Noncommutative Geometry Research Articles

Tuning the separability in noncommutative space

With the help of the generalized Peres–Horodecki separability criterion (Simon’s condition) for a bipartite Gaussian state, we have studied the separability of the noncommutative (NC) space coordinate degrees of freedom. Non-symplectic nature of the transformation between the usual commutative space and NC space restricts the straightforward use of Simon’s condition in NCS. We have transformed the NCS system to an equivalent Hamiltonian in commutative space through the Bopp shift, which enables the utilization of the separability criterion. To make our study fairly general and to analyze the effect of parameters on the separability of bipartite state in NC-space, we have considered a bilinear Hamiltonian with time-dependent (TD) parameters, along with a TD external interaction, which is linear in field modes. The system is transformed (Sp(4,R)) into canonical form keeping the intrinsic symplectic structure intact. The solution of the TD-Schrödinger equation is obtained with the help of the Lewis–Riesenfeld invariant method (LRIM). Expectation values of the observables (thus the covariance matrix) are constructed from the states obtained from LRIM. It turns out that the existence of the NC parameters in the oscillator determines the separability of the states. In particular, for isotropic oscillators, the separability condition for the bipartite Gaussian states depends on specific values of NC parameters. Moreover, particular anisotropic parameter values for the oscillator may cease the separability. In other words, both the deformation parameters (θ, η) and parameter values of the oscillator (mass, frequency) are important characteristics for the separability of bipartite Gaussian states. Thus tuning the parameter values, one can destroy or recreate the separability of states. With the help of a toy model, we have demonstrated how the tuning of a TD-NC space parameter affects the separability.

On noncommutative distributional Khintchine type inequalities

AbstractThe purpose of this paper is to provide distributional estimates for the series of the form with being elements from noncommutative Lorentz spaces and being Rademacher functions. To this end, we introduce a novel class of operators that are closely related to the dual Cesáro operator and construct a new extrapolation theorem that is of independent interest.

Spectral metrics on quantum projective spaces

We show that the noncommutative differential geometry of quantum projective spaces is compatible with Rieffel's theory of compact quantum metric spaces. This amounts to a detailed investigation of the Connes metric coming from the unital spectral triple introduced by D'Andrea and Dąbrowski. In particular, we establish that the Connes metric metrizes the weak-⁎ topology on the state space of quantum projective space. This generalizes previous work by the second author and Aguilar regarding spectral metrics on the standard Podleś spheres.

A geometric representative for the fundamental class in KK-duality of Smale spaces

A fundamental ingredient in the noncommutative geometry program is the notion of KK-duality, often called K-theoretic Poincaré duality, that generalises Spanier-Whitehead duality. In this paper we construct a θ-summable Fredholm module that represents the fundamental class in KK-duality between the stable and unstable Ruelle algebras of a Smale space. To find such a representative, we construct dynamical partitions of unity on the Smale space with highly controlled Lipschitz constants. This requires a generalisation of Bowen's Markov partitions. Along with an aperiodic point-sampling technique we produce a noncommutative analogue of Whitney's embedding theorem, leading to the Fredholm module.

The optical features of noncommutative charged 4D-AdS-Einstein–Gauss–Bonnet black hole: shadow and deflection angle

In this paper, we have explored the optical characteristics, namely the shadow and the deflection angle, inherent to the solution of a 4D-AdS-Einstein–Gauss–Bonnet black hole. This solution, which finds its inspiration in noncommutative geometry, had previously been established in our previous work. The radius of the shadow was determined using the Hamilton-Jacobi method and the Carter separation. Our results revealed that the presence of noncommutativity in the background of spacetime impacts the variation of the shadow radius. More specifically, we have demonstrated that an increase in the parameter θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta $$\\end{document} induces a decrease in the radius of the shadow. In a similar way, analogous observations have been made by studying the variation of the electric charge Q. The noncommutative parameter θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta $$\\end{document} and the electric charge Q have been constrained regarding the EHT observation data of the M87* and Sgr A* black holes. Furthermore, the angle of deflection, which is the outcome of lensing by the black hole, has been derived following the Ishihara et al. approach for a receiver and source positioned at finite distances from the black hole in an asymptotically non-flat spacetime. The impact of the noncommutative parameter θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta $$\\end{document} and the charge Q of the black hole are hence analyzed, and our results depict that these parameters have a significant influence on the angle at which light is deflected by the gravitational field of the black hole.

Quantum statistical mechanics and the boundary of modular curves

The theory of limiting modular symbols provides a noncommutative geometric model of the boundary of modular curves that includes irrational points in addition to cusps. A noncommutative space associated to this boundary is constructed, as part of a family of noncommutative spaces associated to different continued fractions algorithms, endowed with the structure of a quantum statistical mechanical system. Two special cases of this family of quantum systems can be interpreted as a boundary of the system associated to the Shimura variety of GL2 and an analog for SL2. The structure of equilibrium states for this family of systems is discussed. In the geometric cases, the ground states evaluated on boundary arithmetic elements are given by pairings of cusp forms and limiting modular symbols.

Random finite noncommutative geometries and topological recursion

In this paper, we investigate a model for quantum gravity on finite noncommutative spaces using the theory of blobbed topological recursion. The model is based on a particular class of random finite real spectral triples {(\mathcal{A}, \mathcal{H}, D, \gamma, J)} , called random matrix geometries of type {(1,0)} , with a fixed fermion space {(\mathcal{A}, \mathcal{H}, \gamma, J)} and a distribution of the form {e^{- \mathcal{S} (D)}\, {\mathrm{d}} D} over the moduli space of Dirac operators. The action functional {\mathcal{S} (D)} is considered to be a sum of terms of the form {\prod_{i=1}^s \mathrm{Tr} ({D^{n_i}})} for arbitrary {s \geqslant 1} . The Schwinger–Dyson equations satisfied by the connected correlators {W_n} of the corresponding multi-trace formal 1-Hermitian matrix model are derived by a differential geometric approach. It is shown that the coefficients {W_{g,n}} of the large N expansion of {W_n} ’s enumerate discrete surfaces, called stuffed maps, whose building blocks are of particular topologies. The spectral curve {( {\Sigma, \omega_{0,1}, \omega_{0,2}} )} of the model is investigated in detail. In particular, we derive an explicit expression for the fundamental symmetric bidifferential {\omega_{0,2}} in terms of the formal parameters of the model.

Thin-shell gravastar in a noncommutative BTZ geometry

In this paper, we build a thin-shell gravastar model within a noncommutative BTZ geometry. For this, we consider a noncommutative BTZ metric in the inner region and a geometry associated with a BTZ solution in the outer region, joined by the generalized join technique. After investigating the inner spacetime, surface and outer spacetime, we observe that there is a surface energy density and surface pressure, such as to make gravastar stable. This effect persists even when the cosmological constant is zero. Besides, we found a bound for the noncommutativity parameter. In addition, we examine the thermodynamics of the noncommutative BTZ black hole in Schwarzschild-type form in three-dimensional spacetime. We also check the stability condition by calculating the specific heat capacity, as well as for the formation of black hole remnants.

Oxidation, reduction and semi-classical limit for quantum matrix geometries

Matrix configurations define noncommutative spaces endowed with extra structure including a generalized Laplace operator, and hence a metric structure. Made dynamical via matrix models, they describe rich physical systems including noncommutative gauge theory and emergent gravity. Refining the construction in [25], we construct a semi-classical limit through an immersed submanifold of complex projective space based on quasi-coherent states. We observe the phenomenon of oxidation, where the resulting semi-classical space acquires spurious extra dimensions. We propose to remove this artifact by passing to a leaf of a carefully chosen foliation, which allows to extract the geometrical content of the noncommutative spaces. This is demonstrated numerically via multiple examples.

Noncommutative Geometry of Random Surfaces

We associate a noncommutative curve to a periodic, bipartite, planar dimer model with polygonal boundary. It determines the inverse Kasteleyn matrix and hence all correlations. It may be seen as a quantization of the limit shape construction of Kenyon and the author. We also discuss various directions in which this correspondence may be generalized.

Non-commutative Wormhole geometries in presence of modified Chaplygin–Jacobi gas and Anton–Schmidt fluid

In this work, we have found Wormhole (WH) solutions in isotropic cosmology in the backdrop of general relativity while utilizing the “Modified Chaplygin–Jacobi Gas” and “Anton–Schmidt fluid” equations of state. As a starting point for our calculations, we have also considered two matter distributions, namely the Gaussian distribution and the Lorentzian distribution. All four energy conditions (i.e. strong, weak, null and dominant) are calculated, and their respective graphs are plotted as well. The shape function and redshift function associated with the wormhole structure are also derived and plotted graphically. After that, the embedding diagram of our wormhole models and proper radial distance are presented graphically. Finally, we have discussed the stability of the wormhole solutions obtained using the generalized Tolman–Oppenheimer–Volkov (TOV) equation and have examined the aspect of vanishing sound speed inside the wormhole throat to study the role of perturbations of the system.

Nonlinear partial differential equations on noncommutative Euclidean spaces

Nonlinear partial differential equations on noncommutative Euclidean spaces

Noncommutative double geometry

We construct noncommutative theories with the Moyal-Weyl product in the double field theory (DFT) framework. We deform the infinitesimal generalized diffeomorphisms and the Leibniz rule in a consistent way. The prescription requires a generalized star metric, which can be thought of as the fundamental double metric, in order to construct the action. Finally we use the generalized scalar field dynamics and the generalized scalar field-perfect fluid correspondence to construct the generalized energy-momentum tensor of a perfect fluid in the noncommutative double geometry. The present formalism paves the way to the study of string cosmologies scenarios including the Moyal-Weyl product in a T-duality invariant way. Published by the American Physical Society 2024

Noncommutative Riemannian geometry of Kronecker algebras

We study aspects of noncommutative Riemannian geometry of the path algebra arising from the Kronecker quiver with N arrows. To start with, the framework of derivation based differential calculi is recalled together with a discussion on metrics and bimodule connections compatible with the ⁎-structure of the algebra. As an illustration, these concepts are applied to the noncommutative torus where examples of torsion free and metric (Levi-Civita) connections are given.In the main part of the paper, noncommutative geometric aspects of (generalized) Kronecker algebras are considered. The structure of derivations and differential calculi is explored, and torsion free bimodule connections are studied together with their compatibility with hermitian forms, playing the role of metrics on the module of differential forms. Moreover, for several different choices of Lie algebras of derivations, non-trivial Levi-Civita connections are constructed.

Quantum Chromodynamics of the Nucleon in Terms of Complex Probabilistic Processes

Despite the obvious progress made by the Feynman, Ravndal, and Kislinger relativistic model in describing the internal motion of a system with confinement of quarks in a nucleon, it turned out to be insufficiently realistic for a number of reasons. In particular, the model does not take into account some cornerstone properties of QCD, namely, gluon exchange between quarks, the influence of the resulting quark sea on valence quarks, and the self-interaction of colored gluons. It is these phenomena that spontaneously break the chiral symmetry of the quark system and form the bulk of the nucleon. To eliminate the above shortcomings of the model, the problem of self-organization of a three-quark dynamical system immersed in a colored quark–antiquark sea is considered within the framework of complex probabilistic processes that satisfy the stochastic differential equation of the Langevin–Kline–Gordon–Fock type. Taking into account the hidden symmetry of the internal motion of a dynamical system, a mathematically closed nonperturbative approach was developed, which makes it possible to construct the mathematical expectation of the wave function and other parameters of the nucleon in the form of multiple integral representations. It is shown that additional subspaces arising in a representation characterized by a noncommutative geometry with topological features participate in the formation of an effective interaction between valence quarks against the background of harmonic interaction between them.

Black hole evaporation and its remnants with the generalized uncertainty principle including a linear term

In recent years, researchers have investigated the evaporation of Schwarzschild black holes using various forms of the generalized uncertainty principle (GUP), metric quantum correction, and non-commutative geometry, respectively. However, there are differences between the GUP correction and the other two methods in terms of describing the later stages of black hole evaporation. Furthermore, some studies argue that the GUP with a negative parameter cannot effectively correct black hole evaporation, while others contend that the positivity or negativity of the GUP parameters should not affect the correction results. Taking the above into consideration, we reconsider black hole evaporation with the generalized uncertainty principle including a linear term (LGUP), and examine the case of negative parameters. The results indicate that the evaporation behavior of both Schwarzschild black holes and Reissner–Nordström black holes, under LGUP correction, is consistent with the results of metric quantum correction and non-commutative geometry. Additionally, the negative parameter LGUP can also effectively correct for black hole evaporation.

Conformally symmetric wormhole solutions supported by non-commutative geometry in f(Q,T) gravity

This paper investigates wormhole solutions within the framework of extended symmetric teleparallel gravity, incorporating non-commutative geometry, and conformal symmetries. To achieve this, we examine the linear wormhole model with anisotropic fluid under Gaussian and Lorentzian distributions. The primary objective is to derive wormhole solutions while considering the influence of the shape function on model parameters under Gaussian and Lorentzian distributions. The resulting shape function satisfies all the necessary conditions for a traversable wormhole. Furthermore, we analyze the characteristics of the energy conditions and provide a detailed graphical discussion of the matter contents via energy conditions. Additionally, we explore the effect of anisotropy under Gaussian and Lorentzian distributions. Finally, we present our conclusions based on the obtained results.

The finite dual coalgebra as a quantization of the maximal spectrum

In pursuit of a noncommutative spectrum functor, we argue that the Heyneman-Sweedler finite dual coalgebra can be viewed as a quantization of the maximal spectrum of a commutative affine algebra, integrating prior perspectives of Takeuchi, Batchelor, Kontsevich-Soibelman, and Le Bruyn. We introduce fully residually finite-dimensional algebras A as those with enough finite-dimensional representations to let A∘ act as an appropriate depiction of the noncommutative maximal spectrum of A; importantly, this class includes affine noetherian PI algebras. In the case of prime affine algebras that are module-finite over their center, we describe how the Azumaya locus is represented in the finite dual. This is used to describe the finite dual of quantum planes at roots of unity as an endeavor to visualize the noncommutative space on which these algebras act as functions. Finally, we discuss how a similar analysis can be carried out for other maximal orders over surfaces.

Noncommutative Coding Theory and Algebraic Sets for Skew PBW Extensions

The classical commutative coding theory has been recently extended to noncommutative rings of polynomial type. There are many interesting works in coding theory over single Ore extensions. In this review article we present the most relevant algebraic tools and properties of single Ore extensions used in noncommutative coding theory. The last section represents the novelty of the paper. We will discuss the algebraic sets arising in noncommutative coding theory but for skew PBW extensions. These extensions conform a general class of noncommutative rings of polynomial type and cover several algebras arising in physics and noncommutative algebraic geometry, in particular, they cover the Ore extensions of endomorphism injective type and the polynomials rings over fields.

Exploring wormhole solutions in curvature–matter coupling gravity supported by noncommutative geometry and conformal symmetry

This article explores new physically viable wormhole solutions within the framework of f(R,ℒm) gravity theory, incorporating noncommutative backgrounds and conformal symmetries. The study investigates the impact of model parameters on the existence and properties of wormholes. The derived shape function is found to obey all the required criteria. Specific attention is given to traceless wormholes with Gaussian and Lorentzian distributions, investigating the behavior of the shape functions and energy conditions. In both cases, the presence of exotic fluid is confirmed.