Deformation quantization on a Hilbert space

We study deformation quantization on an infinite-dimensional Hilbert space $W$ endowed with its canonical Poisson structure. The standard example of the Moyal star-product is made explicit and it is shown that it is well defined on a subalgebra of $C^\infty(W)$. A classification of inequivalent deformation quantizations of exponential type, containing the Moyal and normal star-products, is also given

Deformation Quantization: Genesis, Developments and Metamorphoses

We start with a short exposition of developments in physics and mathematics that preceded, formed the basis for, or accompanied, the birth of deformation quantization in the seventies. We indicate how the latter is at least a viable alternative, autonomous and conceptually more satisfactory, to conventional quantum mechanics and mention related questions, including covariance and star representations of Lie groups. We sketch Fedosov's geometric presentation, based on ideas coming from index theorems, which provided a beautiful frame for developing existence and classification of star-products on symplectic manifolds. We present Kontsevich's formality, a major metamorphosis of deformation quantization, which implies existence and classification of star-products on general Poisson manifolds and has numerous ramifications. Its alternate proof using operads gave a new metamorphosis which in particular showed that the proper context is that of deformations of algebras over operads, while still another is provided by the extension from differential to algebraic geometry. In this panorama some important aspects are highlighted by a more detailed account.Comment: Latex file. 40 pages with 2 figures. To appear in: Proceedings of the meeting between mathematicians and theoretical physicists, Strasbourg, 2001. IRMA Lectures in Math. Theoret. Phys., vol. 1, Walter De Gruyter, Berlin 2002, pp. 9--5

On generalized Abelian deformations

We study sun-products on $\R^n$, i.e. generalized Abelian deformations associated with star-products for general Poisson structures on $\R^n$. We show that their cochains are given by differential operators. As a consequence, the weak triviality of sun-products is established and we show that strong equivalence classes are quite small. When the Poisson structure is linear (i.e., on the dual of a Lie algebra), we show that the differentiability of sun-products implies that covariant star-products on the dual of any Lie algebra are equivalent each other.Comment: LaTeX 16 pages. To be published in Reviews in Mathematical Physic

Nambu mechanics, nn-ary operations and their quantization

We start with an overview of the "generalized Hamiltonian dynamics" introduced in 1973 by Y. Nambu, its motivations, mathematical background and subsequent developments -- all of it on the classical level. This includes the notion (not present in Nambu's work) of a generalization of the Jacobi identity called Fundamental Identity. We then briefly describe the difficulties encountered in the quantization of such $n$-ary structures, explain their reason and present the recently obtained solution combining deformation quantization with a "second quantization" type of approach on ${\Bbb R}^n$. The solution is called "Zariski quantization" because it is based on the factorization of (real) polynomials into irreducibles. Since we want to quantize composition laws of the determinant (Jacobian) type and need a Leibniz rule, we need to take care also of derivatives and this requires going one step further (Taylor developments of polynomials over polynomials). We also discuss a (closer to the root, "first quantized") approach in various circumstances, especially in the case of covariant star products (exemplified by the case of su(2)). Finally we address the question of equivalence and triviality of such deformation quantizations of a new type (the deformations of algebras are more general than those considered by Gerstenhaber).Comment: 23 pages, LaTeX2e with the LaTeX209 option. To be published in the proceedings of the Ascona meeting. Mathematical Physics Studies, volume 20, Kluwe

An algebra of deformation quantization for star-exponentials on complex symplectic manifolds

The cotangent bundle $T^*X$ to a complex manifold $X$ is classically endowed with the sheaf of \cor-algebras \W[T^*X] of deformation quantization, where \cor\eqdot \W[\rmptt] is a subfield of \C[[\hbar,\opb{\hbar}]. Here, we construct a new sheaf of \cor-algebras \TW[T^*X] which contains \W[T^*X] as a subalgebra and an extra central parameter $t$. We give the symbol calculus for this algebra and prove that quantized symplectic transformations operate on it. If $P$ is any section of order zero of \W[T^*X], we show that \exp(t\opb{\hbar} P) is well defined in \TW[T^*X].Comment: Latex file, 24 page

Produit de Moyal stochastique sur l'espace de Wiener

We propose a stochastic extension of deformation quantization on a Hilbert space. The Moyal product is defined in this context on the space of functionals belonging to all of the Sobolev spaces of the Malliavin calculus.Comment: 5 page

The damped harmonic oscillator in deformation quantization

We propose a new approach to the quantization of the damped harmonic oscillator in the framework of deformation quantization. The quantization is performed in the Schr\"{o}dinger picture by a star-product induced by a modified "Poisson bracket". We determine the eigenstates in the damped regime and compute the transition probability between states of the undamped harmonic oscillator after the system was submitted to dissipation.Comment: Plain LaTex file, 11 page

Morphofunctional characterization of a rare extracranial internal carotid artery giant aneurism

Aneurisms of the extracranial tract of the internal carotid artery (EICAA) are extremely rare, accounting for 0,4% to 2% of all carotid procedures (El-Sabrout et al., 2000); in females incidence is 2-11 times lower than in males (Siablis et al., 2004). A giant EICAA (32 x 35 mm) at C3 level was studied in a woman aged 81. Samples were prepared for standard transmission electron microscopy. Semithin sections were stained according to Relucenti et al. (2010). Ultrathin sections were contrasted with uranyl acetate and lead citrate. Images revealed the presence of many microvessels just beneath the tunica intima. They were patent, often showed sprouts and pericytes. Endothelial cells appeared metabolically active, with euchromatin, nucleoli, membrane blebs and junctional complexes. Their basal membrane was thickened and sometimes multilaminated. Leukocytes adhering to the endothelium were observed. The internal elastic lamina was so fragmented that it was very difficult to recognize it. The tunica media showed vascular smooth muscle cells (VSMCs) arranged in bundles encircled by collagen-rich extracellular matrix (EM), as well as scattered in the EM. VSMCs had perinuclear organelles, dense bands and caveolae. Collagen fibrils were arranged in a twisted or coiled fashion. Elongated fibroblasts were scattered among VSMCs. The external elastic lamina was conserved. Literature on EICAA is almost exclusively clinical, so this morphofunctional study elucidates structural and ultrastructural changes in the aneurism wall that can contribute to the knowledge of aneurism etiopathogenesis