This century, with Einstein's general theory of relativity, space-time assumed a dynamical role in the theory of gravitation. It was a revolution in our ideas and has led to momentous discoveries regarding the large scale organization of matter in the universe. Equally important is the progress we have made in exploring the subatomic world and in the search for an all-unifying theory of fundamental interactions. But problems remain and physics is at cross roads of varying ideas at the "End of the Millenium."
The Minnowbrook Symposium took place on May 28-31 and was organized to bring together the exponents of varying ideas, specifically, 1). Classical general relativity, 2). Recent developments in string theory and the emerging view of space-time, 3). Quantum gravity and generalized quantum field theories and, 4). Non-commutative geometry and its perspective on the structure of space-time at short distances.
The first day of the symposium in the morning began with John Stachel's review of the various space-time structures associated with no-relativistic Galileian, and relativistic Minkowskian theories, and showing how generally covariant space-times are fundamentally different from their predecessors and the implication of this difference on the usual starting point of space-time theories. His talk was followed by two talks on Black Hole Thermodynamics and related problems. Robert Wald, after reviewing some of the established results in black hole thermodynamics, spoke about the major unresolved issues such as whether black hole entropy should be viewed as "residing" in its deep interior, on its horizon, or in its "thermal atmosphere." Ted Jacobson gave an overview of some results and open questions concerning the meaning of black hole entropy and the nature of the holographic bound on the entropy contained within a surface of given area. He dealt with issues such as the role of internal states, entanglement, species independence, renormalization of G and the Second Law of Thermodynamics. He ended up providing a derivation of the Einstein's equation from the proportionality of entropy and area. In the afternoon, Ted Newman spoke about his (and his collaborators) alternate formulation of general relativity in terms of families of characteristic surfaces and a scalar function. With these as fundamental variables, the conventional variables such as the metric tensor become derived concepts and the final equations, although they do not resemble the standard version of GR, yield results identical to those of GR. He was followed by Roger Penrose, who expounded his ideas regarding quantum state-vector reduction, viewing it as a gravitational phenomenon. He discussed both theoretical arguments as well as possible experimental tests (now actively pursued) and presented some new theoretical developments.
The first day ended with a lively panel discussion led by Lee Smolin. Different subgroups got together first to discuss important questions in light of the day's proceedings and issues to be addressed the following days.
The second day began with a session on String Theory and String Theorists' view of Space-time. Due to illness, Joe Polchinski had to cancel his participation at the last minute, but we had two excellent talks by Michael Douglas and Brian Greene that covered some aspects of Polchinski's subject matter. Speaking about D-Geometry, Matrix theory and Noncommutative geometry, Douglas surveyed recent developments in D-branes theory and the nature of space-time as seen by D-branes. Greene presented new geometrical ideas that have emerged out of recent researches in String Theory, ideas involving dualities, mirror symmetry, topology change, and non-commutative geometry, giving rise to an evolving "quantum geometry" or "string geometry." In the afternoon session, Abhay Ashtekar discussed the features of quantum geometry through two specific examples, namely, Einstein-Maxwell theory in 2+1 dimensions and quantum geometry of black hole horizons in 3+1 dimensions. With non-perturbative, complete solution of the problem, one finds unexpected limitations of the classical and semi-classical theory in the first case. In the second case, the horizon geometry is described by the quantum Chern-Simons theory on a punctured 2-sphere, giving rise to states that account for the black hole entropy. John Baez explained how spin network techniques have provided a mathematically rigorous and intuitively compelling picture of the kinematical aspects of loop quantum gravity. For a true understanding of the dynamical aspects of gravity, he said, one needs a model for 'quantum 4-geometry", that is a truly quantum mechanical description of the geometry of space-time. He discussed the notion of " spin foam", as a probable candidate for such a description.
The last and final day began with Alain Connes who described the foundations of non-commutative geometry in terms of spectral triplets and how it has been successful in predicting some features of the standard model. He outlined a "Spectral Action Principle" from which one could derive elementary particle interactions as fluctuations of he metric. Ali Chamseddine took it from there and discussed in some detail examples of non-commutative spaces appearing in the standard model and string theory. He showed how when the spectral action principle is applied to the standard model, internal symmetries merge with space-time symmetries, unifying gravity with the other interactions. Dirk Kreimer discussed recent developments in perturbative quantum field theories pertaining to the role of a Hopf algebra governing their renormalization and how this algebra is related to the Hopf algebra structure of diffemorphisms in the context of non-commutative geometry. The afternoon session was devoted to Discrete as opposed to a continuum space-time picture at short distances. Klaus Fredenhagen spoke about fundamental uncertainty relations for space-time coordinates and how they can be modeled in the framework of non-commutative geometry. Rafael Sorkin described " A Causal Set Dynamics," showing how, starting from certain causality conditions and a discrete form of general covariance, one can derive a very general family of classically stochastic, sequential growth dynamics for causal sets. The resulting theories, according to Sorkin , provide a relatively accessible "half-way house" to full quantum gravity. He noted that non-gravitating matter can also arise dynamically in such theories. The final talk of the session was due to David Finkelstein on Spin, Statistics and Space-time Structure. His basic idea was that below the quark scale, matter, measurement, and space-time can no longer be separated operationally. They fuse into one variable. The final session organized by Carlo Rovelli was devoted to short presentations on some important topics that were not covered in the other sessions. These included among others, Amanda Peet who spoke briefly on dualities in string theories and their implications, Julius Wess on some recent developments in Quantum Groups and Roger Penrose on Twistors.
From all accounts, the symposium achieved its main purpose in that it provided an opportunity for the presentation of varying points and in-depth discussions on fundamental questions regarding space-time structure. Minnowbrook Conference center with its idyllic surroundings, its seclusion and comfort generated a stimulating and friendly atmosphere among the forty or so participants.
For those interested, the complete information regarding the symposium is located at the following URL:
If you click on the speaker's name in the program section, you can link to his transparencies (if the click does not work, add the " speaker name.html" (i.e. Penrose.html) to the /minnowbrook link given above.