Approaches to Quantum Gravity

Quantum Regge calculus 361is given byS R =∑V h δ h , (19.2)hinges hwhere V h is the volume of the hinge h and δ h is the deficit angle there. The principleof stationary action leads to the Regge calculus equivalent to Einstein’s equations;the action S R is varied with respect to edge lengths, giving∑h∂V h∂l iδ h = 0. (19.3)This is particularly simple because, as Regge showed, the variation of the deficitangles vanishes when summed over each simplex. In principle, Eq. (19.3) givesacomplete set of equations, one for each edge, for determining the edge lengths andthus the simplicial geometry. In practice, the discrete analogues of the contractedBianchi identities (see below) mean that the equations are not in general linearlyindependent, so there is freedom to specify some of the variables, as for the lapseand shift in the (3+1) version of continuum general relativity.The Bianchi identities in Regge calculus were given a very simple topologicalinterpretation by Regge [59] (seealso[63]). It is simplest to see this interpretationin three dimensions, but the generalisation to higher dimensions is straightforward.(In two dimensions, both in Regge calculus and in continuum general relativity,there are no Bianchi identities.) In three-dimensional Regge calculus, a vectorparallel-transported round an edge with non-zero deficit angle rotates. If it isparallel-transported along a path which does not enclose an edge, it does not rotate.Consider a number of edges meeting at a vertex. A path can be constructed whichencircles each edge once but is topologically trivial: it can be deformed withoutcrossing any edges into a path which obviously encloses no edges. (Try it with aloop of string and your fingers!) Consequently a vector parallel-transported alongthis path will not rotate. This means that there is a relation among the deficit anglesat the edges: the product of the rotation matrices on each edge is the identity matrix.This is precisely the discrete Bianchi identity. Regge showed that in the limit ofsmall deficit angles, it gives just the usual Bianchi identity of general relativity.The four-dimensional Bianchi identity in Regge calculus states that the productof the rotation matrices on all the triangles meeting along an edge is the identitymatrix. The discrete Bianchi identities have been discussed further [57; 12; 67] anddetailed forms given [29].Closely connected with the Bianchi identities is the existence of diffeomorphisms.There are differing points of view on how to define diffeomorphismsin Regge calculus. One is that diffeomorphisms are transformations of the edgelengths which leave the geometry invariant. In this case, diffeomorphisms exist