Ivancevic_Applied-Diff-Geom

302 Applied Differential Geometry: A Modern Introductionwhere we have neglected some coefficients for simplicity. They also providedarguments indicating that this sum is independent from the triangulationof the manifold.The formula (3.148) is simple and elegant, and the idea has recently hadmany surprising and interesting developments. 3D GR was quantized as atopological field theory by Ed Witten in [Witten (1988c)] and using loopquantum gravity in [Ashtekar et. al. (1989)]. The Ponzano–Regge quantizationbased on equation (3.148) was shown to be essentially equivalent tothe TQFT quantization in [Ooguri (1992a)], and to the loop quantum gravityin [Rovelli (1993)] (for an extensive discussion of 3D quantum gravity,see [Carlip and Nelson (1995)]).It turns out that the Ponzano–Regge ansatz (3.147) can be derived fromloop quantum gravity [Rovelli (1993)]. Indeed, (3.147) is the 2D version ofthe 3D formula (3.146), which gives the quantization of the area. Therefore,a key result of quantum gravity of the last years, namely the quantizationof the geometry, derived in the loop formalism from a full fledgednon–perturbative quantization of GR, was anticipated as an ansatz by theintuition of Ponzano and Regge.Hawking’s Euclidean Quantum GravityHawking’s Euclidean quantum gravity is the approach based on his formalsum over Euclidean geometries (i.e., an Euclidean path integral, see chapter6 below)∫Z ∼ N D[g] e − R d 4 x √ gR[g] . (3.149)As far as we understand, Hawking and his close collaborators do not anymoreview this approach as an attempt to directly define a fundamentaltheory. The integral is badly ill defined, and does not lead to any knownviable perturbation expansion. However, the main ideas of this approachare still alive in several ways.First, Hawking’s picture of quantum gravity as a sum–over–space–times,continues to provide a powerful intuitive reference point for most of the researchrelated to quantum gravity. Indeed, many approaches can be seenas attempts to replace the ill–defined and non–renormalizable formal integral(3.149) with a well defined expression. The dynamical triangulationapproach (see above) and the spin foam approach (see below) are examplesof attempts to realize Hawking’s intuition. Influence of Euclidean quantumgravity can also be found in the Atiyah axioms for TQFT.