Approaches to Quantum Gravity

19Quantum Regge calculusR. WILLIAMS19.1 IntroductionWhen Regge formulated the first discrete version of general relativity in 1961, oneof his motivations was to set up a numerical scheme for solving Einstein’s equationsfor general systems without a large amount of symmetry. The hope was thatthe formulation would also provide ways of representing complicated topologiesand of visualising the resulting geometries. Regge calculus, as it has come to beknown, has not only been used in large scale numerical calculations in classicalgeneral relativity but has also provided a basis for attempts at formulating a theoryof Quantum Gravity.The central idea in Regge calculus [59] is to consider spaces with curvature concentratedon codimension-two subspaces, rather than with continuously distributedcurvature. This is achieved by constructing spaces from flat blocks glued togetheron matching faces. The standard example in two dimensions is a geodesic dome,where a network of flat triangles approximates part of a sphere. The curvatureresides at the vertices, and the deficit angle, given by 2π minus the sum of thevertex angles of the triangles at that point, gives a measure of it. In general dimensionn,flatn-simplices meet on flat (n − 1)-dimensional faces and the curvature isconcentrated on the (n − 2)-dimensional subsimplices or hinges. The deficit angleat a hinge is given by 2π minus the sum of the dihedral angles of the simplicesmeeting at that hinge. The use of simplices is important because specification oftheir edge lengths determines their shapes exactly, and in Regge calculus the edgelengths are the fundamental variables, by analogy with the metric tensor in thecontinuum theory.The analogue of the Einstein actionS = 1 2∫R √ gd d x (19.1)Approaches 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.