Quantum Gravity

QUANTUM-GRAVITATIONAL ASPECTS 287·· ºººFig. 9.1. The first two contributions to the scattering of two closed strings.How can the path integral (9.27) be evaluated? The sum over all ‘paths’contains, in particular, a sum over all worldsheets, that is, a sum over all Riemannsurfaces. In this sum, all topologies have to be taken into account. Figure 9.1shows as an example the first two topologies which arise in the scattering oftwo closed strings. It is in this way that string interactions arise—as amplitudesin the path integral. Unlike the situation in four dimensions, the classificationof these surfaces in two dimensions is well known. Consider as an example thedilaton part of the action (9.26),S φ = 14πIf Φ were constant, Φ(X) =λ, thiswouldyield∫d 2 σ √ h (2) R Φ(X) . (9.34)S φ = χλ = λ(2 − 2g) , (9.35)where χ is the Euler number and g the genus of the surface. (We assume forsimplicity here that only handles are present and no holes or cross-caps.) Thisthen gives the contributione −2λ(1−g) ≡ α g−1to the path integral, and we have introducedα =e 2λ ≡ g 2 c , (9.36)which plays the role of the ‘fine-structure constant’ for the loop expansion; g cdenotes the string-coupling constant for closed strings. Adding a handle correspondsto emission and re-absorption of a closed string. 2 The parameter g(meaning g c or g o , depending on the situation) is the expansion parameter forstring loops. It must be emphasized that one has only one diagram at each orderof the perturbation theory, in contrast to Feynman diagrams in quantum fieldtheory. The reason is that point-like interactions are avoided. Such a ‘smearing’can be done consistently in string theory, and it somehow resembles the ‘smearing’of the spin-network states discussed in Section 6.1. In this way the usual2 For the open string one finds g 2 o ∝ eλ .