John Rawnsley

Quantization of Kähler manifolds I: geometric interpretation of Berezin's quantization

Abstract We give a geometric interpretation of Berezins symbolic calculus on Kahler manifolds in the framework of geometric quantization. Berezins covariant symbols are defined in terms of coherent states and we study a function ϴ on the manifold which is the central object of the theory. When this function is constant Berezins quantization rule coincides with the prescription of geometric quantization for the quantizable functions. It is defined on a larger class of functions. We show in the compact homogeneous case how to extend Berezins procedure to a dense subspace of the algebra of smooth functions.

Quantization of Kähler manifolds II

We use Berezin’s dequantization procedure to define a formal *- product on a dense subalgebra of the algebra ofsmooth functions on a compact homogeneous Kahler manifold M. We prove that this formal *-product isconvergent when M isa hermitian symmetric space.

Quantization of Kähler manifolds. III

We use Berezins dequantization procedure to define a formal *-product on the algebra of smooth functions on the unit disk in ℂ. We prove that this formal *-product is convergent on a dense subalgebra of the algebra of smooth functions.

Singular unitary representations and indefinite harmonic theory

Abstract Square-integrable harmonic spaces are defined and studied in a homogeneous indefinite metric setting. In the process, Dolbeault cohomologies are unitarized, and singlar unitary representations are obtained and studied.

Natural Star Products on Symplectic Manifolds and Quantum Moment Maps

We define a natural class of star products: those which are given by a series of bidifferential operators which at order k in the deformation parameter have at most k derivatives in each argument. This class includes all the standard constructions of star products. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We parametrise such star products, study their invariance properties and give necessary and sufficient conditions for them to have a quantum moment map. We show that Kravchenkos sufficient condition for a moment map for a Fedosov star product is also necessary.

Quantization of Khler manifolds. IV

We use Berezins dequantization procedure to define a formal *-product on the algebra of smooth functions on the bounded symmetric domains. We prove that this formal *-product is convergent on a dense subalgebra of the algebra of smooth functions.

On tangential star products for the coadjoint Poisson structure

We derive necessary conditions on a Lie algebra from the existence of a star product on a neighbourhood of the origin in the dual of the Lie algebra for the coadjoint Poisson structure which is both differential and tangential to all the coadjoint orbits. In particular we show that when the Lie algebra is semisimple there are no differential and tangential star products on any neighbourhood of the origin in the dual of its Lie algebra.

Representations of a semi-direct product by quantization

1. Introduction . The purpose of this note is to apply the Kostant-Souriau quantization theory (2, 3, 4, 5, 7) to construct representations of a semi-direct product.

On the Pairing of Polarizations

AbstractIfF is a positive Lagrangian sub-bundle of a symplectic vector bundle (E, ω) we show by elementary means that the Chern classes ofF are determined by ω. The notions of metaplectic structure for (E, ω), metalinear structure forF and square root ofKF, the canonical bundle ofF are shown to be essentially the same. IfF andG are two positive Lagrangian sub-bundles with