32 research outputs found
Infinitesimal deformations of a formal symplectic groupoid
Given a formal symplectic groupoid over a Poisson manifold ,
we define a new object, an infinitesimal deformation of , which can be
thought of as a formal symplectic groupoid over the manifold equipped with
an infinitesimal deformation of the Poisson bivector
field . The source and target mappings of a deformation of are
deformations of the source and target mappings of . To any pair of natural
star products having the same formal symplectic groupoid
we relate an infinitesimal deformation of . We call it the deformation
groupoid of the pair . 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
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
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
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
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
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
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 ,
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
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 .Comment: 27+1 pages, harvmac file, no figure
Felix Alexandrovich Berezin and his work
This is a survey of Berezin's work focused on three topics: representation
theory, general concept of quantization, and supermathematics.Comment: LaTeX, 27 page
Toeplitz operators on symplectic manifolds
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