Some naturally defined star products for Kähler manifolds by (original) (raw)
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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 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
On representations of star product algebras over cotangent spaces on Hermitian line bundles
Journal of Functional Analysis, 2003
For every formal power series B = B 0 + λB 1 + O(λ 2 ) of closed two-forms on a manifold Q and every value of an ordering parameter κ ∈ [0, 1] we construct a concrete star product ⋆ B κ on the cotangent bundle π : T * Q → Q. The star product ⋆ B κ is associated to the formal symplectic form on T * Q given by the sum of the canonical symplectic form ω and the pull-back of B to T * Q. Deligne's characteristic class of ⋆ B κ is calculated and shown to coincide with the formal de Rham cohomology class of π * B divided by iλ. Therefore, every star product on T * Q corresponding to the Poisson bracket induced by the symplectic form ω + π * B 0 is equivalent to some ⋆ B κ . It turns out that every ⋆ B κ is strongly closed. In this paper we also construct and classify explicitly formal representations of the deformed algebra as well as operator representations given by a certain global symbol calculus for pseudodifferential operators on Q. Moreover, we show that the latter operator representations induce the formal representations by a certain Taylor expansion. We thereby obtain a compact formula for the WKB expansion. *
Quantization of Kähler manifolds. III
Letters in Mathematical Physics, 1994
We use Berezin's dequantization procedure to dene 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.
Berezin-Toeplitz quantization and Berezin symbols for arbitrary compact Kähler manifolds
1999
For phase-space manifolds which are compact Kähler manifolds relations between the Berezin-Toeplitz quantization and the quantization using Berezin's coherent states and symbols are studied. First, results on the Berezin-Toeplitz quantization of arbitrary quantizable compact Kähler manifolds due to Bordemann, Meinrenken and Schlichenmaier are recalled. It is shown that the covariant symbol map is adjoint to the Toeplitz map. The Berezin transform for compact Kähler manifolds is discussed.
Quantization of K�ahler manifolds II
1993
We use Berezin's dequantization procedure to dene 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.
Classical and Quantum Gravity, 1997
We give an elementary proof of the fact that equivalence classes of smooth or differentiable star products on a symplectic manifold M are parametrized by sequences of elements in the second de Rham cohomology space of the manifold. The parametrization is given explicitly in terms of Fedosov's construction which yields a star product when one chooses a symplectic connection and a sequence of closed 2-forms on M. We also show how derivations of a given star product, modulo inner derivations, are parametrized by sequences of elements in the first de Rham cohomology space of M.
1998
This paper is part II of a series of papers on the deformation quantization on the cotangent bundle of an arbitrary manifold Q. For certain homogeneous star products of Weyl ordered type (which we have obtained from a Fedosov type procedure in part I, see q-alg/9707030) we construct differential operator representations via the formal GNS construction (see q-alg/9607019). The positive linear functional is integration over Q with respect to some fixed density and is shown to yield a reasonable version of the Schrödinger representation where a Weyl ordering prescription is incorporated. Furthermore we discuss simple examples like free particle Hamiltonians (defined by a Riemannian metric on Q) and the implementation of certain diffeomorphisms of Q to unitary transformations in the GNS (pre-)Hilbert space and of time reversal maps (involutive anti-symplectic diffeomorphisms of T * Q) to anti-unitary transformations. We show that the fixed-point set of any involutive time reversal map is either empty or a Lagrangean submanifold. Moreover, we compare our approach to concepts using integral formulas of generalized Moyal-Weyl type. Furthermore we show that the usual WKB expansion with respect to a projectable Lagrangean submanifold can be formulated by a GNS construction. Finally we prove that any homogeneous star product on any cotangent bundle is strongly closed, i. e. the integral over T * Q w.r.t. the symplectic volume vanishes on star-commutators. An alternative Fedosov type deduction of the star product of standard ordered type using a deformation of the algebra of symmetric contravariant tensor fields is given.