Brian C. Hall
Ph.D., Cornell University, 1993
Analysis and Partial Differential Equations, Differential Geometry, Mathematical Physics
Mathematical Physics and Lie Groups
My research is in mathematical physics, specifically mathematical problems motivated by quantum mechanics and quantum field theory. My work involves several different types of mathematics, including functional analysis, Lie group theory, and probability theory.
One main theme of my work concerns 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.
More recently, I have been interested a particular sort of quantum field theory known as two-dimensional Yang-Mills theory. In four space-time dimensions, Yang-Mills theory is a major ingredient in the standard model of particle physics. The two-dimensional case may be seen as a warm-up to the four-dimensional theory, but is also of interest in its own right, especially because of connections to string theory. I have been interested in the so-called "large-N limit," in which one considers the structure group U(N) and lets N tend to infinity. A key tool in analyzing this limit is the Makeenko-Migdal equation.
B. K. Driver, F. Gabriel, B. C. Hall, and T. Kemp, The Makeenko-Migdal equation for Yang-Mills theory on compact surfaces. Comm. Math. Phys. 352 (2017), 967–978.
B. K. Driver, B. C. Hall, and T. Kemp, Three proofs of the Makeenko-Migdal equation for Yang-Mills theory on the plane. Comm. Math. Phys. 351 (2017), 741–774.
B. K. Driver, B. C. Hall, and T. Kemp, The large-N limit of the Segal-Bargmann transform on U(N). J. Funct. Anal. 265 (2013), 2585–2644.
B. C. Hall, Lie groups, Lie algebras, and representations. An elementary introduction. Second edition. Graduate Texts in Mathematics, 222. Springer, 2015.
B. C. Hall, Quantum theory for mathematicians. Graduate Texts in Mathematics, 267. Springer, 2013.