The Elegant Universe - Brian Greene [155]
This realization leads us to the next crucial question: What is the value of the string coupling constant (or, more precisely, what are the values of the string coupling constants in each of the five string theories)? At present, no one has been able to answer this question. It is one of the most important unresolved issues in string theory. We can be sure that conclusions based on a perturbative framework are justified only if the string coupling constant is less than 1. Moreover, the precise value of the string coupling constant has a direct impact on the masses and charges carried by the various string vibrational patterns. Thus, we see that much physics hinges on the value of the string coupling constant. And so, let's take a closer look at why the important question of its value—in any of the five string theories—remains unanswered.
The Equations of String Theory
The perturbative approach for determining how strings interact with one another can also be used to determine the fundamental equations of string theory. In essence, the equations of string theory determine how strings interact and, conversely, the way strings interact directly determines the equations of the theory.
As a prime example, in each of the five string theories there is an equation that is meant to determine the value of the theory's coupling constant. Currently, however, physicists have been able to find only an approximation to this equation, in each of the five string theories, by mathematically evaluating a small number of relevant string diagrams using a perturbative approach. Here is what the approximate equations say: In any of the five string theories, the string coupling constant takes on a value such that if it is multiplied by zero the result is zero. This is a terribly disappointing equation; since any number times zero yields zero, the equation can be solved with any value of the string coupling constant. Thus, in any of the five string theories, the approximate equation for its string coupling constant gives us no information about its value.
While we are at it, in each of the five string theories there is another equation that is supposed to determine the precise form of both the extended and the curled-up spacetime dimensions. The approximate version of this equation that we currently have is far more restrictive than the one dealing with the string coupling constant, but it still admits many solutions. For instance, four extended spacetime dimensions together with any curled-up, six-dimensional Calabi-Yau space provide a whole class of solutions, but even this does not exhaust the possibilities, which also allow for a different split between the number of extended and curled-up dimensions.6
What can we make of these results? There are three possibilities. First, starting with the most pessimistic possibility, although each string theory comes equipped with equations to determine the value of its coupling constant as well as the dimensionality and precise geometrical form of spacetime—something no other theory can claim—even the as-yet-unknown exact form of these equations may admit a vast spectrum of solutions, substantially weakening their predictive power. If true, this would be a setback, since the promise of string theory is that it will be able to explain these features of the cosmos, rather than require us to determine them from experimental observation and, more or less arbitrarily, insert them into the theory. We will return to this possibility in Chapter 15. Second, the unwanted flexibility in the approximate