On the Representation Theory of Deformation Quantization

  title={On the Representation Theory of Deformation Quantization},
  author={S. Waldmann},
  journal={arXiv: Quantum Algebra},
  • S. Waldmann
  • Published 2001
  • Mathematics, Physics
  • arXiv: Quantum Algebra
In this contribution to the proceedings of the 68eme Rencontre entre Physiciens Theoriciens et Mathematiciens on Deformation Quantization I shall report on some recent joint work with Henrique Bursztyn on the representation theory of *-algebras arising from deformation quantization as I presented this in my talk. 
In this review we discuss various aspects of representation theory in deformation quantization starting with a detailed introduction to the concepts of states as positive functionals and the GNSExpand
Remarks on modules over deformation quantization algebras
We study an asymptotic version of the Maslov-Hormander construction of Lagrangian distributions in terms of deformation quantization.
Noncommutative Field Theories from a Deformation Point of View
In this review we discuss the global geometry of noncommutative field theories from a deformation point of view: The space-times under consideration are deformations of classical space-time manifoldsExpand
Deformation Quantization: From Quantum Mechanics to Quantum Field Theory
The aim of this paper is to give a basic overview of Deformation Quantization (DQ) to physicists. A summary is given here of some of the key developments over the past thirty years in the context ofExpand
In this note we recall some recent progress in understanding the representation theory of ∗ -algebras over rings C = R(i) where R is ordered and i 2 = −1. The representation spaces are modules overExpand
Towards Adelic Noncommutative Quantum Mechanics
A motivation of using noncommutative and nonarchimedean geometry on very short distances is given. Besides some mathematical preliminaries, we give a short introduction in adelic quantum mechanics.Expand
A personal view on Julius Wess’s human and scientific legacy in Serbia and the Balkan region is given. Motivation for using noncommutative and nonarchimedean geometry on very short distances isExpand
Morita Equivalence of Fedosov Star Products and Deformed Hermitian Vector Bundles
Based on the usual Fedosov construction of star products for a symplectic manifold M, we give a simple geometric construction of a bimodule deformation for the sections of a vector bundle over MExpand
Fedosov observables on constant curvature manifolds and the Klein–Gordon equation
Abstract In this paper we construct the Fedosov star-algebra of observables on the phase–space of a single particle in the case of all (finite-dimensional) constant curvature manifolds imbeddable inExpand
(Bi)modules, morphismes et réduction des star-produits: le cas symplectique, feuilletages et obstructions
(Bi)modules, morphisms and reduction of star-products are studied in a framework of multidifferential operators along maps: morphisms deform Poisson maps and representations on functions spacesExpand


A Remark on the Deformation of GNS Representations of *-Algebras
Abstract Motivated by deformation quantization we investigate an algebraic GNS construction of *-representations of deformed *-algebras over ordered rings and compute their classical limit. TheExpand
Deformation of Hermitian Vector Bundles and Non-Commutative Field Theory
In this note I will relate some recent results on the * -representation theory of star product algebras to the deformation quantization of Hermitian vector bundles. Moreover, the physicalExpand
A Remark on Formal KMS States in Deformation Quantization
Within the framework of deformation quantization, we define formal KMS states on the deformed algebra of power series of functions with compact support in phase space as C[[λ]]-linear functionalsExpand
On Fedosov's approach to deformation quantization with separation of variables
It was shown in our earlier paper that the deformation quantizations with separation of variables on a Kaehler manifold are parametrized by the formal deformations of the Kaehler form. The FedosovExpand
A Path Integral Approach¶to the Kontsevich Quantization Formula
Abstract: We give a quantum field theory interpretation of Kontsevich's deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the pathExpand
Weyl manifolds and deformation quantization
This paper deals with non-commutative objects based on the Weyl algebra from the differential geometric point of view. We propose to extend familiar notions from manifolds to non-commutative objects;Expand
Locality in GNS Representations of Deformation Quantization
Abstract: In the framework of deformation quantization we apply the formal GNS construction to find representations of the deformed algebras in pre-Hilbert spaces over ℂ[[λ]] and establish the notionExpand
Deformation quantizations with separation of variables on a Kähler manifold
We give a simple geometric description of all formal differentiable deformation quantizations on a Kähler manifoldM such that for each open subsetU⊂M ⋆-multiplication from the left by a holomorphicExpand
D-branes and deformation quantization
In this note we explain how world-volume geometries of D-branes can be reconstructed within the microscopic framework where D-branes are described through boundary conformal field theory. We extractExpand
Noncommutative gauge theory for Poisson manifolds
Abstract A noncommutative gauge theory is associated to every Abelian gauge theory on a Poisson manifold. The semi-classical and full quantum version of the map from the ordinary gauge theory to theExpand