The PI's research concerns the quantization of certain special classical systems, those whose classical configuration space is a compact symmetric space, such as a sphere. The simplest physical example of such a system is (the rotational degrees of freedom of) a rigid body, whose configuration space is the rotation group SO(3). The phase space of any such system, namely, the cotangent bundle of the compact symmetric space, has a natural complex structure that makes the phase space into a Kahler manifold. Thus the quantization of such a system can be done in two ways, one using the usual position Hilbert space and the other using a Hilbert space of holomorphic functions. The latter space generalizes the classical Segal-Bargmann space. The two possible quantum Hilbert spaces are related by a unitary transform, the generalized Segal-Bargmann transform, developed by the PI and M. Stenzel. The unitarity of this transform can be re-formulated as a resolution of the identity for the associated "coherent states," as shown in detail by the PI and J. Mitchell. These results have been applied to the quantization of two-dimensional Yang-Mills theory and to the classical limit of Thiemann's quantum gravity theory. The PI is continuing to investigate several aspects of the theory, including the semiclassical localization properties of the coherent states, properties of the associated quantization schemes (generalized Wick, anti-Wick, and Weyl quantizations), and the relationship of the theory to geometric quantization.
Broadly speaking the PI's research is in the boundary region between classical and quantum mechanics. Quantum mechanics is the theory that governs the world at the atomic scale. Although classical (Newtonian) mechanics works well for macroscopic phenomena, it cannot account for the structure of atoms and molecules--at this level the quantum theory takes over. For the two theories to be consistent with one another the predictions of quantum mechanics must pass smoothly into those of classical mechanics as the scale passes from microscopic to macroscopic. On the other hand, the mathematical structure of the two theories is very different, so it is challenging to understand how this quantum-to-classical transition takes place. The PI's research concerns a reformulation of quantum mechanics which is equivalent to the usual one but which brings the description of quantum mechanics closer to that of classical mechanics. Specifically, the PI's work takes one standard reformulation of quantum mechanics, the Segal-Bargmann transform, and extends it to apply to systems with more complicated degrees of freedom, such as rotations. This work has been applied in a simplified model of the strong interaction in particle physics and in an ambitious program of T. Thiemann and collaborators to develop a quantum theory of gravity.
