32 research outputs found

    Infinitesimal deformations of a formal symplectic groupoid

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    Given a formal symplectic groupoid GG over a Poisson manifold (M,π0)(M, \pi_0), we define a new object, an infinitesimal deformation of GG, which can be thought of as a formal symplectic groupoid over the manifold MM equipped with an infinitesimal deformation π0+ϵπ1\pi_0 + \epsilon \pi_1 of the Poisson bivector field π0\pi_0. The source and target mappings of a deformation of GG are deformations of the source and target mappings of GG. To any pair of natural star products (,~)(\ast, \tilde\ast) having the same formal symplectic groupoid GG we relate an infinitesimal deformation of GG. We call it the deformation groupoid of the pair (,~)(\ast, \tilde\ast). We give explicit formulas for the source and target mappings of the deformation groupoid of a pair of star products with separation of variables on a Kaehler- Poisson manifold. Finally, we give an algorithm for calculating the principal symbols of the components of the logarithm of a formal Berezin transform of a star product with separation of variables. This algorithm is based upon some deformation groupoid.Comment: 22 pages, the paper is reworked, new proofs are adde

    On Gammelgaard's formula for a star product with separation of variables

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    We show that Gammelgaard's formula expressing a star product with separation of variables on a pseudo-Kaehler manifold in terms of directed graphs without cycles is equivalent to an inversion formula for an operator on a formal Fock space. We prove this inversion formula directly and thus offer an alternative approach to Gammelgaard's formula which gives more insight into the question why the directed graphs in his formula have no cycles.Comment: 29 pages, changes made in the last two section

    On a formula of Gammelgaard for Berezin-Toeplitz quantization

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    We give a proof of a slightly refined version of Gammelgaard's graph theoretic formula for Berezin-Toeplitz quantization on (pseudo-)Kaehler manifolds. Our proof has the merit of giving an alternative approach to Karabegov-Schlichenmaier's identification theorem. We also identify the dual Karabegov-Bordemann-Waldmann star product.Comment: 18 page

    Identification of Berezin-Toeplitz deformation quantization

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    We give a complete identification of the deformation quantization which was obtained from the Berezin-Toeplitz quantization on an arbitrary compact Kaehler manifold. The deformation quantization with the opposite star-product proves to be a differential deformation quantization with separation of variables whose classifying form is explicitly calculated. Its characteristic class (which classifies star-products up to equivalence) is obtained. The proof is based on the microlocal description of the Szegoe kernel of a strictly pseudoconvex domain given by Boutet de Monvel and Sjoestrand.Comment: 26 page

    An explicit formula for the Berezin star product

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    We prove an explicit formula of the Berezin star product on Kaehler manifolds. The formula is expressed as a summation over certain strongly connected digraphs. The proof relies on a combinatorial interpretation of Englis' work on the asymptotic expansion of the Laplace integral.Comment: 19 pages, to appear in Lett. Math. Phy

    Singular projective varieties and quantization

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    By the quantization condition compact quantizable Kaehler manifolds can be embedded into projective space. In this way they become projective varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This allows for generalization to the case of singular varieties. The set-up is explained in the first part of the contribution. The second part of the contribution is of tutorial nature. Necessary notions, concepts, and results of algebraic geometry appearing in this approach to quantization are explained. In particular, the notions of projective varieties, embeddings, singularities, and quotients appearing in geometric invariant theory are recalled.Comment: 21 pages, 3 figure

    Deformation Quantization of Geometric Quantum Mechanics

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    Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is CC^{\infty}, both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is CPCP^{\infty} endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equation are obtained. As is shown this new "second quantization" leads to essentially different results than the former one. For instance, each state is an eigenstate of the total number particle operator and the corresponding eigenvalue is always 1{1 \over \hbar}.Comment: 27+1 pages, harvmac file, no figure

    Toeplitz operators on symplectic manifolds

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    We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for non-compact manifolds and orbifolds are also established.Comment: 40 page
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