Quantization, Symmetric Spaces, and Symplectic Reduction

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This proposal studies two issues within the area of quantization theory. The first issue, building on the PI's earlier work, is the holomorphic quantization of symmetric spaces. Attention will be given mainly to the noncompact case, where singularities arise that one does not see in the compact case. Building on work of the PI with J. Mitchell and work of B. Krotz, G. Olafsson, and R. Stanton, the PI intends to find appropriate ways to cancel out these singularities. He intends to develop inversion and isometry formulas for the Segal-Bargmann transform on noncompact symmetric spaces, with the goal of making these formulas as parallel as possible to the results in the dual compact case. The PI hopes to combine the approach used in his work with Mitchell with the shift-operator method of Krotz, Olafsson, and Stanton. The second issue concerns the relationship of quantization to reduction. In work with W. Kirwin, the PI will investigate the unitarity (or lack thereof) of the Guillemin-Sternberg map between the "first quantize then reduce" space and the "first reduce and then quantize space." We hope to demonstrate that this map is not unitary, even to leading order in Planck's constant, but that inclusion of the half-form correction yields leading-order unitarity. This proposal concerns quantization, namely, the construction of a quantum-mechanical theory corresponding to a given classical theory. In modern physics, the relevant theories to be quantized often have interesting geometric properties involving various sorts of symmetries. This proposal attempts to understand how this geometry manifests itself in the quantum theory. The first part of the proposal is an attempt to extend a standard tool in quantum theory, the Segal-Bargmann transform (closely related to the ubiquitous concept of coherent states), to more geometrically interesting situations where nice symmetries are present. The PI's earlier work in this area has already been applied in several different ways by workers in loop quantum gravity. The second part of the proposal concerns the way symmetries interact with quantum mechanics. Most modern physical theories (e.g., gravity, electromagnetism, relativity) use symmetry in an essential way. The PI's work addresses the subtle but important question of whether imposing the symmetries before quantizing gives the same result as imposing them after quantizing.