THE ELEGANT UNIVERSE, Brian Greene, 1999, 2003
```(annotated and with added bold highlights by Epsilon=One)

Chapter 12 - Beyond Strings: In Search of M-Theory

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.

Figure 12.3 Strings interact by joining and splitting.

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.

Figure 12.4 The quantum frenzy can cause a string/antistring pair to erupt (b) and annihilate (c), yielding a more complicated interaction.

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

Figure 12.5 The quantum frenzy can cause numerous sequences of
string/antistring pairs to erupt and annihilate.

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.

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.

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?