This work is a contribution to the area of Strict Quantization (in the sense of Rieffel) in the presence of curvature and non-Abelian group actions. More precisely, we use geometry to obtain explicit oscillatory integral formulae for strongly invariant strict deformation quantizations of a class of solvable symplectic symmetric spaces. Each of these quantizations gives rise to a field of (pre)-C*-algebras whose fibers are function algebras which are closed under the deformed product. The symmetry group of the symmetric space acts on each fiber by C*-algebra automorphisms.
We present explicit universal strict deformation quantization formulae for actions of Iwasawa subgroups AN of SU(1,n). This answers a question raised by Rieffel.
Ricci-type symplectic manifolds have been introduced and extensively studied by M. Cahen et al.. In this note, we describe their deformation quantizations in the split solvable symmetric case. In particular, we introduce the notion of non-formal tempered deformation quantization on such a space. We show that the set of tempered deformation quantizations is in one-to-one correspondence with the space of Schwartz operator multipliers on the real line. Moreover we prove that every invariant formal star product on a split Ricci-type solvable symmetric space is an asymptotic expansion of a tempered…
We give a new construction of strict deformation quantization of symplectic manifolds equipped with a proper Lagrangian fiber bundle structure, whose representation spaces are the quantum Hilbert spaces obtained by geometric quantization. The construction can be regarded as a "lattice approximation of the correspondence between differential operators and principal symbols". We analyze the corresponding formal deformation quantization. We also investigate into relations between our construction and Berezin-Toeplitz deformation quantization.
In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative definition of noncommutative non-principal torus bundles.
We address one of the open problems in quantization theory recently listed by Rieffel. By developping in detail Connes' tangent groupoid principle and using previous work by Landsman, we show how to construct a strict, flabby quantization, which is moreover an asymptotic morphism and satisfies the reality and traciality constraints, on any oriented Riemannian manifold. That construction generalizes the standard Moyal rule. The paper can be considered as an introduction to quantization theory from Connes' point of view.
In this paper, we initiate the study of nonassociative strict deformation quantization of C*-algebras with a torus action. We shall also present a definition of nonassociative principal torus bundles, and give a classification of these as nonassociative strict deformation quantization of ordinary principal torus bundles. We then relate this to T-duality of principal torus bundles with $H$-flux. We also show that the Octonions fit nicely into our theory.
We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on $\mathbb R^d$, generalizing known results for constant and linear Poisson structures to polynomial Poisson structures of arbitrary degree. We give several examples of nonlinear Poisson structures and construct explicit formal star products whose deformation parameter can be evaluated to any real value of $\hbar$, giving strict quantizations on the space of analytic functions on $\mathbb R^d$ with infinite radius of convergence. We also address further questions suc…
The aim of this paper is two-fold. Firstly we provide necessary and sufficient criteria for the existence of a strict deformation quantization of algebraic tensor products of Poisson algebras, and secondly, we discuss the existence of products of KMS states. As an application, we discuss the correspondence between quantum and classical Hamiltonians in spin systems and we provide a relation between the resolvent of Schödinger operators for non-interacting many-particle systems and quantization maps.
In this paper, we use the parametrised strict deformation quantization of C*-bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H_3-twisted noncommutative torus bundles over a locally compact space. This is extended to the case of general torus bundles and their parametrised strict deformation quantization. Rieffel's basic construction of an algebra deformation can be mimicked to deform a monoidal category, which deforms not only algebras but also modules. As a special case, we consider the parametrised stric…
We apply methods from strict quantization of solvable symmetric spaces to obtain universal deformation formulae for actions of every three-dimensional solvable Lie group. We also study compatible co-products by generalizing the notion of smash product in the context of Hopf algebras. We investigate in particular the dressing action of the `book' group on SU(2).
In this work various symbol spaces with values in a sequentially complete locally convex vector space are introduced and discussed. They are used to define vector-valued oscillatory integrals which allow to extend Rieffel's strict deformation quantization to the framework of sequentially complete locally convex algebras and modules with separately continuous products and module structures, making use of polynomially bounded actions of $\mathbb{R}^n$. Several well-known integral formulas for star products are shown to fit into this general setting, and a new class of examples involving compactl…
This paper shows how to construct classical and quantum field C*-algebras modeling a $U(1)^n$-gauge theory in any dimension using a novel approach to lattice gauge theory, while simultaneously constructing a strict deformation quantization between the respective field algebras. The construction starts with quantization maps defined on operator systems (instead of C*-algebras) associated to the lattices, in a way that quantization commutes with all lattice refinements, therefore giving rise to a quantization map on the continuum (meaning ultraviolet and infrared) limit. Although working with op…
In the standard example of strict deformation quantization of the symplectic sphere $S^2$, the set of allowed values of the quantization parameter $\hbar$ is not connected; indeed, it is almost discrete. Li recently constructed a class of examples (including $S^2$) in which $\hbar$ can take any value in an interval, but these examples are badly behaved. Here, I identify a natural additional axiom for strict deformation quantization and prove that it implies that the parameter set for quantizing $S^2$ is never connected.
We obtain a family of strict $\hat G$-invariant products on the space of holomorphic functions on a semisimple coadjoint orbit of a complex connected semisimple Lie group $\hat G$. By restriction, we also obtain strict $G$-invariant products $*_\hbar$ on a space $A(O)$ of certain analytic functions on a semisimple coadjoint orbit $O$ of a real connected semisimple Lie group $G$. The space $A(O)$ endowed with one of the products $*_\hbar$ is a Fréchet algebra, and the formal expansion of the products around $\hbar = 0$ determines a formal deformation quantization of $O$, which is of Wick type i…
We apply methods from strict quantization of solvable symmetric spaces to obtain universal deformation formulae for actions of a class of solvable Lie groups. We also study compatible co-products by generalizing the notion of smash product in the context of Hopf algebras.
This work is the final version of my master thesis. Many, but not all of its key results are already available as a preprint with Chiara Esposito and Stefan Waldmann on arxiv.org under the title "Convergence of the Gutt Star Product", which has been submitted to a journal. The main goal of this work is to understand a certain deformation of the commutative product on the symmetric tensor algebra over a given (locally convex) Lie algebra in a topological setting. Therefore, the tensor algebra over the Lie algebra is topologized in a locally convex manner. This finally provides a new example of …
In 1979, M. Kashiwara and M. Vergne formulated a conjecture on a Lie group G which implies that the Duflo isomorphism of Z(g) and S(g)^g extends to a natural module isomorphism between the spaces of germs of invariant distributions on G and g=Lie(G), respectively. They also proved their conjecture for G solvable. Using Kontsevich's deformation quantization we establish directly this result for distributions on any real Lie group G. In turn this gives a new proof of Duflo's result on the local solvability of bi-invariant differential operators on G.
$C^*$-algebraic Weyl quantization is extended by allowing also degenerate pre-symplectic forms for the Weyl relations with infinitely many degrees of freedom, and by starting out from enlarged classical Poisson algebras. A powerful tool is found in the construction of Poisson algebras and non-commutative twisted Banach-$*$-algebras on the stage of measures on the not locally compact test function space. Already within this frame strict deformation quantization is obtained, but in terms of Banach-$*$-algebras instead of $C^*$-algebras. Fourier transformation and representation theory of the mea…
We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C*-algebra may be regarded as a result of a quantization procedure. The C*-algebra of the tangent groupoid of a given Lie groupoid G (with Lie algebra g) is the C*-algebra of a continuous field of C*-algebras over R with fibers A_0=C*(g)=C_0(g*) and A_h=C*(G) for nonzero h. The same is true for the corresponding reduced C*-algebras. Our results have applications to, e.g., transformation group C*-algebras, K-theory, and inde…
Deformation quantization and geometric quantization on Kähler manifolds give the mathematical description of the algebra of quantum observables and the Hilbert spaces respectively, where the later forms a representation of quantum observables asymptotically via Toeplitz operators. When there is a Hamiltonian $G$-action on a Kähler manifold, there are associated symmetries on both the quantum algebra and representation aspects. We show that in nice cases of coadjoint orbits and Kähler-Einstein manifolds, these symmetries are strictly compatible (not only asymptotically).
A perturbative quantization procedure for Lie bialgebras is introduced and used to classify all three dimensional complex quantum algebras compatible with a given coproduct. The role of elements of the quantum universal enveloping algebra that, analogously to generators in Lie algebras, have a distinguished type of coproduct is discussed, and the relevance of a symmetrical basis in the universal enveloping algebra stressed. New quantizations of three dimensional solvable algebras, relevant for possible physical applications for their simplicity, are obtained and all already known related resul…
For a complex or real algebraic group G, with g:=Lie(G), quantizations of global type are suitable Hopf algebras F_q[G] or U_q(g) over C[q,q^{-1}]. Any such quantization yields a structure of Poisson group on G, and one of Lie bialgebra on g : correspondingly, one has dual Poisson groups G^* and a dual Lie bialgebra g^*. In this context, we introduce suitable notions of quantum subgroup and of quantum homogeneous space, in three versions: weak, proper and strict (also called "flat" in the literature). The last two notions only apply to those subgroups which are coisotropic, and those homogeneo…
The standard (Berezin-Toeplitz) geometric quantization of a compact Kaehler manifold is restricted by integrality conditions. These restrictions can be circumvented by passing to the universal covering space, provided that the lift of the symplectic form is exact. I relate this construction to the Baum-Connes assembly map and prove that it gives a strict quantization of the manifold. I also propose a further generalization, classify the required structure, and provide a means of computing the resulting algebras. These constructions involve twisted group C*-algebras of the fundamental group whi…
Based on work done by Bonechi, Cattaneo, Felder and Zabzine on Poisson sigma models, we formally show that Kontsevich's star product can be obtained from the twisted convolution algebra of the geometric quantization of a Lie 2-groupoid, one which integrates the Poisson structure. We show that there is an analogue of the Poisson sigma model which is valued in Lie 1-groupoids and which can often be defined non-perturbatively; it can be obtained by symplectic reduction using the quantization of the Lie 2-groupoid. We call these groupoid-valued sigma models and we argue that, when they exist, they…
We show that two approaches to equivariant strict deformation quantization of C*-algebras by actions of negatively curved Kahlerian Lie groups, one based on oscillatory integrals and the other on quantizations maps defined by dual 2-cocycles, are equivalent.
As a generalization of the linear Poisson bracket on the dual space of a Lie algebra, we introduce certain non-linear Poisson brackets which are ``cocycle perturbations'' of the linear Poisson bracket. We show that these special Poisson brackets are equivalent to Poisson brackets of central extension type, which resemble the central extensions of an ordinary Lie bracket via Lie algebra cocycles. We are able to formulate (strict) deformation quantizations of these Poisson brackets by means of twisted group C*-algebras. We also indicate that these deformation quantizations can be used to constru…
