Approaches to Quantum Gravity

16Three-dimensional spin foam Quantum GravityL. FREIDEL16.1 IntroductionLoop quantum gravity provides a background independent approach to QuantumGravity. In this context the kinematical Hilbert space is spanned by spin networks(graph labeled by Lorentz group representations) which are eigenstates of geometricaloperators. The dynamics of such theories is encoded in a set of transitionsamplitudes between initial and final spin network states which carry the informationabout the physical inner product of the theory. Generically, these amplitudesare constructed in terms of spin foam models which are local state sum modelsassociated with a sum of colored 2-complexes interpolating between initial andfinal spin network states.There are many important questions that need to be addressed in this frameworksuch as the proper choice of the dynamics, the construction and interpretation ofthe spin foam amplitude, the coupling to matter and the description of the semiclassicalregime of such a theory. The contributions of D. Oriti and A. Perez in thisvolume address some of these issues.In this contribution we will focus on the simple case of three-dimensionalgravity and its quantization via spin foam models. The advantage of using thespin foam framework is twofold. First, this framework is not specifically tailoredto three dimensions, unlike Chern–Simons quantization for instance, andsome of the lessons and techniques used there can be useful for the more realisticfour-dimensional case. Second, this way of quantizing gravity agrees withother quantizations when they apply but is in general applicable to a largerclass of problems (like Lorentzian gravity and computation of topology changingamplitudes).The proper way to encode the dynamic of 3d gravity in terms of spin foammodel was discovered a long time ago by Ponzano and Regge [1]. In the followingwe summarize a set of recent results [2; 3; 4; 5] concerning the Ponzano–ReggeApproaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, ed. Daniele Oriti.Published by Cambridge University Press. c○ Cambridge University Press 2009.