The Elegant Universe - Brian Greene [218]
3. When we speak of "exact" answers in this chapter, such as the "exact" motion of the earth, what we really mean is the exact prediction for some physical quantity within some chosen theoretical framework. Until we truly have the final theory—perhaps we now do, perhaps we never will—all of our theories will themselves be approximations to reality. But this notion of approximate has nothing to do with our discussion in this chapter. Here we are concerned with the fact that within a chosen theory, it is often difficult, if not impossible, to extract the exact predictions that the theory makes. Instead, we have to extract such predictions using approximation methods based on a perturbative approach.
4. These diagrams are string theory versions of the so-called Feynman diagrams, invented by Richard Feynman for performing perturbative calculations in point-particle quantum field theory.
5. More precisely, every virtual string pair, that is, every loop in a given diagram, contributes—among other more complicated terms—a multiplicative factor of the string coupling constant. More loops translate into more factors of the string coupling constant. If the string coupling constant is less than 1, repeated multiplications make the overall contribution ever smaller; if it is 1 or larger, repeated multiplications yield a contribution with the same or larger magnitude.
6. For the mathematically inclined reader, we note that the equation states that spacetime must admit a Ricci-flat metric. If we split spacetime into a Cartesian product of four-dimensional Minkowski spacetime and a six-dimensional compact Kähler space, Ricci-flatness is equivalent to the latter being a Calabi-Yau manifold. This is why Calabi-Yau spaces play such a prominent role in string theory.
7. Of course, nothing absolutely ensures that these indirect approaches are justified. For example, just as some faces are not left-right symmetric, it might be that the laws of physics are different in other far-flung regions of the universe, as we will discuss briefly in Chapter 14.
8. The expert reader will recognize that these statements require so-called N=2 supersymmetry.
9. To be a little more precise, if we call the Heterotic-O coupling constant gHO and the Type I coupling constant gI, then the relation between the two theories states that they are physically identical so long as gHO = 1/gI, which is equivalent to gI = 1/gHO. When one coupling constant is big the other is small.
10. This is a close analog of the R, 1/R duality discussed previously. If we call the Type IIB string coupling constant gIIB then the statement that appears to be true is that the values gIIB and 1/gIIB describe the same physics. If gIIB is big, 1/gIIB is small, and vice versa.
11. If all but four dimensions are curled up, a theory with more than eleven total dimensions necessarily gives rise to massless particles with spin greater than 2, something that both theoretical and experimental considerations rule out.
12. A notable exception is the important 1987 work of Duff, Paul Howe, Takeo Inami, and Kelley Stelle in which they drew on earlier insights of Eric Bergshoeff, Ergin Sezgin, and Townsend to argue that ten-dimensional string theory should have a deep eleven-dimensional connection.
13. More precisely, this diagram should be interpreted as saying that we have a single theory that depends on a number of parameters. The parameters include coupling constants as well as geometrical size and shape parameters. In principle, we should be able to use the theory to calculate particular values for all of these parameters—a particular value for its coupling constant and a particular form for the spacetime geometry—but within our current theoretical understanding, we do not know how to accomplish this. And so, to understand the theory better