Approaches to Quantum Gravity

364 R. Williamswhere V is the volume of the corresponding tetrahedron and θ i the exterior dihedralangle at edge i. In the sum over edge lengths, the large values dominate, so the sumover the j i in the state sum can be replaced by an integral over the edge lengths,and the asymptotic formula used. By writing the cosine as a sum of exponentials,and interchanging the orders of summation over tetrahedra and over edges within agiven tetrahedron in the expression for the state sum, we can show that it containsa term of the form∫ ( ∏ ∏dj i (2 j i + 1)itets k∫ ∏=i) ⎛1√ exp ⎝iVk( ∏dj i (2 j i + 1)tets k∑edges l(j l 2π − ∑tets k∋l(π − θ kl ) ) ⎞ ⎠1√Vk)exp(iS R ), (19.7)which looks precisely like a Feynman sum over histories with the Regge action inthree dimensions, and with the other terms contributing to the measure.This result was rather puzzling and there seemed to be no obvious way to generaliseit to four dimensions, so it was virtually ignored for twenty years. Then, inthe early 1990s, Turaev and Viro [68] wrote down a very similar expression whichwas made finite by the use of representations of the quantum group Sl q (2), ratherthan SU(2). It was then realised that a regularised version of the Ponzano–Reggestate sum provided a model of Quantum Gravity in three dimensions, with zerocosmological constant. These three-dimensional models then led on to the developmentof spin foam models which currently play an important role in the searchfor a theory of Quantum Gravity.19.3 Quantum Regge calculus in four dimensions: analytic calculationsThe complicated dependence on the edge lengths of the deficit angles in the Reggecalculus action means that calculations have mainly involved either highly symmetricconfigurations or perturbation theory about a classical background. Theearliest work was a comparison between the Regge propagator in the weak fieldlimit and the continuum propagator [64]. This will be described in some detail asthe formalism has been the basis for many calculations of this type.Consider a lattice of four-dimensional unit hypercubes, with vertices labelled bycoordinates (n 1 , n 2 , n 3 , n 4 ), where each n i is an integer. Each hypercube is dividedinto 24 4-simplices, by drawing in appropriate “forward-going” diagonals. Thewhole lattice can be generated by the translation of a set of edges based on theorigin. We interpret the coordinates of vertices neighbouring the origin as binarynumbers (so for example, (0, 1, 0, 0) is vertex 4, (1, 1, 1, 1) is vertex 15). The edges