Gauge Theories and Quantum Gravity

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At the frontier of theoretical physics lies the task of reconciling gravity with elementary particle physics. One effective way to make progress in this difficult problem is the development of toy models for the interaction of gravity with matter in a quantum theory. These models often have fewer space dimensions than the three of our real world; this makes them easier to manipulate mathematically. Ideally, this gives us answers to questions in this setting that are at present unattainable in more realistic theories. Over the next few years I intend to pursue three projects in this area. The first project is a re-formulation of one promising theory, string theory, one of the premier ways to to reconcile gravity and elementary particle physics. String theories as usually formulated require more than three dimensions. This is both a strength and a weakness- the former because the extra dimensions might be associated with the unification of the fundamental interactions; the latter because there has been no satisfactory resolution of the unobservability of the extra dimensions. In my work on this problem, I have the string evolve in a world where there is dust evolving as well, and this permits a world with 3 space dimensions. In my second project, I will use a mathematical tool, the Yang-Mill flow, which is related to the heat equation, in that it uniformizes the flowing quantity. I will use the Yang-Mills flow to probe the spacetime foam picture of quantum gravity in which at very small distances, the structure of spacetime fluctuates wildly. The Yang-Mills flow will show how the spacetime foam smooths out at larger distance scales to yield the more uniform spacetime structure predicted by Einstein's theory of gravity. In my final project, I probe a universe that is spherically symmetric- say far from all bits of matter, where gravity can be described by a theory in a world of one space dimension. I will use a construction that enables one to simply describe the interaction of matter with spherically symmetric gravity to explore the formation of a black hole, its decay and its eventual end point.