arXiv subjects
Publications and source records attributed to Brian C. Hall.
This paper gives necessary conditions and slightly stronger sufficient conditions for a holomorphic function to be the Segal-Bargmann transform of a function in L^p(R^d) with respect to a Gaussian measure. The proof relies on a family of inversion formulas for the Segal-Bargmann transform, which can be "tuned" to give the best estimates for a given value of p.
This set of lecture notes gives an introduction to holomorphic function spaces as used in mathematical physics. The emphasis is on the Segal-Bargmann space and the canonical commutation relations. Later sections describe more advanced topics such as the Segal-Bargmann transform for compact Lie groups and the infinite-dimensional theory.
I review certain results in harmonic analysis for systems whose configuration space is a compact Lie group. The results described involve a heat kernel measure, which plays the same role as a Gaussian measure on Euclidean space. The main constructions are group analogs of the Hermite expansion, the Segal-Bargmann transform, and the Taylor expansion. The results are related to geometric quantization, to stochastic analysis, and to the quantization of 1+1-dimensional Yang-Mills theory.
These notes give an elementary introduction to Lie groups, Lie algebras, and their representations. Designed to be accessible to graduate students in mathematics or physics, they have a minimum of prerequisites. Topics include definitions and examples of Lie groups and Lie algebras, the relationship between Lie groups and Lie algebras via the exponential mapping, the basics of representations theory, the Baker-Campbell-Hausdorff formula, a detailed study of the representations of SU(3), and a brief survey of the representation theory of general semisimple groups.
This paper explains some of the ideas behind a prior joint work of the author with Bruce Driver on the canonical quantization of Yang-Mills theory on a spacetime cylinder. The idea is that the generalized Segal-Bargmann transform for a compact group can be obtained from the ordinary Segal-Bargmann transform by imposing gauge symmetry.
We use a variant of the classical Segal-Bargmann transform to understand the canonical quantization of Yang-Mills theory on a space-time cylinder. This transform gives a rigorous way to make sense of the Hamiltonian on the gauge-invariant subspace. Our results are a rigorous version of the widely accepted notion that on the gauge-invariant subspace the Hamiltonian should reduce to the Laplacian on the compact structure group. We show that the infinite-dimensional classical Segal-Bargmann transform for the space of connections, when restricted to the gauge-invariant subspace, becomes the generalized Segal-Bargmann transform for the the structure group.