Abstract not found

For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finite-dimensional matrix algebras gl(N), N → ∞.

Revised Version 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 Wick type on every Kähler manifold by a straight forward generalization of the corresponding star product in C n: 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.

In this paper we construct homogeneous star products of Weyl type on every cotangent bundle T ∗ Q by means of the Fedosov procedure using a symplectic torsion-free connection on T ∗ Q homogeneous of degree zero with respect to the Liouville vector field. By a fibrewise equivalence transformation we construct a homogeneous Fedosov star product of standard ordered type equivalent to the homogeneous Fedosov star product of Weyl type. Representations for both star product algebras by differential operators on functions on Q are constructed leading in the case of the standard ordered product to the usual standard ordering prescription for smooth complex-valued functions on T ∗ Q polynomial in the momenta (where an arbitrary fixed torsion-free connection ∇0 on Q is used). Motivated by the flat case T ∗ R n another homogeneous star product of Weyl type corresponding to the Weyl ordering prescription is constructed. The example of the cotangent bundle of an arbitrary Lie group is explicitly computed and the star product given by Gutt is rederived in our approach.

The purpose of the paper is to study various aspects of star products on a symplectic manifold related to the Fedosov method. By introducing the notion of “quantum exponential maps”, we give a criterion characterizing Fedosov connections. As a consequence, a geometric realization is obtained for the equivalence between an arbitrary ∗-product and a Fedosov one. Every Fedosov ∗-product is shown to be a Vey ∗-product. Consequently, one obtains that every ∗-product is equivalent to a Vey ∗-product, a classical result of Lichnerowicz. Quantization of a hamiltonian G-space, and in particular, quantum momentum maps are studied. Lagrangian submanifolds are also studied under a deformation quantization. 1

We conjecture that the Sen–Seiberg limit of the Type IIA D2–brane action in a flat spacetime background can be resummed, at all orders in α ′, to define an associative star product on the membrane. This star product can be independently constrained from the equivalent Matrix theory description of the corresponding M2–brane, by carefully analyzing the known BPS conditions. Higher derivative corrections to the Born–Infeld action on the IIA side are reinterpreted, after the Sen–Seiberg limit, as higher derivative corrections to a field theory on the membrane, which itself can be resummed to yield the known Matrix theory quantum mechanics action. Conversely, given the star product on the membrane as a formal power series in α ′, one can constrain the higher derivative corrections to the Born–Infeld action, in the Sen–Seiberg limit. This claim is explicitly verified to first order. Finally, we also comment on the possible application of this method to the derivation of the Matrix theory action for membranes in a curved background. 1

Abstract. We give a complete identification of the deformation quantization which was obtained from the Berezin-Toeplitz quantization on an arbitrary compact Kähler 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 Szegö kernel of a strictly pseudoconvex domain given by Boutet de Monvel and Sjöstrand.

In this paper we develop a method of constructing Hilbert spaces and the representation of the formal algebra of quantum observables in deformation quantization which is an analog of the well-known GNS construction for complex C ∗-algebras: in this approach the corresponding positive linear functionals (‘states’) take their values not in the field of complex numbers, but in (a suitable extension field of) the field of formal complex Laurent series in the formal parameter. By using the algebraic and topological properties of these fields we prove that this construction makes sense and show in physical examples that standard representations as the Bargmann and Schrödinger representation come out correctly, both formally and in a suitable convergence scheme. For certain Hamiltonian functions (contained in the Gel’fand ideal of the positive functional) a formal solution to the time-dependent Schrödinger equation is shown to exist. Moreover, we show that for every Kähler manifold equipped with the Fedosov star product of Wick type all the classical delta functionals are positive and give rise to some formal Bargmann representation of the deformed algebra.

Abstract. For arbitrary compact quantizable Kähler manifolds it is shown how a natural formal deformation quantization (star product) can be obtained via Berezin-Toeplitz operators. Results on their semi-classical behaviour (their asymptotic expansion) due to Bordemann, Meinrenken and Schlichenmaier are used in an essential manner. It is shown that the star product is null on constants and fulfills parity. A trace is constructed and the relation to deformation quantization by geometric quantization is given. dedicated to the memory of Moshe Flato

I was asked by the organisers to present some aspects of Deformation Quantization. Moshé has pursued, for more than 25 years, a research program based on the idea that physics progresses in stages, and one goes from one level of the theory to the next one by a deformation, in the mathematical sense of the word, to be defined in an appropriate category. His study of deformation theory applied to mechanics started in 1974 and led to spectacular developments with the deformation quantization programme. I first met Moshé at a conference in Liège in 1977. A few months later he became my thesis “codirecteur”. Since then he has been one of my closest friends, present at all stages of my personal and mathematical life. I miss him.... I have chosen, in this presentation of Deformation Quantization, to focus on 3 points: the uniqueness –up to equivalence – of a universal star product (universal in the sense of Kontsevich) on the dual of a Lie algebra, the cohomology classes introduced by Deligne for equivalence classes of differential star products on a symplectic manifold and the construction of some convergent star products on Hermitian symmetric spaces. Those subjects will appear in a promenade through the history of existence and equivalence in deformation quantization.