Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization (original) (raw)

Deformation quantization of compact Kähler manifolds via Berezin-Toeplitz operators

arXiv: Quantum Algebra, 1996

This talk reports on results on the deformation quantization (star products) and on approximative operator representations for quantizable compact Kahler manifolds obtained via Berezin-Toeplitz operators. After choosing a holomorphic quantum line bundle the Berezin-Toeplitz operator associated to a differentiable function on the manifold is the operator defined by multiplying global holomorphic sections of the line bundle with this function and projecting the differentiable section back to the subspace of holomorphic sections. The results were obtained in (respectively based on) joint work with M. Bordemann and E. Meinrenken.

Deformation quantization of compact Kaehler manifolds by Berezin-Toeplitz quantization

1999

For arbitrary compact quantizable Kähler manifolds it is shown how a natural formal deformation quantization (star product) can be obtained via Berezin-Toeplitz operators. Results on their semi-classical behaviour (their asymptotic expansion) due to Bordemann, Meinrenken and Schlichenmaier are used in an essential manner. It is shown that the star product is null on constants and fulfills parity. A trace is constructed and the relation to deformation quantization by geometric quantization is given. dedicated to the memory of Moshe Flato Date: 22.10.99. 1 2 MARTIN SCHLICHENMAIER

Berezin-Toeplitz quantization of compact Kähler manifolds

Arxiv preprint q-alg/9601016, 1996

In this lecture results are reviewed obtained by the author together with Martin Bordemann and Eckhard Meinrenken on the Berezin-Toeplitz quantization of compact Kähler manifolds. Using global Toeplitz operators, approximation results for the quantum operators are shown. From them it follows that the quantum operators have the correct classical limit. A star product deformation of the Poisson algebra is constructed.

Berezin-Toeplitz quantization and star products for compact Kähler manifolds

Contemporary Mathematics, 2012

For compact quantizable Kähler manifolds certain naturally defined star products and their constructions are reviewed. The presentation centers around the Berezin-Toeplitz quantization scheme which is explained. As star products the Berezin-Toeplitz, Berezin, and star product of geometric quantization are treated in detail. It is shown that all three are equivalent. A prominent role is played by the Berezin transform and its asymptotic expansion. A few ideas on two general constructions of star products of separation of variables type by Karabegov and by Bordemann-Waldmann respectively are given. Some of the results presented is work of the author partly joint with Martin Bordemann, Eckhard Meinrenken and Alexander Karabegov. At the end some works which make use of graphs in the construction and calculation of these star products are sketched.

Berezin-Toeplitz Quantization of compact Kaehler manifolds

1996

Invited lecture at the XIV-th workshop on geometric methods in physics, Bialowieza, Poland, July 9-15, 1995. In this lecture results are reviewed obtained by the author together with Martin Bordemann and Eckhard Meinrenken on the Berezin-Toeplitz quantization of compact Kaehler manifolds. Using global Toeplitz operators, approximation results for the quantum operators are shown. From them it follows that the quantum operators have the correct classical limit. A star product deformation of the Poisson algebra is constructed.

Almost-Kähler deformation quantization

Letters in Mathematical Physics, 2001

We use a natural affine connection with nontrivial torsion on an arbitrary almost-Kähler manifold which respects the almost-Kähler structure to construct a Fedosov-type deformation quantization on this manifold. 1 2 A.V. KARABEGOV AND M. SCHLICHENMAIER parameterized by the elements of the affine vector space (1/iν)[ω] + H 2 (M, C)[[ν]] via the mapping * → cl( * ).

Berezin–Toeplitz quantization on Kähler manifolds

Journal für die reine und angewandte Mathematik (Crelles Journal), 2011

We study the Berezin-Toeplitz quantization on Kähler manifolds. We explain first how to compute various associated asymptotic expansions, then we compute explicitly the first terms of the expansion of the kernel of the Berezin-Toeplitz operators, and of the composition of two Berezin-Toeplitz operators. As application we estimate the norm of Donaldson's É-operator.

Berezin-Toeplitz Quantization and Star Products for Compact Kaehler Manifolds

2012

For compact quantizable Kähler manifolds certain naturally defined star products and their constructions are reviewed. The presentation centers around the Berezin-Toeplitz quantization scheme which is explained. As star products the Berezin-Toeplitz, Berezin, and star product of geometric quantization are treated in detail. It is shown that all three are equivalent. A prominent role is played by the Berezin transform and its asymptotic expansion. A few ideas on two general constructions of star products of separation of variables type by Karabegov and by Bordemann--Waldmann respectively are given. Some of the results presented is work of the author partly joint with Martin Bordemann, Eckhard Meinrenken and Alexander Karabegov. At the end some works which make use of graphs in the construction and calculation of these star products