Brian C. Hall

Lie Groups, Lie Algebras, and Representations

An important concept in physics is that of symmetry, whether it be rotational symmetry for many physical systems or Lorentz symmetry in relativistic systems. In many cases, the group of symmetries of a system is a continuous group, that is, a group that is parameterized by one or more real parameters. More precisely, the symmetry group is often a Lie group, that is, a smooth manifold endowed with a group structure in such a way that operations of inversion and group multiplication are smooth. The tangent space at the identity in a Lie group has a natural “bracket” operation that makes the tangent space into a Lie algebra. The Lie algebra of a Lie group encodes many of the properties of the Lie group, and yet the Lie algebra is easier to work with because it is a linear space.

Geometric Quantization¶and the Generalized Segal--Bargmann Transform¶for Lie Groups of Compact Type

Abstract: Let K be a connected Lie group of compact type and let T*(K) be its cotangent bundle. This paper considers geometric quantization of T*(K), first using the vertical polarization and then using a natural Kähler polarization obtained by identifying T*(K) with the complexified group Kℂ. The first main result is that the Hilbert space obtained by using the Kähler polarization is naturally identifiable with the generalized Segal–Bargmann space introduced by the author from a different point of view, namely that of heat kernels. The second main result is that the pairing map of geometric quantization coincides with the generalized Segal–Bargmann transform introduced by the author. This means that the pairing map, in this case, is a constant multiple of a unitary map. For both results it is essential that the half-form correction be included when using the Kähler polarization.These results should be understood in the context of results of K. Wren and of the author with B. Driver concerning the quantization of (1+1)-dimensional Yang–Mills theory. Together with those results the present paper may be seen as an instance of “quantization commuting with reduction”.

Yang-Mills Theory and the Segal-Bargmann Transform

Abstract:We use a variant of the Segal–Bargmann transform to study canonically quantized Yang–Mills theory on a space-time cylinder with a compact structure group K. The non-existent Lebesgue measure on the space of connections is “approximated” by a Gaussian measure with large variance. The Segal–Bargmann transform is then a unitary map from the L2 space over the space of connections to a holomorphicL2 space over the space of complexified connections with a certain Gaussian measure. This transform is given roughly by followed by analytic continuation. Here is the Laplacian on the space of connections and is the Hamiltonian for the quantized theory.On the gauge-trivial subspace, consisting of functions of the holonomy around the spatial circle, the Segal–Bargmann transform becomes followed by analytic continuation, where ΔK is the Laplacian for the structure group K. This result gives a rigorous meaning to the idea that reduces to ΔK on functions of the holonomy. By letting the variance of the Gaussian measure tend to infinity we recover the standard realization of the quantized Yang–Mills theory on a space-time cylinder, namely, −½ΔK is the Hamiltonian and L2(K) is the Hilbert space. As a byproduct of these considerations, we find a new one-parameter family of unitary transforms from L2(K) to certain holomorphic L2-spaces over the complexification of K. This family of transformations interpolates between the two previously known unitary transformations.Our work is motivated by results of Landsman and Wren and uses probabilistic techniques similar to those of Gross and Malliavin.

Regulation of LKB1/STRAD Localization and Function by E-Cadherin

LKB1 kinase is a tumor suppressor that is causally linked to Peutz-Jeghers syndrome. In complex with the pseudokinase STRAD and the scaffolding protein MO25, LKB1 phosphorylates and activates AMPK family kinases, which mediate many cellular processes. The prototypical family member AMPK regulates cell energy metabolism and epithelial apicobasal polarity. This latter event is also dependent on E-cadherin-mediated adherens junctions (AJs) at lateral borders. Strikingly, overexpression of LKB1/STRAD can also trigger establishment of epithelial polarity in the absence of cell-cell or cell-matrix contacts. However, the upstream factors that normally govern LKB1/STRAD function are unknown. Here we show by immunostaining and fluorescence resonance energy transfer that active LKB1/STRAD kinase complex colocalizes with E-cadherin at AJs. LKB1/STRAD localization and AMPK phosphorylation require E-cadherin-dependent maturation of AJs. However, LKB1/STRAD complex kinase activity is E-cadherin independent. These data suggest that in polarized epithelial cells, E-cadherin regulates AMPK phosphorylation by controlling the localization of the LKB1 complex. The LKB1 complex therefore appears to function downstream of E-cadherin in tumor suppression.

Harmonic analysis with respect to heat kernel measure

This paper surveys developments over the last decade in harmonic analysis on Lie groups relative to a heat kernel measure. These include analogs of the Hermite expansion, the Segal-Bargmann transform, and the Taylor ex- pansion. Some of the results can be understood from the standpoint of geo- metric quantization. Others are intimately related to stochastic analysis.

Coherent states on spheres

We describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d-dimensional sphere Sd. The coherent states are labeled by points in the associated phase space T*(Sd). These coherent states are not of Perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators. We describe as well the Segal–Bargmann representation for the system, the associated unitary Segal–Bargmann transform, and a natural inversion formula. Although many of these results are in principle special cases of the results of Hall and Stenzel, we give here a substantially different description based on ideas of Thiemann and of Kowalski and Rembielinski. All of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space. We focus on the sphere case in order to carry out the calculations in a self-contained and explicit way.

A New Form of the Segal-Bargmann Transform for Lie Groups of Compact Type

I consider a two-parameter family Bs,t of unitary transforms mapping an L2-space over a Lie group of compact type onto a holomorphic L 2 -space overthe complexified group. These were studied usinginfinite- dimensional analysis in joint work with B. Driver, but are treated here by finite-dimensional means. These transforms interpolate between two previously known transforms, and all should be thought of as generaliza- tions of the classical Segal-Bargmann transform. I consider also the limiting cases s→∞ and s→ t/2.

HOLOMORPHIC SOBOLEV SPACES AND THE GENERALIZED SEGAL-BARGMANN TRANSFORM

We consider the generalized Segal-Bargmann transform Ct for a compact group K, introduced in B. C. Hall, J. Funct. Anal. 122 (1994), 103-151. Let KC denote the complexification of K. We give a necessary-and- sufficient pointwise growth condition for a holomorphic function on KC to be in the image under Ct of C 1 (K). We also characterize the image under Ct of Sobolev spaces on K. The proofs make use of a holomorphic version of the Sobolev embedding theorem.

Representations of Semisimple Lie Algebras

In this chapter, we prove the theorem of the highest weight for irreducible, finite-dimensional representations of a complex semisimple Lie algebra \(\mathfrak{g}.\)

Bounds on the Segal-Bargmann transform ofL p functions

This article gives necessary conditions and slightly stronger sufficient conditions for a holomorphic function to be the Segal-Bargmann transform of a function inLp (ℝd, ρ) where ρ is 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 article also gives a single necessary-and-sufficient condition for a holomorphic function to be the transform of a function f such that any derivative of f multiplied by any polynomial is in Lp (d, ρ). Finally, I give some weaker but dimension-independent conditions.