Deformation Quantization: Twenty Years After

We first review the historical developments, both in physics and in mathematics, that preceded (and in some sense provided the background of) deformation quantization. Then we describe the birth of the latter theory and its evolution in the past twenty years, insisting on the main conceptual developments and keeping here as much as possible on the physical side. For the physical part the accent is put on its relations to, and relevance for, "conventional" physics. For the mathematical part we concentrate on the questions of existence and equivalence, including most recent developments for general Poisson manifolds; we touch also noncommutative geometry and index theorems, and relations with group theory, including quantum groups. An extensive (though very incomplete) bibliography is appended and includes background mathematical literature.Comment: 39 pages; to be published with AIP Press in Proceedings of the 1998 Lodz conference "Particles, Fields and Gravitation". LaTeX (compatibility mode) with aipproc styl

Topological Hopf algebras, quantum groups and deformation quantization

After a presentation of the context and a brief reminder of deformation quantization, we indicate how the introduction of natural topological vector space topologies on Hopf algebras associated with Poisson Lie groups, Lie bialgebras and their doubles explains their dualities and provides a comprehensive framework. Relations with deformation quantization and applications to the deformation quantization of symmetric spaces are describedComment: 13 pages, to appear in the proceedings of the conference "Hopf algebras in noncommutative geometry and physics" (VUB, Brussels, May 2002

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

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

Some multifaceted aspects of mathematical physics, our common denominator with Elliott Lieb

Mathematical physics has many facets, of which we shall briefly give a (very partial) description, centered around those of main interest for Elliott and us (Moshe Flato and I). In our case these aspects had as a corollary a variety of "parascientific activities", in particular the foundation of IAMP (the International Association of Mathematical Physics) and of the journal LMP (Letters in Mathematical Physics), both of which were strongly impacted by Elliott, and Elliott's long insistence that publishers do not demand "copyright transfer" as a precondition for publication but are satisfied with "consent to publish", which is increasingly becoming standard. Since this article is testimony to the huge scientific impact of Elliott, the latter intertwined aspects constitute the core of the present contribution.Comment: 12 pages, dedicated to our friend Elliott Lieb on the occasion of the ninetieth anniversary of his birt

Quantized anti de Sitter spaces and non-formal deformation quantizations of symplectic symmetric spaces

We realize quantized anti de Sitter space black holes, building Connes spectral triples, similar to those used for quantized spheres but based on Universal Deformation Quantization Formulas (UDF) obtained from an oscillatory integral kernel on an appropriate symplectic symmetric space. More precisely we first obtain a UDF for Lie subgroups acting on a symplectic symmetric space M in a locally simply transitive manner. Then, observing that a curvature contraction canonically relates anti de Sitter geometry to the geometry of symplectic symmetric spaces, we use that UDF to define what we call Dirac-isospectral noncommutative deformations of the spectral triples of locally anti de Sitter black holes. The study is motivated by physical and cosmological considerations.Comment: 24 pages, to appear in Contemporary Mathematics (AMS) in the volume of the proceedings of the conference Poisson 2006 held at Keio Univ (Japan

Charge screening and magnetic anisotropy in metallic rare-earth systems

The calculation of magnetic anisotropy constants is performed beyond the point charge model for a continuous charge density distribution of screening conduction electrons. An important role of the non-uniform electron density, in particular, of the Friedel oscillations, in the formation of crystal field is demonstrated. Such effects can modify strongly the effective ion (impurity) charge and even change its sign. This enables one to justify the anion model, which is often used at discussing experimental data on hydrogen-containing systems. Possible applications to the pure rare-earth metals and RCo5 compounds are discussed. The deformation of magnetic structure near the interstitial positive muon owing to the strong local anisotropy, and the corresponding contribution to the dipole field at the muon are considered.Comment: 10 pages, RevTeX, 3 figure

Remembering Ludwig Dmitrievich Faddeev, Our Lifelong Partner in Mathematical Physics

International audienceWe briefly recount the long friendship that developed between Ludwig and us (Moshé Flato and I), since we first met at ICM 1966 in Moscow. That friendship extended to his school and family, and persists to this day. Its strong personal impact and main scientific components are sketched, including reflections on what mathematical physics is (or should be)

The geometry of space-time and its deformations, from a physical perspective, From Geometry to Quantum Mechanics – in honor of Hideki Omori

Abstract We start with an epistemological introduction on the evolution of the concepts of space and time and more generally of physical concepts in the context of the relation between mathematics and physics from the point of view of deformation theory. The concepts of relativity, including anti de Sitter space-time, and of quantization, are important paradigms; we briefly present these and some consequences. The importance of symmetries and of space-time in fundamental physical theories is stressed. The last section deals with "composite elementary particles" in anti de Sitter space-time and ends with speculative ideas around possible quantized anti de Sitter structures in some parts of the universe