Holomorphic function spaces and quantization

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There has been much interest in the physics literature in the study of mathematical models involving matrices of large size. This work began with the pioneering results Eugene Wigner in the 1950's, in which the energy levels of certain models in nuclear physics were modeled as the "eigenvalues" of large random matrices. The beautiful result of Wigner is that the energy levels actually become nonrandom in the limit as the size of the matrices goes to infinity and are described by Wigner's famous semicircular distribution. This result was the first indication that certain complicated calculations actually become simpler as the size of the matrices involved gets larger. This mathematics research project by Brian Hall is in the same spirit. Hall and his collaborator T. Kemp study an important tool in quantum mechanics (the "Segal--Bargmann transform") on groups of matrices. Their first results indicate that the transform does indeed simplify substantially in the limit as the size of the matrices goes to infinity, to the point that the calculations can be carried out efficiently on a computer. There are many beautiful results still to investigate, including the problem of determining the appropriate analog of Wigner's semicircular law in this setting. This work has close connections to the two-dimensional version of strong nuclear force. That subject, in turn, has been connected to string theory by the Nobel Prize winner David Gross, since the "worldsheet" swept out by a string is a two-dimensional surface. This mathematics research project by Brian Hall concerns the mathematical theory of quantum mechanics. Quantum mechanics is the fundamental physical theory describing the behavior of matter at the atomic scale. Quantum mechanics is foundational to many areas of science and engineering, including solid state physics and the design of computer chips. Ideas from quantum mechanics also have had a profound impact in mathematics, as exemplified by the awarding of the Fields Prize (the so-called Nobel Prize for mathematics) to a physicist, Ed Witten, in 1990. A key issue in quantum mechanics concerns its connection with classical mechanics, the theory that governs the behavior of matter on the macroscopic scale. The Segal--Bargmann transform is a mathematical tool that facilitates a comparison between classical and quantum mechanics. Hall's earlier work has extended the range of application of the Segal--Bargmann transform, and has been cited extensively in both the physics and mathematics literature. Hall's ongoing research will expand the scope of the transform still farther, by connecting it to models derived from study of the strong nuclear force and string theory. This work has potential applications in both mathematics itself, with connections to the exciting new field of free probability theory, and in physics, with possible connections to both string theory and loop quantum gravity.