Brian C. Hall
Professor
Ph.D., Cornell University, 1993
Email: hall.79@nd.edu
Office: 226 Hayes-Healy
Phone: (574) 631-8698
Fax: (574) 631-6579
For additional information see Brian Hall’s Personal Page.
Research Interests
My research is in mathematical physics, specifically mathematical problems motivated by quantum mechanics. My work involves several different type of mathematics, including functional analysis, Lie group theory, and probability theory.
I work on generalizations of the Segal-Bargmann transform. The ordinary Segal-Bargmann transform was developed by I.E. Segal and V. Bargmann in the early 1960’s and provides a unitary transformation from the “position Hilbert space” to a new space called the Segal-Bargmann space. This space is a certain Hilbert space of holomorphic functions and using it as the quantum Hilbert space brings quantum mechanics closer to the underlying classical mechanics. The ordinary Segal-Bargmann transforms was for a quantum particle moving in Euclidean space; my work has been to generalize this to a quantum particle moving on a compact symmetric space. The generalized Segal-Bargmann transform that I have developed can also be understood as a system of generalized coherent states.
The generalized Segal-Bargmann transform has been applied to the quantization of (1+1) -dimensional Yang-Mills theory (a “toy model” of quantum chromodynamics) by N. P. Landsman and K.K. Wren. It has also been used in the “Quantum geometry” approach to quantum gravity, most notably by T. Thiemann and co-authors.
Selected Publications
- B. Hall, Harmonic analysis with respect to heat kernel measure, Bull. Amer. Math. Soc. (N.S.) 38 (2001), 43-78. http://www.ams.org/bull/2001-38-01/
- B. Hall, Geometric Quantization and the generalized Segal-Bargmann transform for Lie groups of compact type, Commun. Math. Phys. 226 (2002), 233-268. http://link.springer.de/link/service/journals/00220/bibs/2226002/22260233.htm
- B. Hall, Coherent states and the quantization of (1 + 1)-dimensional Yang-Mills theory, Rev. Math. Phys. 13 (2001), 1281-1305. http://ejournals.wspc.com.sg/rmp/13/1310/S0129055X011310.html
- B. Hall and J. Mitchell, Coherent states on spheres, J. Math. Phys. 43 (2002), 1211-1236. http://ojps.aip.org/jmp/
- B. Hall, The Segal-Barqmann “coherent state” transform for compact Lie groups, J. Funct. Anal. 122 (1994), 103-151.
Please direct questions and comments to: hall.79@nd.edu