Guth’s
original inflationary scenario, discussed in Chapters 5 and 6). The interiors of the bubbles will inflate at a smaller rate, and bubbles of still-lower energy will pop out inside them (see
Figure 15.6
).
bg
As a result, the entire string theory landscape will be explored—countless bubbles will be formed, filled with every possible kind of vacuum.
8
original inflationary scenario, discussed in Chapters 5 and 6). The interiors of the bubbles will inflate at a smaller rate, and bubbles of still-lower energy will pop out inside them (see
Figure 15.6
).
bg
As a result, the entire string theory landscape will be explored—countless bubbles will be formed, filled with every possible kind of vacuum.
8
Figure 15.6
.
Bubbles filled with lower-energy vacua nucleate in the inflating high-energy background, and still-lower energy bubbles nucleate inside them.
.
Bubbles filled with lower-energy vacua nucleate in the inflating high-energy background, and still-lower energy bubbles nucleate inside them.
We live in one of the bubbles, but the theory does not tell us which one. Only a tiny fraction of the bubbles are hospitable to life, and we must find ourselves in one of these rare bubbles. Much to the dismay of many string theorists, this is precisely the kind of picture that is assumed in anthropic arguments. If string theory is indeed the ultimate theory of reality, then it appears that the anthropic worldview is inevitable.
It needs to be said that the landscape of string theory is far from being fully mapped. In order to yield a realistic cosmology, some of the slopes have to be very gentle, allowing for slow-roll inflation. Recent work indicates
that there are indeed such regions in the landscape. We should also search for even gentler slopes, required by Linde’s scalar field model of a variable cosmological “constant” (discussed in Chapter 13). None have been found so far. But Bousso and Polchinski suggest that googles of vacua in the landscape provide a suitable alternative.
that there are indeed such regions in the landscape. We should also search for even gentler slopes, required by Linde’s scalar field model of a variable cosmological “constant” (discussed in Chapter 13). None have been found so far. But Bousso and Polchinski suggest that googles of vacua in the landscape provide a suitable alternative.
Instead of a continuum of vacuum energy densities in Linde’s model, the landscape gives a discrete set of values. Normally, this would be a problem, because only a tiny fraction of these values (about 1 in 10
120
) fall in the small anthropically allowed range. If we had less than 10
120
vacua, this range would most probably be empty. But with 10
500
vacua in the landscape, the set of values is so dense that it is almost continuous, and we expect that googles of vacua will have the cosmological constant in the anthropically allowed interval. The principle of mediocrity can then be applied in the same manner as before, and the successful prediction of the observed cosmological constant is unaffected.
120
) fall in the small anthropically allowed range. If we had less than 10
120
vacua, this range would most probably be empty. But with 10
500
vacua in the landscape, the set of values is so dense that it is almost continuous, and we expect that googles of vacua will have the cosmological constant in the anthropically allowed interval. The principle of mediocrity can then be applied in the same manner as before, and the successful prediction of the observed cosmological constant is unaffected.
The paper by Bousso and Polchinski, which appeared in 2000, did make a stir, but the landslide began three years later, when they were joined by one of the inventors of string theory, Leonard Susskind of Stanford University. Susskind is a fiercely independent thinker and is also a man of great charm and charisma. His power of persuasion is phenomenal; this is the man you want to have on your side.
Susskind was still unconvinced when Bousso and Polchinski’s paper first came out. He felt that the existence of a multitude of vacua assumed in the paper relied more on conjecture than on mathematical fact. But the developments of the following few years showed that the conjectures were basically sound, and in 2003 Susskind came out in full force promoting what he called “the anthropic landscape of string theory.” He argued that the diversity of vacua in string theory provided, for the first time, a solid scientific basis for anthropic arguments. String theorists, he said, should therefore embrace the anthropic principle, instead of fighting against it.
In less than a year, everybody was talking about “the landscape.” The number of papers discussing multiple vacua and other anthropic-related issues grew from four in 2002 to thirty-two in 2004. Of course, not everybody
was pleased with this turn of events. “I hate this recent landscape idea,” says Paul Steinhardt, “and I am hopeful it will go away.”
9
David Gross, the 2004 Nobel Prize winner, who regards the use of the anthropic principle as giving up the ideal of uniqueness, paraphrased Winston Churchill, saying “Never, never, never, never give up!” When I talked to him at a meeting in Cleveland, he complained that the anthropic principle is like a virus. Once you get it, you are lost to the community. “Ed Witten
bh
dislikes this idea intensely,” says Susskind, describing the situation, “but I’m told he’s very nervous that it might be right. He’s not happy about it, but I think he knows that things are going in that direction.”
10
was pleased with this turn of events. “I hate this recent landscape idea,” says Paul Steinhardt, “and I am hopeful it will go away.”
9
David Gross, the 2004 Nobel Prize winner, who regards the use of the anthropic principle as giving up the ideal of uniqueness, paraphrased Winston Churchill, saying “Never, never, never, never give up!” When I talked to him at a meeting in Cleveland, he complained that the anthropic principle is like a virus. Once you get it, you are lost to the community. “Ed Witten
bh
dislikes this idea intensely,” says Susskind, describing the situation, “but I’m told he’s very nervous that it might be right. He’s not happy about it, but I think he knows that things are going in that direction.”
10
If the landscape ideas are correct, explaining the observed constants of nature is not going to be easy. First, we will need to map the landscape. What kinds of vacua are there, and how many of each kind? We cannot realistically hope to obtain a detailed characterization of all 10
500
vacua, so some kind of statistical description will be necessary. We will also need to estimate the probabilities for bubbles of one vacuum to form amidst another vacuum. Then we will have all the ingredients to develop a model of an eternally inflating universe with bubbles inside bubbles inside bubbles, as illustrated in
Figure 15.6
. Once we have this model, the principle of mediocrity can be used to determine the probability for us to live in one of the vacua or another.
500
vacua, so some kind of statistical description will be necessary. We will also need to estimate the probabilities for bubbles of one vacuum to form amidst another vacuum. Then we will have all the ingredients to develop a model of an eternally inflating universe with bubbles inside bubbles inside bubbles, as illustrated in
Figure 15.6
. Once we have this model, the principle of mediocrity can be used to determine the probability for us to live in one of the vacua or another.
We are now making our first, tentative steps in this program, and formidable challenges lie ahead. “But,” writes Leonard Susskind, “I would bet that at the turn of the 22nd century, philosophers and physicists will look nostalgically at the present and recall a golden age in which the narrow provincial 20th century concept of the universe gave way to a bigger better megaverse, populating a landscape of mind-boggling proportions.”
11
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BOOK: Many Worlds in One: The Search for Other Universes
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