Approaches to Quantum Gravity

374 R. Williams
dealt with include tetrad variables [46], the constraint structure [47], matter fields
[49] and the continuum limit [50].
A recent development is the application by Gambini and Pullin of their consistent
discretisation [19] to Regge calculus. Their method is algebraic rather than
geometric, and it seems to solve the problems of preservation of the constraints
in numerical relativity and closure of the constraints in the quantum theory. Its
application to Regge calculus [24] is valid in both the Euclidean and Lorentzian
domains, and there is a natural elimination of spikes, which seemed to cause trouble
in Regge calculus in the past [2] (but see also [34]). The method involves
first formulating Regge calculus as a classical unconstrained canonical system, and
then quantising it by implementing canonical transformations which give the discrete
time evolution as a unitary quantum operator. The framework can incorporate
topology change, in particular the evolution from a “no boundary” initial state.
19.8 Conclusions
Regge calculus was the first discretisation scheme in general relativity and the
first form of simplicial Quantum Gravity (for more references, see the bibliography
by Williams and Tuckey [71]). From it have developed a number of
important and highly topical approaches to discrete Quantum Gravity, including
Lorentzian dynamical triangulations and spin foam models [60]. It remains to be
seen which approach will give rise to a fully satisfactory diffeomorphism-invariant
and background-independent theory of Quantum Gravity. There are strong reasons
for taking seriously Regge calculus and the other theories just mentioned,
since they go to the basic level and seek to study the dynamical nature of quantum
space-time.
Acknowledgements
This work has been supported in part by the UK Particle Physics and Astronomy
Research Council. The author thanks Herbert Hamber for helpful comments on this
chapter. She is particularly grateful to Girton College for its continuing support.
References
[1] J. Ambjørn, M. Carfora & A. Marzuoli, The Geometry of Dynamical Triangulations
(Berlin, Springer, 1997).
[2] J. Ambjørn, J. Nielsen, J. Rolf & G. Savvidy, Spikes in quantum Regge calculus,
Class. Quantum Grav., 14 (1997) 3225–41.
[3] M. Bander, Functional measure for lattice gravity, Phys. Rev. Lett., 57 (1986)
1825–7.