In all cases, one observes a transition on the boundary of
this phase, but so far no convincing evidence of long-range
correlations has been found in its vicinity. Within the accuracy
of the numerical simulations, this main conclusion is not altered
by the inclusion of determinantal factors
in the measure, the inclusion of higher-order derivative terms,
or the addition of matter fields.
Why
does this happen? Each of the approaches can claim that its
state space represents, at least roughly, an approximation to the
space of smooth Riemannian metrics or geometries. This leaves
only the path-integral measure as a possible culprit. The
measures used up to now were the simplest ones compatible with
considerations of locality and gauge-invariance. It seems
premature to blame the absence of diffeomorphism invariance
(whose status in the gauge-theoretic formulation and the Regge
calculus program remains unclear), since the explicitly
diffeomorphism-invariant dynamical triangulations approach
suffers from similar problems. Further analytical insights are
needed to understand which modifications of the measure would
make these models more interacting.
There are a number of loop holes which could change the picture just presented. It is possible that adding enough matter of the correct type could have a non-trivial effect, or that Regge calculus with the inclusion of higher-order curvature terms does indeed possess a second-order phase transition. Since we have very little experience with universality properties of 4d generally covariant theories, it is not a priori clear whether the choice of measure and the initial restrictions on the lattice geometry can affect the final results.
One may of course take the attitude that something is fundamentally wrong with trying to construct a theory of quantum gravity via a statistical field theory approach, and that a different starting point is needed, an obvious candidate being a non-perturbative theory of superstrings, or of more general extended objects. In any case, these different approaches need not be mutually exclusive, and one may therefore take the results of the discrete approaches presented here as an indication that other attempts of constructing quantum gravity non-perturbatively may run into similar difficulties.
A further unresolved problem is the ``analytic continuation'' of the path-integral results to Lorentzian signature. The Hamiltonian ansatz circumvents this problem, and some progress has been made in the Hamiltonian gauge-theoretic discrete approach. Although the kinematical structure is in place and some information on the constraint algebra has been obtained, the physical state space has not yet been identified. Its results are therefore not sufficiently complete to admit comparison with the path-integral simulations. For the simplicial formulations, only little is known about their canonical counterparts. One would hope that future research will throw further light on these issues.
Acknowledgement. I am indebted to R. Williams, P. Menotti, H. Hamber, W. Beirl and J. Ambjørn for comments and criticism, to E. Schlenk for help with the references, and to R. Helling for assistance in reformatting them.
![]() |
Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll http://www.livingreviews.org/lrr-1998-13 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |