My most active research area at the moment is in gauge symmetries of generally covariant theories. First let me give some definitions for the physics layperson. Then I'll give a fuller technical description below for the experts.
General covariance means that the form of the dynamical equations is the same regardless of the coordinate system which is used. Einstein's general relativity, which is the successor to Newton's universal law of gravity, is generally covariant. This means that there is no special, preferred choice of spatial or time coordinates. It is still an open question as to how and to what extent this freedom to choose coordinates will be preserved in a quantum theory of gravity. In fact, it is precisely this lack of a fixed spacetime arena (with associated coordinates) which is the obstacle to the construction of a quantum theory of gravity. A Quantum theory of gravity, which would be applicable on extremely small spatial scales (of the order of 10E-33 centimeters) and time scales (of the order of 10E-45 seconds) does not yet exist.
In May of 2001 I was invited to give one of the inaugural presentations to the Austin College faculty highlighting faculty research. My powerpoint presentation available here was entitled Toward a Quantum Theory of Time. In February of 2007 I was a principle organizer of a two-day event at Austin College entitled The Nature of Time: A Minisymposium on Lessons from the Foundations of Relativity and Quantum Physics. This meeting brought together professional physicists and philosphers, and was unusual in that roughly a half of the presentations were intended for a general audience.
With collaborators Josep Pons of the University of Barcelona and Larry Shepley of the Center for Relativity at the University of Texas at Austin we have written a series of papers in which we investigate the preservation of general coordianate transformation and additional gauge symmetries in the transition from a Lagrangian to a Hamiltonian description. Our initial papers establish the general framework in which gauge variables are retained as canonical variables [gr-qc/9612037], and we also describe an alternative algorithm to the Dirac-Bergmann constraint procedure for constructing a self-consistent Hamiltonian model [math-ph/9811029]. We also discussed gauge symmetries in Einstein-Yang-Mills models, a real triad version of canonical gravity, and Ashtekar's complex connection approach to gravity. The links are to preprint versions at the Los Alamos archive. Josep and I also analyzed canonical symmetries in connection approaches with arbitrary Immiri parameter (of which the Ashtekar connection is a special case). A summary preprint version, subsequently published in the proceedings of the Ninth Marcel Grossmann meeting in Rome ("The gauge group in the Ashtekar-Barbero formulation of canonical gravity", in Proceedings of the Ninth Marcel Grossmann Meeting, edited by V.G. Gurzadyan, R. T. Jantzen and R. Ruffini, (World Scientific, New Jersey, 2002), 1298-1299 (with J. Pons)) is available here. The full version is one of those completed projects that begs to be written up!
Our current work addresses the nature of observables in classical general relativity, and their potential usefulness in the construction of an eventual quantum theory of gravity. I published three preliminary suggestions. The first proposed a gauge averaging procedure modeled after an approach of Rovelli's, though retaining gauge variables and recognizing the essential distinction between time evolution and realizeable canonical gauge symmetries. It appears in the proceedings of Third Conference on Constrained Dynamics and Quantum Gravity held in Sardinia in September, 1999. A preprint is available here. The second work was a preliminary exploration into the construction of diffeomorphism invariants using dynamical field-dependent finite gauge transformations. It appears in the Proceedings of the Second Meeting of the International Association for Relativistic Dynamics held in Tel Aviv in June, 2000. A preprint is available here. The third continued the exploration of finite gauge transformations and was published in the Proceedings of the Ninth Marcel Grossmann Meeting held in Rome in 2000 ("Quantum general invariance", in Proceedings of the Ninth Marcel Grossmann Meeting, edited by V.G. Gurzadyan, R. T. Jantzen and R. Ruffini, (World Scientific, New Jersey, 2002), 1300-1301). A preprint is available here.
Josep and I have recently published an exhaustive work dealing with the construction of diffeomorphism invariants (observables) in general relativity. A preprint is available here. We construct local invariants through the use of intrinsic coordinates. This can be accomplished in the canonical framework in general relativity using Weyl curvature scalars, as was first suggested by Komar and Bergmann. I reported on this work at the Balfest in Italy in the summer of 2003, and also gave a summary description in a Theoretical Relativity Seminar at the University of Marylandin June, 2003 and also in a seminar at the Max Planck Institute for Gravitational Physics in Golm in 2005. One essential new observation in this work is the recognition that gauge variables become functionals of the non-gauge variables, and consequently in the quantum theory they become subject to fluctuations. In particular, in canonical quantum gravity the light cone is itself fluctuating.
At the Eleventh Marcel Grossmann Meeting held in Berlin in the summer of 2006 I proposed a first quantum mechanical application of the use of intrinsic time in a simple isotropic cosmological model with a massless scalar material source. The Proceedings preprint is available here. This is a further elaboration of an Austin College Physics Honors Thesis completed by Allison Schmitz.