Approaches to Quantum Gravity

String theory, holography and Quantum Gravity 207terms of the spinor variables on the cosmological horizon. The geometry of thehorizon was a fuzzy sphere of the de Sitter radius, but the geometry of the momentumspace of a single particle had a more severe cut off, scaling like the square rootof the de Sitter radius. This suggests that if the infinite radius limit is super-Poincaréinvariant, the gravitino mass will scale like 1/4 .The holographic approach to the quantum theory of gravity incorporates insightsfrom string theory about the importance of supersymmetry and the holographicprinciple to the definition of the quantum generalization of a Lorentzian geometry.It has not yet made explicit contact with string theory. The route toward suchcontact branches into two: kinematics and dynamics. The first step is to show howthe kinematical variables S a (n) of the causal diamond approach, converge to thenatural asymptotic variables of the boundary description of string theory: Fockspaces of scattering states for asymptotically flat space-times, and conformal fieldsfor asymptotically AdS space times. The second is to relate the boundary dynamicsto the consistency condition of the causal diamond approach. In the AdS casethe problem is essentially kinematic. Once we have established that the boundaryvariables satisfy the locality axiom of field theory, the dynamics must be that ofa CFT. The relation to the causal diamond approach will help us to understandhow to describe local processes, and the inevitable gauge dependence of any suchdescription, in AdS/CFT.For the asymptotically flat case we only have a non-perturbative dynamical principlefor those space-times in which Matrix Theory applies. Even there one musttake a difficult large N limit to establish the symmetry properties of the boundarytheory. It would be more attractive to have a non-perturbative equation whichdetermined the super-Poincaré invariant S-matrix directly. In ancient times it wasshown that unitarity, holomorphy, and some information about high energy behavior,completely determined the scattering matrix in perturbation theory, up to localcounterterms. Even in the maximally symmetric case of eleven dimensions, theseprinciples do not seem to uniquely determine the counterterms. They also sufferfrom a lack of elegance and a vagueness of definition. One can hope that the consistencycondition of the causal diamond formulation can lead to an elegant andprecise form of the holomorphy of the S-matrix, which will completely determineit. The first, kinematic, step of relating the causal diamond formalism to Fock spacewill be discussed in [31].References[1] A. Strominger and C. Vafa, Microscopic origin of the Bekenstein–Hawking entropy,Phys. Lett. B 379 (1996) 99 [arXiv:hep-th/9601029].[2] J. M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory,JHEP 9712 (1997) 002 [arXiv:hep-th/9711053].