Quantization on Cotangent Bundles

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PI: Brian C. Hall, University of Notre DameDMS-0200649Abstract:The PI's research concerns the quantization of certainspecial classical systems, those whose classical configurationspace is a compact symmetric space, such as a sphere. Thesimplest physical example of such a system is (the rotationaldegrees of freedom of) a rigid body, whose configuration space isthe rotation group SO(3). The phase space of any such system,namely, the cotangent bundle of the compact symmetric space, hasa natural complex structure that makes the phase space into aKahler manifold. Thus the quantization of such a system can bedone in two ways, one using the usual position Hilbert space andthe other using a Hilbert space of holomorphic functions. Thelatter space generalizes the classical Segal-Bargmann space. Thetwo possible quantum Hilbert spaces are related by a unitarytransform, the generalized Segal-Bargmann transform, developed bythe PI and M. Stenzel. The unitarity of this transform can bere-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 oftwo-dimensional Yang-Mills theory and to the classical limit ofThiemann's quantum gravity theory. The PI is continuing toinvestigate several aspects of the theory, including thesemiclassical localization properties of the coherent states,properties of the associated quantization schemes (generalizedWick, anti-Wick, and Weyl quantizations), and the relationship ofthe theory to geometric quantization. Broadly speaking the PI's research is in the boundaryregion between classical and quantum mechanics. Quantum mechanicsis the theory that governs the world at the atomic scale.Although classical (Newtonian) mechanics works well formacroscopic phenomena, it cannot account for the structure ofatoms and molecules--at this level the quantum theory takes over.For the two theories to be consistent with one another thepredictions of quantum mechanics must pass smoothly into those ofclassical mechanics as the scale passes from microscopic tomacroscopic. On the other hand, the mathematical structure of thetwo theories is very different, so it is challenging tounderstand how this quantum-to-classical transition takes place.The PI's research concerns a reformulation of quantum mechanicswhich is equivalent to the usual one but which brings thedescription of quantum mechanics closer to that of classicalmechanics. Specifically, the PI's work takes one standardreformulation of quantum mechanics, the Segal-Bargmann transform,and extends it to apply to systems with more complicated degreesof freedom, such as rotations. This work has been applied in asimplified model of the strong interaction in particle physicsand in an ambitious program of T. Thiemann and collaborators todevelop a quantum theory of gravity.