Berezin-Toeplitz quantization and naturally defined star products for Kähler manifolds

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Abstract

For compact quantizable Kähler manifolds the Berezin-Toeplitz quantization schemes, both operator and deformation quantization (star product) are reviewed. The treatment includes Berezin’s covariant symbols and the Berezin transform. The general compact quantizable case was done by Bordemann–Meinrenken–Schlichenmaier, Schlichenmaier, and Karabegov–Schlichenmaier. For star products on Kähler manifolds, separation of variables, or equivalently star product of (anti-) Wick type, is a crucial property. As canonically defined star products the Berezin-Toeplitz, Berezin, and the geometric quantization are treated. It turns out that all three are equivalent, but different.

Keywords

Berezin-Toeplitz quantization Geometric quantization Deformation quantization Kähler manifolds Star products Separation of variables type of star product 

Mathematics Subject Classification

Primary 53D55 Secondary 32J27 47B35 53D50 81S10 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics Research Unit, FSTCUniversity of LuxembourgEsch-sur-AlzetteLuxembourg

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