by Brian Greene
These deviations from inevitability might seem to be unfortunate fundamental characteristics of string theory. But research since the mid-1990s has given us dramatic new hope that these features may be merely reflections of the way string theorists have been analyzing the theory. Briefly put, the equations of string theory are so complicated that no one knows their exact form. Physicists have managed to write down only approximate versions of the equations. It is these approximate equations that differ significantly from one string theory to the next. And it is these approximate equations, within the context of any one of the five string theories, that give rise to an abundance of solutions, a cornucopia of unwanted universes.
Since 1995 (the start of the second superstring revolution), there has been a growing body of evidence that the exact equations, whose precise form is still beyond our reach, may resolve these problems, thereby helping to give string theory the stamp of inevitability. In fact, it has already been established to the satisfaction of most string theorists that, when the exact equations are understood, they will show that all five string theories are actually intimately related. Like the appendages on a starfish, they are all part of one connected entity whose detailed properties are currently under intense investigation. Rather than having five distinct string theories, physicists are now convinced that there is one theory that sews all five into a unique theoretical framework, And like the clarity that emerges when hitherto hidden relationships are revealed, this union is providing a powerful new vantage point for understanding the universe according to string theory.
To explain these insights we must engage some of the most difficult, cutting-edge developments in string theory. We must understand the nature of the approximations used in studying string theory and their inherent limitations. We must gain some familiarity with the clever techniques—collectively called dualities—that physicists have invoked to circumvent some of these approximations. And then we must follow the subtle reasoning that makes use of these techniques to find the remarkable insights alluded to above. But don't worry. The really hard work has already been done by string theorists and we will content ourselves here with explaining their results.
Nevertheless, as there are many seemingly separate pieces that we must develop and assemble, in this chapter it is especially easy to lose the forest for the trees. And so, if at any time in this chapter the discussion gets a little too involved and you feel compelled to rush on to black holes (Chapter 13) or cosmology (Chapter 14), take a quick glance back at the following section, which summarizes the key insights of the second superstring revolution.
A Summary of the Second Superstring Revolution
The primary insight of the second superstring revolution is summarized by Figures 12.1 and 12.2. In Figure 12.1 we see the situation prior to the recent ability to go (partially) beyond the approximation methods physicists have traditionally used to analyze string theory. We see that the five string theories were thought of as being completely separate. But, with the newfound insights emerging from recent research, as indicated in Figure 12.2, we see that, like the starfish's five arms, all of the string theories are now viewed as a single, all-encompassing framework. (In fact, by the end of this chapter we will see that even a sixth theory—a sixth arm-will be merged into this union.) This overarching framework has provisionally been called M-theory, for reasons that will become clear as we proceed. Figure 12.2 represents a landmark achievement in the quest for the ultimate theory~ Seemingly disconnected threads of research in string theory have now been woven together into a single tapestry—a unique, all-encompassing theory that may well be the long-sought theory of everything.
Although much work remains to be done, there are two essential features of M-theory that physicists have already uncovered. First, M-theory has eleven dimensions (ten space and one time). Somewhat as Kaluza found that one additional spatial dimension allowed for an unexpected merger of general relativity and electromagnetism, string theorists have realized that one additional spatial dimension in string theory–beyond the nine space and one time dimensions discussed in preceding chapters—allows for a deeply satisfying synthesis of all five versions of the theory. Moreover, this extra spatial dimension is not pulled out of thin air; rather, string theorists have realized that the reasoning of the 1970s and 1980s that led to one time and nine space dimensions was approximate, and that exact calculations, which can now be completed, show that one spatial dimension had hitherto been overlooked.
The second feature of M-theory that has been discovered is that it contains vibrating strings, but it also includes other objects: vibrating two-dimensional membranes, undulating three-dimensional blobs (called "three-branes"), and a host of other ingredients as well. As with the eleventh dimension, this feature of M-theory emerges when calculations are freed from reliance on the approximations used prior to the mid-1990s.
Beyond these and a variety of other insights attained over the last few years, much of the true nature of M-theory remains mysterious–one suggested meaning for the "M." Physicists worldwide are working with great vigor to acquire a full understanding of M-theory, and this may well constitute the central problem of twenty-first-century physics.
An Approximation Method
The limitations of the methods physicists have been using to analyze string theory are bound up with something called perturbation theory. Perturbation theory is an elaborate name for making an approximation to try to give a rough answer to a question, and then systematically improving this approximation by paying closer attention to fine details initially ignored. It plays an important part in many areas of scientific research, has been an essential element in understanding string theory, and, as we now illustrate, is also something we encounter frequently in our day-to-day lives.
Imagine that one day your car is acting up, so you go see a mechanic to have it checked out. After giving your car a once-over, he gives you the bad news. The car needs a new engine block, for which parts and labor typically run in the $900 range. This is a ballpark approximation that you expect to be refined as the finer details of the work required become apparent. A few days later, having had time to run additional tests on the car, the mechanic gives you a more precise estimate, $950. He explains that you also need a new regulator, which with parts and labor costs about $50. Finally, when you go to pick up the car, he has added together all of the detailed contributions and presents you with a bill of $987.93. This, he explains, includes the $950 for the engine block and regulator, an additional $27 covering a fan belt, $10 for a battery cable, and $.93 for an insulated bolt. The initial approximate figure of $900 has been refined by including more and more details. In physics terms, these details are referred to as perturbations to the initial estimate.
When perturbation theory is properly and effectively applied, the initial estimate will be reasonably close to the final answer; when incorporated, the fine details ignored in the initial estimate make small differences in the final result. But sometimes when you go to pay a final bill it is shockingly different from the initial estimate. Although you might use other, more emotive terms, technically this is called a failure of perturbation theory. This means that the initial approximation was not a good guide to the final answer because the "refinements," rather than causing relatively small deviations, resulted in large changes to the ballpark estimate.
As indicated briefly in earlier chapters, our discussion of string theory to this point has relied on a perturbative approach somewhat analogous to that used by the mechanic. The "incomplete understanding" of string theory that we have referred to from time to time has its roots, in one way or another, in this approximation method. Let's build up to an understanding of this important remark by discussing perturbation theory in a context that is less abstract than string theory but closer to its string theory application than the example of the mechanic.
A Classical Example of Perturbation Theory
Understanding the motion of the earth through the solar system provi
des a classic example of using a perturbative approach. On such large distance scales, we need consider only the gravitational force, but unless further approximations are made, the equations encountered are extremely complicated. Remember that according to both Newton and Einstein, everything exerts a gravitational influence on everything else, and this quickly leads to a complex and mathematically intractable gravitational tug-of-war involving the earth, the sun, the moon, the other planets, and, in principle, all other heavenly bodies as well. As you can imagine, it is impossible to take all of these influences into account and determine the exact motion of the earth. In fact, even if there were only three heavenly participants, the equations become so complicated that no one has been able to solve them in full.3
Nevertheless, we can predict the motion of the earth through the solar system with great accuracy by making use of a perturbative approach. The enormous mass of the sun, in comparison to that of every other member of our solar system, and its proximity to the earth, in comparison to that of every other star, makes it by far the dominant influence on the earth's motion. And so, we can get a ballpark estimate by considering only the sun's gravitational influence. For many purposes this is perfectly adequate. If necessary, we can refine this approximation by sequentially including the gravitational effects of the next-most-relevant bodies, such as the moon and whichever planets are passing closest by at the moment. The calculations can start to become difficult as the emerging web of gravitational influences gets complicated, but don't let this obscure the perturbative philosophy: The sun-earth gravitational interaction gives us an approximate explanation of the earth's motion, while the remaining complex of other gravitational influences offers a sequence of ever smaller refinements.
A perturbative approach works in this example because there is a dominant physical influence that admits a relatively simple theoretical description. This is not always the case. For example, if we are interested in the motion of three comparable-mass stars orbiting one another in a trinary system, there is no single gravitational relationship whose influence dwarfs the others. Correspondingly, there is no single dominant interaction that provides a ballpark estimate, with the other effects yielding small refinements. If we tried to use a perturbative approach by, say, singling out the gravitational attraction between two stars and using it to determine our ballpark approximation, we would quickly find that our approach had failed. Our calculations would reveal that the "refinement" to the predicted motion arising from the inclusion of the third star is not small, but in fact is as significant as the supposed ballpark approximation. This is familiar: The motion of three people dancing the hora bears little resemblance to two people dancing the tango. A large refinement means that the initial approximation was way off the mark and the whole scheme was built on a house of cards. You should note that it is not simply a matter of including the large refinement due to the third star. There is a domino effect: The large refinement has a significant impact on the motion of the other two stars, which in turn has a large impact on the motion of the third star, which then has a substantial impact on the other two, and so on. All strands in the gravitational web are equally important and must be dealt with simultaneously. Oftentimes, in such cases, our only recourse is to make use of the brute power of computers to simulate the resulting motion.
This example highlights the importance, when using a perturbative approach, of determining whether the supposedly ballpark estimate really is in the ballpark, and if it is, which and how many of the finer details must be included in order to achieve a desired level of accuracy. As we now discuss, these issues are particularly crucial for applying perturbative tools to physical processes in the microworld.
A Perturbative Approach to String Theory
Physical processes in string theory are built up from the basic interactions between vibrating strings. As we discussed toward the end of Chapter 6,* these interactions involve the splitting apart and joining together of string loops, such as in Figure 6.7, which we reproduce in Figure 12.3 for convenience. String theorists have shown how a precise mathematical formula can be associated with the schematic portrayal of Figure 12.3—a formula that expresses the influence that each incoming string has on the resulting motion of the other. (The details of the formula differ among the five string theories, but for the time being we will ignore such subtle features.) If it weren't for quantum mechanics, this formula would be the end of the story of how the strings interact. But the microscopic frenzy dictated by the uncertainty principle implies that string/antistring pairs (two strings executing opposite vibrational patterns) can momentarily erupt into existence, borrowing energy from the universe, so long as they annihilate one another with sufficient haste, thereby repaying the energy loan. Such pairs of strings, born of the quantum frenzy but which live on borrowed energy and hence must shortly recombine into a single loop, are known as virtual string pairs. And even though it is only momentary, the transient presence of these additional virtual string pairs affects the detailed properties of the interaction.
*Those readers who skipped over the "More Precise Answer" section of Chapter 6 might find it helpful to skim the beginning part of that section.
This is schematically depicted in Figure 12.4. The two initial strings slam together at the point marked (a), where they merge together into a single loop. This loop travels a bit, but at (b) frenzied quantum fluctuations result in the creation of a virtual string pair that travels along and then subsequently annihilates at (c), producing, once again, a single string. Finally, at (d), this string gives up its energy by dissociating into a pair of strings that head off in new directions. Because of the single loop in the center of Figure 12.4, physicists call this a "one-loop" process. As with the interaction depicted in Figure 12.3, a precise mathematical formula can be associated with this diagram to summarize the effect the virtual string pair has on the motion of the two original strings.
But that's not the end of the story either, because quantum jitters can cause momentary virtual string eruptions to occur any number of times, producing a sequence of virtual string pairs. This gives rise to diagrams with more and more loops, as illustrated in Figure 12.5. Each of these diagrams provides a handy and simple way of depicting the physical processes involved: The incoming strings merge together, quantum jitters cause the resulting loop to split apart into a virtual string pair, these travel along and then annihilate one another by merging together into a single loop, which travels along and produces another virtual string pair, and on and on. As with the other diagrams, there is a corresponding mathematical formula for each of these processes that summarizes the effect on the motion of the original pair of strings.4
Moreover, just as the mechanic determined your final car-repair bill through a refinement of his original estimate of $900 by adding to it $50, $27, $10, and $.93, and just as we arrived at an ever more precise understanding of the motion of the earth through a refinement of the sun's influence by including the smaller effects of the moon and other planets, string theorists have shown that we can understand the interaction between two strings by adding together the mathematical expressions for diagrams with no loops (no virtual string pairs), with one loop (one pair of virtual strings), with two loops (two pairs of virtual strings), and so forth, as illustrated in Figure 12.6.
An exact calculation requires that we add together the mathematical expressions associated with each of these diagrams, with an increasingly large number of loops. But, since there are an infinite number of such diagrams and the mathematical calculations associated with each get increasingly difficult as the number of loops grows, this is an impossible task. Instead, string theorists have cast these calculations into a perturbative framework based on the expectation that a reasonable ballpark estimate is given by the zero-loop processes, with the loop diagrams resulting in refinements that get smaller as the number of loops increases.
In fact, almost everything we know about string theory—including much of the material covered in
previous chapters—has been discovered by physicists performing detailed and elaborate calculations using this perturbative approach. But to trust the accuracy of the results found, one must determine whether the supposedly ballpark approximations that ignore all but the first few diagrams in Figure 12.6 are really in the ballpark. This leads us to ask the crucial question: Are we in the ballpark?
Is the Ballpark in the Ballpark?
It depends. Although the mathematical formula associated with each diagram becomes very complicated as the number of loops grows, string theorists have recognized one basic and essential feature. Somewhat as the strength of a rope determines the likelihood that vigorous pulling and shaking will cause it to tear into two pieces, there is a number that determines the likelihood that quantum fluctuations will cause a single string to split into two strings, momentarily yielding a virtual pair. This number is known as the string coupling constant (more precisely, each of the five string theories has its own string coupling constant, as we will discuss shortly). The name is quite descriptive: The size of the string coupling constant describes how strongly the quantum jitters of three strings (the initial loop and the two virtual loops into which it splits) are related—how tightly, so to speak, they are coupled to one another. The calculational formalism shows that the larger the string coupling constant, the more likely it is that quantum jitters will cause an initial string to split apart (and subsequently rejoin); the smaller the string coupling constant, the less likely it is for such virtual strings to erupt momentarily into existence.
We will shortly take up the question of determining the value of the string coupling constant within any of the five string theories, but first, what do we really mean by "small" or "large" when assessing its size? Well, the mathematics underlying string theory shows that the dividing line between "small" and "large" is the number 1, in the following sense. If the string coupling constant has a value less than 1, then—like multiple strikes of lightning—larger numbers of virtual string pairs are increasingly unlikely to erupt momentarily into existence. If the coupling constant is 1 or greater, however, it is increasingly likely that ever-larger numbers of such virtual pairs will momentarily burst on the scene.5 The upshot is that if the string coupling constant is less than 1, the loop diagram contributions become ever smaller as the number of loops grows. This is just what is needed for the perturbative framework, since it indicates that we will get reasonably accurate results even if we ignore all processes except for those with just a few loops. But if the string coupling constant is not less than 1, the loop diagram contributions become more important as the number of loops increases. As in the case of a trinary star system, this invalidates a perturbative approach. The supposed ballpark approximation—the process with no loops—is not in the ballpark. (This discussion applies equally well to each of the five string theories—with the value of the string coupling constant in any given theory determining the efficacy of the perturbative approximation scheme.)