Schrödinger Institute Workshop on Mathematical Problems of Quantum Gravity

Abhay Ashtekar, Penn State

A 2-month workshop was held at the Erwin Schrödinger International Institute for Mathematical Sciences in Vienna during July and August, '96. It was jointly organized by Peter Aichelburg and myself.

There were 23 participants from outside Austria, mostly young physicists who have been working on various aspects of quantum gravity. In addition, about a dozen faculty and students from Vienna actively participated in the seminars and discussions. While the focus of this effort was on non-perturbative quantum general relativity, there were several experts from string theory, supergravity, quantum cosmology, quantum field theory, as well as mathematical physics in a broad sense of the term. Unfortunately, there was a rather severe desk-space limitation in July and so the workshop had to make do without the participation of a number of experts who had time-constraints of their own. There were two weekly ``official seminars" which were widely announced --one entitled ``fundamental issues", and the other ``advanced topics". They enhanced the scientific interaction between workshop participants and the local physics and mathematics community. In addition, there were ``discussion seminars" (the remaining) three days a week. The afternoons were left open for further informal discussions (and real work!).

On the scientific front, the workshop elevated the subject to a new level of maturity. It enabled the participants to take stock of a number of areas to obtain a global picture of issues that are now well-understood and also opened new directions for several other key issues. Because of the space limitation, I will restrict myself here only to a few illustrative highlights. A more detailed discussion of the (July) activities can be found in John Baez's ``This Week's Finds" series, weeks 85-88 ( which also contains many references. A Schrödinger Institute pre-print containing abstracts of seminars will be available early October. Further information on the workshop as well as pre-prints of research carried out during the workshop can be obtained from the Schrödinger Institute home page

In the list that follows, the names in parenthesis refer to people who gave seminars or led discussions (although almost everyone present made significant contributions to all the discussions).

Quantum Hamiltonian constraint. (Hans-Jürgen Matschull, Jorge Pullin, Carlo Rovelli, Thomas Thiemann)

Quantum geometry. (AA, Jerzy Lewandowksi, Renate Loll, Thiemann)

Lattice methods and skeletonization in loop quantum gravity. (Loll, Michael Reisenberger)

Super-selection rules in quantum gravity. (AA, Lewandowski, Donald Marolf, Jose Mourão, Thiemann)

Degenerate metrics: extensions of GR. (Ted Jacobson, Lewandowski, Matschull)

Global issues, Hamiltonian formulations. (Fernanado Barbero, Domenico Giulini)

Mathematical issues in quantum field theory and quantum gravity. (John Baez, Matthias Blau, Herbert Balasin, Rodolfo Gambini, Mourao, Marolf)

Exactly soluble midisuperspaces. (AA, Hermann Nicolai)

Lessons from low dimensional gravity. (AA, Giulini, Lewandowski, Marolf, Mourao, Thiemann, Strobl).

Black-hole entropy. (Jacobson, Kirill Krasnov, Marolf, Rob Myers, Rovelli)

Topological quantum field theories (Baez, Reisenberger)

String duality, conformal field theories (Jürgen Fuchs, Krzysztof Meissner, Myers, Strobl)

Foundations of quantum mechanics and quantum cosmology (AA, Giulini, Jonathan Halliwell, Franz Embacher)

If participants were to single out one topic that generated most excitement, it would probably be the regularization of the Hamiltonian constraint by Thiemann (gr-qc/9606088, 89, 90, 91). This has significantly deepened our understanding of the mathematical problems underlying quantum dynamics of general relativity. (For details, see Baez's article in this issue.) However, a number of important problems remain. In particular, during the workshop it was realized that these regularized quantum constraints have the feature that they strongly commute not only on diffeomorphism invariant states (which is to be expected physically) but also on a rather large class of states which are not diffeomorphism invariant (which is alarming from a physical viewpoint). A related potential difficulty is with the semi-classical limit: it is not clear if all the quantum constraints, taken together, admit a sufficient number of semi-classical states. Analogous calculations in 2+1 dimensions indicate that the appropriate semi-classical sector does exist. In 3+1 dimensions, further work is needed. This will no doubt be an area of much research and new effort in the coming year.

Jorge Pullin
Sun Sep 1 16:45:26 EDT 1996