Approaches to Quantum Gravity

362 R. Williamsonly in flat space and correspond to changes in the edge lengths as the verticesmove around in that flat space [52; 42]. If the space is almost flat, then one candefine approximate diffeomorphisms. The other view is that diffeomorphisms leavethe action invariant and this gives rather more flexibility. It is easy to imaginechanges in the edge lengths which could decrease the deficit angles in one regionand compensatingly increase them in another region, producing no overall changein the action. The invariance could even be local in the sense that changes in thelengths of the edges meeting at one vertex could be made so that the action isunchanged.In three dimensions, it is possible to construct transformations which are exactinvariances of the action. This relies on the uniqueness of the embedding of thestar of a vertex of a three-skeleton in a flat four-dimensional space. By a detailedcounting argument [64], one can show that the number of degrees of freedom (i.e.the edge lengths in the star) is exactly equal to the number of coordinates for itsembedding in four dimensions. Thus exact diffeomorphisms are defined locally ateach point and consist of the three-parameter family of motions of the point (in theflat four-dimensional space defined by its star) which leave the action invariant. Thecorresponding argument does not go through in four dimensions because there isno unique embedding of a four-dimensional star in a higher dimensional flat space.Attempts have been made to find alternative definitions in four dimensions butnone is completely satisfactory. Of course it is always possible to find approximatediffeomorphisms, in particular ones where the invariance holds to third order in thedeficit angles [33].A “gauge-fixed” version of Regge calculus was constructed by Römer andZähringer [65] in which the simplices were all taken to be equilateral. This workwas a forerunner to the scheme known as dynamical triangulations, in whichall edge lengths are identical and the sum over histories involves the sum overtriangulations [1] (see chapter 18 by Ambjørn et al.).Another basic type of transformation in general relativity is a conformal transformation.One way to define this in Regge calculus [64] is to define a scalar functionφ at each vertex. The procedure which guarantees that at least locally, the conformaltransformations form a group, is to require that l xy , the length of the edgejoining vertices x and y, transforms intol ′ xy = φ xφ y l xy . (19.4)However, the edge lengths are constrained: they must be such that the hypervolumesof all four-simplices are real. One can show that the product of two suchconformal transformations, such that each separately preserves these constraints,is a transformation which will in general violate the constraints. Thus globally thegroup property is violated. Furthemore, no subset of the transformations forms a