1,016 research outputs found
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
Patterns in Wigner-Weyl approach
We present a family of methods, which can describe behaviour of quantum
ensembles and demonstrate the creation of nontrivial (meta) stable states
(patterns), localized, chaotic, entangled or decoherent from basic localized
modes in collective models arising from the quantum hierarchy of Wigner-von
Neumann-Moyal equations.Comment: LaTeX2e, w-art.cls, w-thm.sty, w-pamm.clo, 3 pages, 2 figures,
presented at GAMM Meeting, 2006, Berlin, German
Nambu mechanics, -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 -ary structures, explain their
reason and present the recently obtained solution combining deformation
quantization with a "second quantization" type of approach on . 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
Energy Release in Air Showers
A simulation study of the energy released in air due to the development of an
extensive air shower has been carried out using the CORSIKA code. The
contributions to the energy release from different particle species and
energies as well as the typical particle densities are investigated. Special
care is taken of particles falling below the energy threshold of the simulation
which contribute about 10% to the total energy deposition. The dominant
contribution to the total deposition stems from electrons and positrons from
sub-MeV up to a few hundred MeV, with typical transverse distances between
particles exceeding 1 mm for 10 EeV showers.Comment: 12 pages, 3 figures, accepted by Astropart. Phys.; small content
changes in final versio
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
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