Domenico Giulini, University of Zürich

giulini@physik.unizh.ch

On December 13.-14., just prior to GR15, the Raman Research Institute at Bangalore in India hosted a discussion meeting on quantum general relativity as part of its Golden Jubilee celebrations. The plan was to have three talks each morning and one in the afternoon, then followed by longer discussion sessions. The beautiful setting of the institute, together with the un-forced and smooth organization indeed created a perfect atmosphere for inspiring discussions. The topics covered a fairly wide range, from (2+1)-dimensional quantum gravity, loop gravity, lattice approaches and 3-dimensional topology to the quantum theory of black holes and, in particular, the issues associated with black hole entropy. Canonical approaches dominated the scene, but this was partly due to the unfortunate fact that Ashoke Sen had to cancel his talk on string calculations of black hole entropy.

The first speaker was Steve Carlip who presented five main lessons that could so far be learned from (2+1)-dimensional gravity. He listed numerous consistent ways for quantization and pointed out their partial inequivalences. For example, consistent quantizations with or without topology change exist, hence topology change is consistent with, but not required by, quantum gravity. Another striking lesson concerns the euclidean path integral approach. In (2+1)-dimensions it can be shown that the contribution from the many arbitrarily complicated interpolating topologies cannot be neglected (as is sometimes assumed). Once more it became clear that, despite all differences to (3+1)-dimensions, (2+1)-dimensional gravity is an important and useful test bed to study concepts and expectations in quantum gravity.

Carlo Rovelli gave a large scale survey on progress and problems in loop quantum gravity. Recent progress in physical predictions at the Planck scale mainly originate from calculations of spectra of operators (on the auxiliary Hilbert space of pure gravity) representing area and volume of two- and three-dimensional subsets. In absence of any matter degrees of freedom these subsets are mathematically specified in a non diffeomorphism invariant fashion. Progress on the mathematical side was also reported. The long standing problems concerning the lack of a scalar product, overcompleteness of the loop basis and the implementation of the reality conditions seem to be settled now. Anomaly free regularizations of the super-hamiltonian have been constructed, but there is still an ongoing debate as to its physical correctness, since it does not define a deformation of the classical constraint algebra and hence seems to reproduce the wrong classical limit. Rovelli ended by emphasizing the complementary strengths and weaknesses of loop quantum gravity and string theory.

Renate Loll reported on the status of discrete approaches to 4-dimensional quantum gravity based on the Einstein action. She discussed results from Hamiltonian path-integral approaches with connection variables and dynamical triangulations. The common open problem is the absence of appropriate measures on the discretized configuration spaces. The choices explored so far seem too simple to lead to an interacting, diffeomorphism-invariant field theory.

There were two talks on topological issues in (3+1)-dimensional canonical gravity. Domenico Giulini started with discussing the role and significance of three-dimensional topology in the classical and quantum theories. One of the issues addressed was whether and how classical topology leaves its fingerprints in the quantum theory. In this context the mapping class groups of three-dimensional manifolds were argued to be the natural objects to look at, since they carry significant amounts of topological information and also enter the quantum theory through the reduction procedure. Giulini concluded by listing some general properties of 3d mapping class groups, like finite presentations, residual finiteness and semi-direct product structures. Sumati Surya reported on some work using Mackey theory to find interesting representations of 3d mapping class groups and discussed their physical implications. Thinking of the 3-manifold as configuration of elementary `geons' (i.e. prime-manifolds), she showed and discussed the general absence of spin-statistics correlations at the kinematical level, and also the possibility of novel `cyclic' statistics types which she encountered with three RP-3 geons.

Two talks and an additional discussion session -- filling the gap that the cancellation of Ashoke Sen's talk left -- were devoted to black hole entropy. V. Frolov's talk centered around the problem of universality of black hole entropy which, despite some impressive derivations, like e.g. by counting states of D-branes, is still an open one. He discussed the idea of entanglement entropy, some of its problems, and how they can be solved in some models of induced gravity. He reported on recent work on such models showing that universality exists within a special class. In Parthasarathi Majumdar's talk the different approaches to understand black hole entropy were compared. In particular, the string calculations and viewpoints now came to their right. A final discussion session, solely devoted to all kinds of questions relating to black hole entropy, marked the end of this most pleasant meeting.

Sun Feb 8 20:46:09 EST 1998