Approaches to Quantum Gravity

282 A. PerezjjjkjkmkmjjjjmkmkmkmkkjjjmkmkmkmjFig. 15.8. A different representation of the transition of Fig. 15.5. This spin foamis obtained by a different ordering choice in (15.15).6g−66g−76g−86g−96g−106g−116g−136g−14534126g−12Fig. 15.9. A spin-network basis of physical states for an arbitrary genus g Riemannsurface. There are 6g − 6 spins labels (recall that 4-valent nodes carry anintertwiner quantum number).One such basis is illustrated in Fig. 15.9. The number of quantum numbers necessaryto label the basis element is 6g − 6, corresponding to the dimension of themoduli space of SU(2) flat connections on a Riemann surface of genus g. Thisis the number of degrees of freedom of the classical theory. In this way we arriveat a fully combinatorial definition of the standard H phys by reducing the infinitedegrees of freedom of the kinematical phase space to finitely many by the actionof the generalized projection operator P.15.3.4 Quantum spacetime as gauge-historiesWhat is the geometric meaning of the spin foam configurations? Can we identifythe spin foams with “quantum spacetime configurations”? The answer to the abovequestions is, strictly speaking, in the negative in agreement with our discussion at