Quantization Map Research Articles

Gauge theory on ρ-Minkowski space-time

We construct a gauge theory model on the 4-dimensional ρ-Minkowski space-time, a particular deformation of the Minkowski space-time recently considered. The corresponding star product results from a combination of Weyl quantization map and properties of the convolution algebra of the special Euclidean group. We use noncommutative differential calculi based on twisted derivations together with a twisted notion of noncommutative connection. The twisted derivations pertain to the Hopf algebra of ρ-deformed translations, a Hopf subalgebra of the ρ-deformed Poincaré algebra which can be viewed as defining the quantum symmetries of the ρ-Minkowski space-time. The gauge theory model is left invariant under the action of the ρ-deformed Poincaré algebra. The kinetic part of the action is found to coincide with the one of the usual (commutative) electrodynamics.

Quantization of locally compact groups associated with essentially bijective 1-cocycles

Given an extension [Formula: see text] of locally compact groups, with [Formula: see text] abelian, and a compatible essentially bijective [Formula: see text]-cocycle [Formula: see text], we define a dual unitary [Formula: see text]-cocycle on [Formula: see text] and show that the associated deformation of [Formula: see text] is a cocycle bicrossed product defined by a matched pair of subgroups of [Formula: see text]. We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang–Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of [Formula: see text] on [Formula: see text] and a unitary quantization map [Formula: see text] of Kohn–Nirenberg type.

An Energy Consumption Model for SRAM-Based In-Memory-Computing Architectures

In this paper, a mathematical model for obtaining energy consumption of IMC architectures is constructed. This model provides energy estimation based on the distribution of a specific dataset. In addition, the estimation reduces the required simulation time to create an energy consumption model of SRAM-based IMC architectures. To validate our model with realistic data, the energy consumption of IMC is compared by using NeuroSim V3.0 for the CIFAR-10 and MNIST-like datasets. Furthermore, an application is created with our model to select highest performing quantization mapping based upon the parameters of energy consumption and accuracy.

Lie groupoid deformations and convolution algebras

We define a morphism from the deformation complex of a Lie groupoid to the Hochschild complex of its convolution algebra, and show that it maps the class of a geometric deformation to the algebraic class of the induced deformation in Hochschild cohomology. Applied to the adiabatic groupoid, we show that the van Est map to deformation cohomology of Lie algebroids is induced by taking the classical limit of a quantization map on the dual of the Lie algebroid.

A Hybrid Weight Quantization Strategy for Memristive Neural Networks

Due to the ability to store data and process information, the memristor-based neuromorphic system has attracted extensive attention. Its efficient parallel computing approach allows it to implement neural networks in hardware. However, due to the limitation of the range of memristor conductance, it is difficult to represent high-precision weights in memristive neural network. During the off-chip learning, it is crucial to find an efficient weight quantization scheme and map it to the memristor array. Therefore, a hybrid weight quantization strategy that combines uniform and non-uniform quantization is proposed to overcome these problems. Specifically, the curve fitting of pulse modulation for tantalum oxide-based memristor is carried out, and the mapping rules of weights are proposed to simplify the process of reading verification. Furthermore, the hybrid quantization strategy is proposed and applied to a multilayer perceptron and a convolutional neural network, respectively. The effectiveness and robustness of the hybrid quantization scheme are verified in the MNIST dataset. Experiments show that the proposed hybrid quantization scheme can achieve 99.26% accuracy at 4 bits and tolerate 20% noise interference. The simulation results in this paper also provide an effective solution for the hardware implementation of memristive neural networks.

Science Mapping and Country Clustering Regarding Challenges of Public Governance to Ensure Societal Well-Being

This study maps the challenges public governance faces in its mission to ensure a high level of societal well-being in European countries due to complex and multidimensional analysis, both on scientific and economic levels: bibliometric analysis, vector quantisation mapping, and clustering analysis of countries. The proposed research advocates considering relevant descriptors of the two phenomena, namely the six dimensions of public governance and the composite quality of life index, to analyse their interdependence by 2020, the year for which official statistics reveal data. Our research proposes a classification of the European Union Member States from the perspective of the progress made at the governmental decision-making level for the multidimensional approach to quality of life to identify the models of good practice. The methodological support was offered by cluster analysis and a vector quantisation method, namely K-means. KNIME software has allowed us to connect to various data sources visually. Clustering of European countries has revealed several disparities thus, Denmark and Finland (which are countries with a high level of quality of life) are examples of good practices, while countries such as Romania and Bulgaria are facing difficulties in significantly improving their quality of life due to the deficiencies of the governing act. Finally, the research highlights the channels through which public governance can substantially contribute to societal well-being, keeping in mind the relatively low importance of welfare given to formal aspects of democratic representation compared to the extent of quality governance.

Learning Vector Quantization, Hebbian Learning, and Self-Organizing Map for Classification

Deep Learning has been rapidly developed. Almost all proposed methods already have very high accuracy. Most of these methods still use techniques from the past with some modifications to adapt to existing modules. Sometimes it is necessary to understand past methods to produce new methods. Therefore, this research examines past models that have the potential to improve the performance of existing deep learning models. The methods to be examined include Learning Vector Quantization (LVQ), Hebbian learning, and Self-Organizing Map (SOM). The iris dataset available on Scikit-learn (SKlearn) is used here for testing in cases of supervised learning and unsupervised learning (especially SOM). The results show that LVQ has a good accuracy of 93%, while Hebbian learning has an accuracy of 56%. SOM fluctuates between 88% and 93%. Although the accuracy of SOM does not exceed LVQ, this model does not require labels in its training process.

Automatic Image Annotation Using Adaptive Convolutional Deep Learning Model

Every day, websites and personal archives create more and more photos. The size of these archives is immeasurable. The comfort of use of these huge digital image gatherings donates to their admiration. However, not all of these folders deliver relevant indexing information. From the outcomes, it is difficult to discover data that the user can be absorbed in. Therefore, in order to determine the significance of the data, it is important to identify the contents in an informative manner. Image annotation can be one of the greatest problematic domains in multimedia research and computer vision. Hence, in this paper, Adaptive Convolutional Deep Learning Model (ACDLM) is developed for automatic image annotation. Initially, the databases are collected from the open-source system which consists of some labelled images (for training phase) and some unlabeled images {Corel 5 K, MSRC v2}. After that, the images are sent to the pre-processing step such as colour space quantization and texture color class map. The pre-processed images are sent to the segmentation approach for efficient labelling technique using J-image segmentation (JSEG). The final step is an automatic annotation using ACDLM which is a combination of Convolutional Neural Network (CNN) and Honey Badger Algorithm (HBA). Based on the proposed classifier, the unlabeled images are labelled. The proposed methodology is implemented in MATLAB and performance is evaluated by performance metrics such as accuracy, precision, recall and F1_Measure. With the assistance of the proposed methodology, the unlabeled images are labelled.

Star product approach for loop quantum cosmology

Guided by recent developments toward the implementation of the deformation quantization program within the loop quantum cosmology (LQC) formalism, in this paper we address the introduction of both the integral and differential representation of the star product for LQC. To this end, we consider the Weyl quantization map for cylindrical functions defined on the Bohr compactification of the reals. The integral representation contains all of the common properties that characterize a star product which, in the case under study here, stands for a deformation of the usual pointwise product of cylindrical functions. Our construction also admits a direct comparison with the integral representation of the Moyal product which may be reproduced from our formulation by judiciously substituting the appropriate characters that identify such representation. Further, we introduce a suitable star commutator that correctly reproduces both the quantum representation of the holonomy-flux algebra for LQC and, in the proper limit, the holonomy-flux classical Poisson algebra emerging in the cosmological setup. Finally, we propose a natural way to obtain the quantum dynamical evolution in LQC in terms of this star commutator for cylindrical functions as well as a differential representation of the star product using discrete finite differences. We expect that our findings may contribute to a better understanding of certain issues arising within the LQC program, in particular, those related to the semiclassical limit and the dynamical evolution of quantum states.

Quantization: History and problems

In this work, I explore the concept of quantization as a mapping from classical phase space functions to quantum operators. I discuss the early history of this notion of quantization with emphasis on the works of Schrödinger and Dirac, and how quantization fit into their overall understanding of quantum theory in the 1920's. Dirac, in particular, proposed a quantization map which should satisfy certain properties, including the property that quantum commutators should be related to classical Poisson brackets in a particular way. However, in 1946, Groenewold proved that Dirac's mapping was inconsistent, making the problem of defining a rigorous quantization map more elusive than originally expected. This result, known as the Groenewold-Van Hove theorem, is not often discussed in physics texts, but here I will give an account of the theorem and what it means for potential ``corrections” to Dirac's scheme. Other proposals for quantization have arisen over the years, the first major one being that of Weyl in 1927, which was later developed by many, including Groenewold, and which has since become known as Weyl Quantization in the mathematical literature. Another, known as Geometric Quantization, formulates quantization in differential-geometric terms by appealing to the character of classical phase spaces as symplectic manifolds; this approach began with the work of Souriau, Kostant, and Kirillov in the 1960's. I will describe these proposals for quantization and comment on their relation to Dirac's original program. Along the way, the problem of operator ordering and of quantizing in curvilinear coordinates will be described, since these are natural questions that immediately present themselves when thinking about quantization.

Introduction to coherent quantization

This paper studies coherent quantization, the way operators in the quantum space of a coherent space—defined in the recent book ’Coherent Quantum Mechanics’ by the first author—can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties. The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization of coherent maps. The importance of coherent maps for quantum mechanics is due to the fact that there is a quantization map that associates homomorphically with every coherent map a linear operator from the quantum space into itself. The quantization map generalizes the second quantization procedure for free classical fields to symmetry groups of general coherent spaces. Field quantization is obtained by specialization to Klauder spaces, whose quantum spaces are the bosonic Fock spaces. Implied by the new approach is a short, coordinate-free derivation of all basic properties of creation and annihilation operators in Fock spaces.

Strict deformation quantization of abelian lattice gauge fields

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 operator systems at the finite level, in the continuum limit we obtain genuine C*-algebras. We also prove that the C*-algebras (classical and quantum) are invariant under time evolutions related to the electric part of abelian Yang--Mills. Our classical and quantum systems at the finite level are essentially the ones of [van Nuland and Stienstra, 2020], which admit completely general dynamics, and we briefly discuss ways to extend this powerful result to the continuum limit. We also briefly discuss reduction, and how the current set-up should be generalized to the non-abelian case.

A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids

Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras are characterised by an action of the Grothendieck-Teichm\"uller group via one-colored directed and oriented graphs, respectively. In this note, we study the action of multi-oriented graph complexes on Lie bialgebroids and their "quasi" generalisations. Using results due to T. Willwacher and M. Zivkovi\'c on the cohomology of (multi)-oriented graphs, we show that the action of the Grothendieck-Teichm\"uller group on Lie bialgebras and quasi-Lie bialgebras can be generalised to quasi-Lie bialgebroids via graphs with two colors, one of them being oriented. However, this action generically fails to preserve the subspace of Lie bialgebroids. By resorting to graphs with two oriented colors, we instead show the existence of an obstruction to the quantization of a generic Lie bialgebroid in the guise of a new $\mathsf{Lie}_\infty$-algebra structure non-trivially deforming the "big bracket" for Lie bialgebroids. This exotic $\mathsf{Lie}_\infty$-structure can be interpreted as the equivalent in $d=3$ of the Kontsevich-Shoikhet obstruction to the quantization of infinite-dimensional Poisson manifolds (in $d=2$). We discuss the implications of these results with respect to a conjecture due to P. Xu regarding the existence of a quantization map for Lie bialgebroids.

Graph Signal Compression by Joint Quantization and Sampling

Graph signals arise in various applications, ranging from sensor networks to social media data. The high-dimensional nature of these signals implies that they often need to be compressed in order to be stored and transmitted. The common framework for graph signal compression is based on sampling, resulting in a set of continuous-amplitude samples, which in turn have to be quantized into a finite bit representation. In this work, we study the joint design of graph signal sampling along with quantization, for graph signal compression. We focus on bandlimited graph signals, and show that the compression problem can be represented as a task-based quantization setup, in which the task is to recover the spectrum of the signal. Based on this equivalence, we propose a joint design of the sampling and recovery mechanisms for a fixed quantization mapping, and present an iterative algorithm for dividing the available bit budget among the discretized samples. Furthermore, we show how the proposed approach can be realized using graph filters combining elements corresponding the neighbouring nodes of the graph, thus facilitating distributed implementation at reduced complexity. Our numerical evaluations on both synthetic and real world data shows that the joint sampling and quantization method yields a compact finite bit representation of high-dimensional graph signals, which allows reconstruction of the original signal with accuracy within a small gap of that achievable with infinite resolution quantizers.

Injective Tensor Products in Strict Deformation Quantization

The aim of this paper is twofold. 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.

The classical limit of Schrödinger operators in the framework of Berezin quantization and spontaneous symmetry breaking as an emergent phenomenon

The algebraic properties of a strict deformation quantization are analyzed on the classical phase space [Formula: see text]. The corresponding quantization maps enable us to take the limit for [Formula: see text] of a suitable sequence of algebraic vector states induced by [Formula: see text]-dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on [Formula: see text], defining a classical algebraic state. The observables are here represented in terms of a Berezin quantization map which associates classical observables (functions on the phase space) to quantum observables (elements of [Formula: see text] algebras) parametrized by [Formula: see text]. The existence of this classical limit is in particular proved for ground states of a wide class of Schrödinger operators, where the classical limiting state is obtained in terms of a Haar integral. The support of the classical state (a probability measure on the phase space) is included in certain orbits in [Formula: see text] depending on the symmetry of the potential. In addition, since this [Formula: see text]-algebraic approach allows for both quantum and classical theories, it is highly suitable to study the theoretical concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by switching off [Formula: see text]. To this end, a detailed mathematical description is outlined and it is shown how this algebraic approach sheds new light on spontaneous symmetry breaking in several physical models.

English

Let $${{\cal D}_{\lambda ,\mu }}$$ be the space of linear differential operators on weighted densities from $${{\cal F}_\lambda }$$ to $${{\cal F}_\mu }$$ as module over the orthosymplectic Lie superalgebra $$\mathfrak{osp}(3\left| 2 \right.)$$ , where $${{\cal F}_\lambda }$$ , $$\lambda \in \mathbb{C}$$ is the space of tensor densities of degree λ on the supercircle S1∣3. We prove the existence and uniqueness of projectively equivariant quantization map from the space of symbols to the space of differential operators. An explicite expression of this map is also given.

Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras

We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed D=4 quantum inhomogeneous conformal Hopf algebras Uθ(su(2,2)⋉T4) and Uθ¯(su(2,2)⋉T¯4), where T4 describes complex twistor coordinates and T¯4 the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently, we introduce the quantum deformations of D=4 Heisenberg-conformal algebra (HCA) su(2,2)⋉Hℏ4,4 (Hℏ4,4=T¯4⋉ℏT4 is the Heisenberg algebra of twistorial oscillators) providing in twistorial framework the basic covariant quantum elementary system. The class of algebras describing deformation of HCA with dimensionfull deformation parameter, linked with Planck length λp, is called the twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in space-time framework. We describe the examples of TDSR algebra linked with Palatial twistors which are introduced by the Drinfeld twist and the quantization map in Hℏ4,4. We also introduce generalized quantum twistorial phase space by considering the Heisenberg double of Hopf algebra Uθ(su(2,2)⋉T4).

Generalized Susskind–Glogower coherent states

Susskind-Glogower coherent states, whose Fock expansion coefficients include Bessel functions, have recently attracted considerable attention for their optical properties. Nevertheless, identity resolution is still an open question, which is an essential mathematical property that defines an overcomplete basis in the Fock space and allows a coherent state quantization map. In this regard, the modified Susskind-Glogower coherent states have been introduced as an alternative family of states that resolve the identity resolution. In the present manuscript, the quantization map related to the modified Susskind-Glogower coherent states is exploited, which naturally leads to a particular representation of the $\mathfrak{su}(1,1)$ Lie algebra in its discrete series. The latter provides evidence about further generalizations of coherent states, built from the Susskind-Glogower ones by extending the indexes of the Bessel functions of the first kind and, alternatively, by employing the modified Bessel functions of the second kind. In this form, the new families of Susskind-Glogower-I and Susskind-Glogower-II coherent states are introduced. The corresponding quantization maps are constructed so that they lead to general representations of elements of the $\mathfrak{su}(1,1)$ and $\mathfrak{su}(2)$ Lie algebras as generators of the SU$(1,1)$ and SU$(2)$ unitary irreducible representations respectively. For completeness, the optical properties related to the new families of coherent states are explored and compared with respect to some well-known optical states.

Learning hidden Markov models for linear Gaussian systems with applications to event-based state estimation

This work attempts to approximate a linear Gaussian system with a finite-state hidden Markov model (HMM), which is found useful in dealing with challenges in designing networked control systems An indirect approach is developed, where a state-space model (SSM) is firstly identified for a Gaussian system and the SSM is then used as an emulator for learning an HMM. In the proposed method, the training data for the HMM are obtained from the data generated by the SSM through building a quantization mapping. Parameter learning algorithms are designed to learn the parameters of the HMM, through exploiting the periodical structural characteristics of the HMM. The convergence and asymptotic properties of the proposed algorithms are analyzed. The HMM learned using the proposed algorithms is applied to event-triggered state estimation, and numerical results on model learning and state estimation demonstrate the validity of the proposed algorithms.