Approaches to Quantum Gravity

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