753 research outputs found
Deformation quantization on a Hilbert space
We study deformation quantization on an infinite-dimensional Hilbert space
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 . 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
An algebra of deformation quantization for star-exponentials on complex symplectic manifolds
The cotangent bundle to a complex manifold 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 . We give the symbol calculus
for this algebra and prove that quantized symplectic transformations operate on
it. If 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
On generalized Abelian deformations
We study sun-products on , i.e. generalized Abelian deformations
associated with star-products for general Poisson structures on . 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, -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
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
Full STEAM Ahead â a Collaborative Colloquium
On February 2, 2012, Contra Costa County Office of Education organized its 2nd Annual STEAM Colloquium: Full STEAM Ahead. This forum brought together over 150 educators, business leaders and community members to discuss and share best practices in Science, Technology, Engineering, Arts and Mathematics (STEAM) educatio
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