Strict Quantization Research Articles

Strict Quantization of Polynomial Poisson Structures

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 such as continuity of the classical limit hbar rightarrow 0, compatibility with ^*-involutions, and the existence of positive linear functionals. The latter can be used to realize the strict quantizations as ^*-algebras of operators on a pre-Hilbert space which we demonstrate in a concrete example.

Strict quantization of coadjoint orbits

For every semisimple coadjoint orbit \hat{\mathcal{O}} of a complex connected semisimple Lie group \hat{G} , we obtain a family of \hat{G} -invariant products \hat{*}_\hbar on the space of holomorphic functions on \hat{\mathcal{O}} . For every semisimple coadjoint orbit \mathcal{O} of a real connected semisimple Lie group G , we obtain a family of G -invariant products *_\hbar on a space \mathcal{A}(\mathcal{O}) of certain analytic functions on \mathcal{O} by restriction. \mathcal{A}(\mathcal{O}) , endowed with one of the products *_\hbar , is a G -Fréchet algebra, and the formal expansion of the products around \hbar=0 determines a formal deformation quantization of \mathcal{O} , which is of Wick type if G is compact. Our construction relies on an explicit computation of the canonical element of the Shapovalov pairing between generalized Verma modules and complex analytic results on the extension of holomorphic functions.

Classical limits of unbounded quantities by strict quantization

This paper extends the tools of C*-algebraic strict quantization toward analyzing the classical limits of unbounded quantities in quantum theories. We introduce the approach first in the simple case of finite systems. Then we apply this approach to analyze the classical limits of unbounded quantities in bosonic quantum field theories with particular attention to number operators and Hamiltonians. The methods take classical limits in a representation-independent manner and so allow one to compare quantities appearing in inequivalent Fock space representations.

Strict Deformation Quantization via Geometric Quantization in the Bieliavsky Plane

Using standard techniques from geometric quantization, we rederive the integral product of functions on ℝ2 (non-Euclidian) which was introduced by Pierre Bieliavsky as a contribution to the area of strict quantization. More specifically, by pairing the nontransverse real polarization on the pair groupoid ℝ2×ℝ¯2, we obtain the well-defined integral transform. Together with a convolution of functions, which is a natural deformation of the usual convolution of functions on the pair groupoid, this readily defines the Bieliavsky product on a subset of L2ℝ2.

Classical limits of gauge-invariant states and the choice of algebra for strict quantization

We analyze the quantization of a system consisting of a particle in an external Yang–Mills field within a C*-algebraic framework. We show that in both the classical and quantum theories of such a system, the kinematical algebra of physical quantities can be obtained by restricting attention to symmetry-invariant states on a C*-algebra. We use this to show that symmetry-invariant quantum states correspond to symmetry-invariant classical states in the classical limit.

Quantization of topological invariants under symmetry-breaking disorder

In the strictly periodic setting, the electric polarization of inversion-symmetric solids with and without time-reversal symmetry and the isotropic magneto-electric response function of time-reversal symmetric insulators are known to be topological invariants displaying an exact $\mathbb Z_2$ quantization. This quantization is stabilized by the symmetries. In the present work, we investigate the fate of such symmetry-stabilized topological invariants in the presence of a disorder which breaks the symmetries but restores them on average. Using a rigorous analysis, we conclude that the strict quantization still holds in these conditions. Numerical calculations confirm this prediction.

Positive quantization in the presence of a variable magnetic field

Starting with a previously constructed family of coherent states, we introduce the Berezin quantization for a particle in a variable magnetic field and we show that it constitutes a strict quantization of a natural Poisson algebra. The phase-space reinterpretation involves a magnetic version of the Bargmann space and leads naturally to Berezin-Toeplitz operators.

Embedded Polar Quantization

Embedded polar quantization can be useful for progressive transmission of circularly symmetric data, e.g., for fine-grain scalable coding of parametric audio. Sets of constrained-resolution embedded quantizers are built recursively by successive refinement processes, that are detailed for strict polar quantization and unrestricted polar quantization. The quadratic error minimization problem is solved using equations similar to those of Max, and the refinement algorithm can, in the unrestricted case, be simplified using a high-rate approximation. For Gaussian data, comparisons with reference non-embedded quantizers show that the embedding property comes at an often negligible cost in terms of rate-distortion performance.

A Note on Q-algebras and Quantization

In this note, we study Schwarz’s conjecture on application of Q-algebras to strict quantization. We prove that in the case of a torus with a constant Poisson structure, Schwarz’s formalism gives the same star product as Rieffel [10]. We construct twisted Fock modules as examples of quantization dg-modules in the case of a compact Kähler manifold. In particular, we relate this construction on $$\mathbb{CP}^1$$ to representations of a fuzzy sphere

Quantization of Multiply Connected Manifolds

The standard (Berezin-Toeplitz) geometric quantization of a compact Kähler 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 original 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 which are determined by a group cocycle constructed from the cohomology class of the symplectic form. This provides an algebraic counterpart to the Morita equivalence of a symplectic manifold with its fundamental group.

Strict Quantizations of Almost Poisson Manifolds

We show the existence of (non-Hermitian) strict quantization for every almost Poisson manifold.

Strict Quantizations of Symplectic Manifolds

We introduce the notion of a strict quantization of a symplectic manifold and show its existence under a topological condition.

Strict Quantization of Solvable Symmetric Spaces

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.

Connes' tangent groupoid and strict quantization

We address one of the open problems in quantization theory recently listed by Rieffel. By developing 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.

Lie Groupoid C * -Algebras and Weyl Quantization

For any Lie groupoid $G$, the vector bundle $g^*$ dual to the associated Lie algebroid $g$ is canonically a Poisson manifold. The (reduced) C*-algebra of $G$ (as defined by A. Connes) is shown to be a strict quantization (in the sense of M. Rieffel) of $g^*$. This is proved using a generalization of Weyl's quantization prescription on flat space. Many other known strict quantizations are a special case of this procedure; on a Riemannian manifold, one recovers Connes' tangent groupoid as well as a recent generalization of Weyl's prescription. When $G$ is the gauge groupoid of a principal bundle one is led to the Weyl quantization of a particle moving in an external Yang-Mills field. In case that $G$ is a Lie group (with Lie algebra $g$) one recovers Rieffel's quantization of the Lie-Poisson structure on $g^*$. A transformation group C*-algebra defined by a smooth action of a Lie group on a manifold $Q$ turns out to be the quantization of the semidirect product Poisson manifold $g^*x Q$ defined by this action.

Strict quantization of coadjoint orbits

A strict quantization of a compact symplectic manifold S on a subset I⊆R, containing 0 as an accumulation point, is defined as a continuous field of C*-algebras {Aℏ}ℏ∈I, with A0=C0(S), and a set of continuous cross sections {Q(f)}f∈C∞(S) for which Q0(f)=f. Here Qℏ(f*)=Qℏ(f)* for all ℏ∈I, whereas for ℏ→0 one requires that i[Qℏ(f),Qℏ(g)]/ℏ→Qℏ({f,g}) in norm. We discuss general conditions which guarantee that a (deformation) quantization in a more physical sense leads to one in the above sense. Using ideas of Berezin, Lieb, Simon, and others, we construct a strict quantization of an arbitrary integral coadjoint orbit Oλ of a compact connected Lie group G, associated to a highest weight λ. Here I=0∪1/N, so that ℏ=1/k, k∈N, and A1/k is defined as the C*-algebra of all matrices on the finite-dimensional Hilbert space Vkλ carrying the irreducible representation Ukλ(G) with highest weight kλ. The quantization maps Q1/k(f) are constructed from coherent states in Vkλ, and have the special feature of being positive maps.

Lateral motion of confined excitonic polaritons

Transient-grating measurements on high-quality ${\mathrm{Z}\mathrm{n}\mathrm{S}\mathrm{e}/\mathrm{Z}\mathrm{n}\mathrm{S}}_{x}{\mathrm{Se}}_{1\ensuremath{-}x}$ heterostructures reveal a strong increase of the lateral polariton-transport coefficient with the ZnSe thickness $L$ in a range from 5 to 30 excitonic Bohr radii. In this weak-confinement regime a strict quantization of the polariton--wave-vector component ${k}_{z}$ normal to the layer occurs, as is experimentally verified for our samples. On this basis, it is shown by use of polariton theory that the observed increase of the lateral transport coefficient can be explained by an intrinsic drastic increase of the lateral polariton group velocity. This is caused by a transition from quasiexcitonic to polaritonic transport when the first quantized mode with ${k}_{z}^{1}=\ensuremath{\pi}/L$ passes through the bottleneck region of the lower polariton branch for increasing $L$.

Twisted Lie group $C^*$-algebras as strict quantizations

A nonzero 2-cocycle Γ∈ Z2(g, R) on the Lie algebra g of a compact Lie group G defines a twisted version of the Lie–Poisson structure on the dual Lie algebra g*, leading to a Poisson algebra C∞ (g*(Γ)). Similarly, a multiplier c∈ Z2(G, U(1)) on G which is smooth near the identity defines a twist in the convolution product on G, encoded by the twisted group C-algebra C*(G,c). Further to some superficial yet enlightening analogies between C∞ (g*(Γ)) and C*(G,c), it is shown that the latter is a strict quantization of the former, where Planck’s constant ħ assumes values in (Z\{0})-1. This means that there exists a continuous field of C*-algebras, indexed by ħ ∈ 0 ∪ (Z\{0})-1, for which A0= C0(g*) and Aħ=C*(G,c) for ħ ≠ 0, along with a cross-section of the field satisfying Dirac’s condition asymptotically relating the commutator in Aħ to the Poisson bracket on C∞(g*(Γ)). Note that the ‘quantization’ of ħ does not occur for Γ=0.

Some examples and applications of Fedosov's star products and deformation quantization

We develop a simple example of Fedosov's method of constructing star products and we suggest a generalization of the method to the framework of Poisson supermanifolds. It is also shown how it is possible to construct a strict flabby quantization using Connes' concept of tangent groupoid.