Approaches to Quantum Gravity

370 R. WilliamsAt a typical interior vertex, the classical equations of motion were obtained forthe variations ɛ (i)j. New variables f (i)jwere introduced byɛ (i)j=ˆɛ (i)j+ f (i)j, (19.29)where the ˆɛ (i)jsatisfied the classical equations of motion. Use of these equationsled to the elimination of the ˆɛ (i)j, leaving Gaussian integrals over the f (i)j, whichcontributed only to the normalisation.The only remaining contributions to the action were those assigned to verticeson the boundary. Fourier transforms were taken in the directions with periodicboundary conditions. The fact that the scalar curvature is constrained to vanish onthe boundary was used to eliminate many terms, and a careful identification ofthe boundary ɛ (i)js with the appropriate continuum h ij s[64] led eventually to theHartle–Kuchar expression.19.5 Matter fields in Regge calculus and the measureThe work described so far has been for spaces devoid of matter, but clearly a theoryof Quantum Gravity must include the coupling of gravity to all types of matter. Ona lattice, it is conventional for a scalar field to be defined at the sites, and for a gaugefield to be associated with edges, and this has been the standard method in Reggecalculus (see for example, [31]). On the other hand, fermions need to be definedwithin the simplices, or rather on the sites of the dual lattice, with their couplingdefined by way of the Lorentz transformation relating the frames in neighbouringsimplices [61]. Following a suggestion of Fröhlich [23], Drummond [20] formulateda way of defining spinors on a Regge manifold, which could be modified toinclude the effect of torsion. It is not clear whether the method would overcomethe problem of fermion doubling.Since most of the quantum applications of Regge calculus involve the path integralapproach, the definition of the measure is obviously very important. In hispaper examining very basic questions in quantum Regge calculus, including matterfields as mentioned above, Fröhlich [23] discussed unitarity and reflection positivity,and also defined a measure on a sequence of incidence matrices and thevolumes of their simplices. The dependence of the proposed measure on the cut-offwould involve renormalization group techniques. The measure was also discussedby Cheeger, Müller and Schrader [14], Hartle [38] and Bander [3].In spite of these early suggestions, there is still controversy over the form ofthe measure. It depends not only on the attitude to simplicial diffeomorphisms butalso on the stage at which translation from the continuum to the discrete takesplace. Hamber and Williams [33] argue that the local gauge invariance propertiesof the lattice action show that no Fadeev–Popov determinant is required in the