The Elegant Universe - Brian Greene [154]
Figure 12.5 The quantum frenzy can cause numerous sequences of string/antistring pairs to erupt and annihilate.
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.
Figure 12.6 The net influence each incoming string has on the other comes from adding together the influences involving diagrams with ever more loops.
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