A Fedosov Star Product of the Wick Type for Kähler Manifolds

In this Letter we compute some elementary properties of the Fedosov star product of Weyl type, such as symmetry and order of differentiation. Moreover, we define the notion of a star product of the Wick type on every Kähler manifold by a straightforward generalization of the corresponding star product in Cn: the corresponding sequence of bidifferential operators differentiates its first argument in holomorphic directions and its second argument in antiholomorphic directions. By a Fedosov type procedure, we give an existence proof of such star products for any Kähler manifold.     

Local ν-Euler Derivations and Deligne's¶Characteristic Class of Fedosov Star Products¶and Star Products of Special Type

In this paper we explicitly construct local ν-Euler derivations , where the ξ_α are local, conformally symplectic vector fields and the are formal series of locally defined differential operators, for Fedosov star products on a symplectic manifold (M,ω) by means of which we are able to compute Deligne's characteristic class of these star products. We show that this class is given by , where is a formal series of closed two-forms on M the cohomology class of which coincides with the one introduced by Fedosov to classify his star products. Moreover, we consider star products that have additional algebraic structures and compute the effect of these structures on the corresponding characteristic classes of these star products. Specifying the constituents of Fedosov's construction we obtain star products with these special properties. Finally, we investigate equivalence transformations between such special star products and prove existence of equivalence transformations being compatible with the considered algebraic structures. Dedicated to the memory of Moshé Flato Received: 28 June 1999 / Accepted: 11 April 2002?Published online: 11 September 2002     

On a Graph Theoretic Formula of Gammelgaard for Berezin–Toeplitz Quantization

We give a proof of (a slightly refined version of) a graph theoretic formula due to Gammelgaard, Karabegov and Schlichenmaier for Berezin–Toeplitz quantization on Kähler manifolds. We obtain the formula by inverting the Berezin transform using a composition formula for the ring of differential operators encoded by linear combinations of strongly connected graphs. The same method is also used to identify the dual Karabegov–Bordemann–Waldmann star product. Our proof has the merit of giving more insight into Karabegov–Schlichenmaier’s identification theorem (Karabegov in J Reine Angew Math 540:49–76, 2001) that the Karabegov classifying form of the Berezin and Berezin–Toeplitz star products are, respectively, obtained by deforming the Kähler metric along the Ricci curvature and the logarithm of the Bergman kernel.     

A Formal Model of Berezin-Toeplitz Quantization

We give a new construction of symbols of the differential operators on the sections of a quantum line bundle L over a Kähler manifold M using the natural contravariant connection on L. These symbols are the functions on the tangent bundle TM polynomial on fibres. For high tensor powers of L, the asymptotics of the composition of these symbols leads to the star product of a deformation quantization with separation of variables on TM corresponding to some pseudo-Kähler structure on TM. Surprisingly, this star product is intimately related to the formal symplectic groupoid with separation of variables over M. We extend the star product on TM to generalized functions supported on the zero section of TM. The resulting algebra of generalized functions contains an idempotent element which can be thought of as a natural counterpart of the Bergman projection operator. Using this idempotent, we define an algebra of Toeplitz elements and show that it is naturally isomorphic to the algebra of Berezin-Toeplitz deformation quantization on M.     

Infinitesimal Deformations of a Formal Symplectic Groupoid

Given a formal symplectic groupoid G over a Poisson manifold (M, π ₀), we define a new object, an infinitesimal deformation of G, which can be thought of as a formal symplectic groupoid over the manifold M equipped with an infinitesimal deformation \({\pi_0 + \varepsilon \pi_1}\) of the Poisson bivector field π ₀. To any pair of natural star products \({(\ast,\tilde\ast)}\) having the same formal symplectic groupoid G we relate an infinitesimal deformation of G. We call it the deformation groupoid of the pair \({(\ast,\tilde\ast)}\) . To each star product with separation of variables \({\ast}\) on a Kähler–Poisson manifold M we relate another star product with separation of variables \({\hat\ast}\) on M. We build an algorithm for calculating the principal symbols of the components of the logarithm of the formal Berezin transform of a star product with separation of variables \({\ast}\) . This algorithm is based upon the deformation groupoid of the pair \({(\ast,\hat\ast)}\) .     

Geodesic symmetries and invariant star products on Kähler symmetric spaces

Starting from work by F. A. Berezin, an earlier paper by the author obtained an invariant star product on every nonexceptional symmetric Kähler space. This would be a generalization to those spaces of the star product on ²ⁿ corresponding to Wick quantization. In this Letter we consider, via geometric quantization, the unitary operators corresponding to geodesic symmetries, and we define a Weyl quantization (first defined by Berezin on rank 1 spaces) in a way similar to the way in which the Weyl quantization can be obtained from the Wick quantization on ²ⁿ . We then calculate every Hochschild 2-cochain of another invariant star product, equivalent to the Wick one, which would be a generalization to those spaces of the Moyal star product on ²ⁿ . M. Cahen and S. Gutt have already provided a theorem of existence and essential unicity of an invariant star product on every irreducible Kähler symmetric space.     

Remarks on the star product of functions on finite and compact groups

We show that the characters of irreducible unitary representations of finite groups and compact Lie groups provide kernels of star-product on complex valued functions f(g) of the group elements g. Examples of permutation groups of two and three elements as well as SU(2) group are considered. The k-deformed star products of the functions of finite and compact Lie groups are presented. The explicit form of the quantizers and dequantizers as well as the duality symmetry of the considered star products of the functions on the finite and compact Lie groups are discussed.     

Some remarks on -invariant Fedosov star products and quantum momentum mappings

In these notes we consider a slightly generalized Fedosov star product * on a symplectic manifold (M,ω), emanating from the fibrewise Weyl product and the triple (,Ω,s) consisting of a symplectic torsion free connection on M, a formal series ΩνZ²_dR(M)[[ν]] of closed two-forms on M, and a certain formal series s of symmetric contravariant tensor fields on M. We prove necessary and sufficient conditions for certain classical symmetries to become symmetries of the star product, only sufficient conditions having been published in special cases when this letter was written (note, however, the different proofs in [S. Gutt, J. Rawnsley, Natural star products on symplectic manifolds and quantum moment maps, 2003. math.SG/0304498 v1]). For a given symplectic vector field X on M, it is well known that (= is a sufficient condition for the Lie derivative to be a derivation of *. We prove that these conditions are in fact necessary ones, also providing a very simple proof for their being sufficient. Moreover, we prove a criterion that has first been presented by Gutt [S. Gutt, Star products and group actions, Contribution to the Bayrischzell Workshop, April 26–29, 2002] (see also [S. Gutt, J. Rawnsley, Natural star products on symplectic manifolds and quantum moment maps, 2003. math.SG/0304498 v1] for a different proof) and which specifies a necessary and sufficient condition for to be a quasi-inner derivation. The statement that this condition is a sufficient one dates back to Kravchenko [O. Kravchenko, Compos. Math. 123 (2000) 131]. Applying our results, we find necessary and sufficient criteria for a Fedosov star product to be -invariant and to admit a quantum Hamiltonian. Finally, supposing the existence of a quantum Hamiltonian, we present a cohomological condition on Ω that is equivalent to the existence of a quantum momentum mapping. In particular, our results show that the existence of a classical momentum mapping in general does not imply the existence of a quantum momentum mapping and thus give a negative answer to Xu’s question posed in [P. Xu, Commun. Math. Phys. 197 (1998) 167].     

A generalization of Weyl's theorem on projectively related affine connections

From a physical point of view, the geodesics in a four-dimensional Lorentzian spacetime are really significant only as point sets. In 1921 Weyl proved that two torsion-free covariant derivative operators D_M and on a manifold M have the same geodesics with possibly different parametrizations if and only if there is a 1-form α on M such that , where 1 is the identity (1,1) tensor on M. By a theorem of Ambrose, Palais and Singer [1], torsion-free covariant derivative operators are generated by affine sprays, and vice versa. More generally, any (not necessarily affine) spray induces a number of covariant derivatives in the tangent bundle τ of M, or in the pull-back bundle τ∗τ. We show that in the context of sprays, similarly to Weyl's relation, a correspondence between the Yano derivatives can be detected.     

Morita Equivalence of Fedosov Star Products and Deformed Hermitian Vector Bundles

Based on the usual Fedosov construction of star products for a symplectic manifold M, we give a simple geometric construction of a bimodule deformation for the sections of a vector bundle over M starting with a symplectic connection on M and a connection for E. In the case of a line bundle, this gives a Morita equivalence bimodule, and the relation between the characteristic classes of the Morita equivalent star products can be found very easily within this framework. Moreover, we also discuss the case of a Hermitian vector bundle and give a Fedosov construction of the deformation of the Hermitian fiber metric.     

Propagation properties of partially coherent Laguerre-Gaussian beams in turbulent atmosphere

To study the propagation properties of partially coherent Laguerre-Gaussian (PLG) beams through turbulent atmosphere, the analytical formulas are derived for the angular width and the beam-propagation factor (M²-factor) of PLG beams by using the extended Huygens-Fresnel principle and the second-order moments of the Wigner distribution function (WDF). The corresponding numerical results are also calculated. When propagation distance increases, the angular width is found to spread faster for PLG beams with higher beam order, smaller correlation length and bigger structure constant The angular width of PLG beams decreases with increase in waist width (w₀).The M²-factor of PLG beams with higher beam order and smaller correlation length is less affected by turbulence with increase in propagation distance. The propagation properties of the M²-factor for PLG beams with the smaller structure constant are better than that with bigger structure constant . The M²-factor of PLG beams decreases with increase in the wavelength λ, and it is also less affected by turbulence for beams with higher order and smaller correlation length. Furthermore, for the PLG beams with the same beam order, the angular width and the M²-factor keep invariable in free space.