Noncommutative Model Research Articles

Dissipation in the 1/D expansion for planar matrix models

We consider the thermal behavior of a large number of matrix degrees of freedom in the planar limit. We work in 0+1 dimensions, with D matrices, and use 1/D as an expansion parameter. This can be thought of as a noncommutative large-D vector model, with two independent quartic couplings for the two different orderings of the matrices. We compute a thermal two-point correlator to O(1/D) and find that the degeneracy present at large D is lifted, with energy levels split by an amount ∼1/D. This implies a timescale for thermal dissipation ∼D. At high temperatures dissipation is predominantly due to one of the two quartic couplings. Published by the American Physical Society 2025

Hamiltonian mapping and quantum perturbation equations in the point matter black hole and noncommutative black hole models

Hamiltonian mapping and quantum perturbation equations in the point matter black hole and noncommutative black hole models

Quantum scalar field on fuzzy de Sitter space. Part I. Field modes and vacua

We study a scalar field on a noncommutative model of spacetime, the fuzzy de Sitter space, which is based on the algebra of the de Sitter group SO(1, d) and its unitary irreducible representations. We solve the Klein-Gordon equation in d = 2, 4 and show, using a specific choice of coordinates and operator ordering, that all commutative field modes can be promoted to solutions of the fuzzy Klein-Gordon equation. To explore completeness of this set of modes, we specify a Hilbert space representation and study the matrix elements (integral kernels) of a scalar field: in this way the complete set of solutions of the fuzzy Klein-Gordon equation is found. The space of noncommutative solutions has more degrees of freedom than the commutative one, whenever spacetime dimension is d > 2. In four dimensions, the new non-geometric, internal modes are parametrised by S2 × W, where W is a discrete matrix space. Our results pave the way to analysis of quantum field theory on the fuzzy de Sitter space.

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.

Generating fuzzy dark matter droplets

Generating fuzzy dark matter droplets

First Experimental Survey of a Whole Class of Non-Commutative Quantum Gravity Models in the VIP-2 Lead Underground Experiment

This study is aimed to set severe constraints on a whole class of non-commutative space-times scenarios as a class of universality for several quantum gravity models. To this end, slight violations of the Pauli exclusion principle—predicted by these models—are investigated by searching for Pauli forbidden Kα and Kβ transitions in lead. The selection of a high atomic number target material allows to test the energy scale of the space-time non-commutativity emergence at high atomic transition energies. As a consequence, the measurement is very sensitive to high orders in the power series expansion of the Pauli violation probability, which allows to set the first constraint to the “triply special relativity” model proposed by Kowalski-Glikman and Smolin. The characteristic energy scale of the model is bound to Λ>5.6·10−9 Planck scales.

Structure of the canonical uniton factorization of a solution of a noncommutative unitary sigma model

Structure of the canonical uniton factorization of a solution of a noncommutative unitary sigma model

Noncommutative Yang model and its generalizations

Long time ago, Yang [Phys. Rev. 72, 874 (1947)] proposed a model of noncommutative spacetime that generalized the Snyder model to a curved background. In this paper, we review his proposal and the generalizations that have been suggested during the years. In particular, we discuss the most general algebras that contain as subalgebras both de Sitter and Snyder algebras, preserving Lorentz invariance, and are generated by a two-parameter deformation of the canonical Heisenberg algebra. We also define their realizations on quantum phase space, giving explicit examples, both exact and in terms of a perturbative expansion in deformation parameters.

Dissipation driven degeneracies

Dissipation alone can produce counterintuitive topological wave transport that is otherwise absent in a non-dissipative system. This work demonstrates the influence of dissipation on degeneracies that arise in the context of elastic wave transport. The conditions on the parameters necessary to observe non-Hermitian degeneracies in the Bloch spectrum are precisely derived. It will be shown—contrary to the expectation from singularity theory of a linear eigenvalue problem—that a proportionally damped system with commutative damping does not exhibit non-Hermitian degeneracies. The necessity of a non-commutative and non-proportional dissipation model to observe non-Hermitian degeneracies (or exceptional points) is emphasized. Non-proportional dissipation is used to induce a non-Hermitian degeneracy in a local resonance sub-Bragg bandgap of a linear chain, without using negative damping. While Bloch waves are chosen to illustrate the influence of dissipation, the results readily extend to waves in non-periodic media as well as other wave and vibration transport problems.

Asymptotic behavior for a commutative and noncommutative in coordinates Mixmaster model

In this work, we find the asymptotic behavior for a commutative and noncommutative Mixmaster model in a Quantum Cosmology (QC) and semiclassical scenario, showing explicitly the calculus and applying the asymptotically equivalent functions that allow us to analyze the equivalence between Bianchi IX and Bianchi II models.

Strongest Atomic Physics Bounds on Noncommutative Quantum Gravity Models.

Investigations of possible violations of the Pauli exclusion principle represent critical tests of the microscopic space-time structure and properties. Space-time noncommutativity provides a class of universality for several quantum gravity models. In this context the VIP-2 lead experiment sets the strongest bounds, searching for the Pauli exclusion principle violating atomic transitions in lead, excluding the θ-Poincaré noncommutative quantum gravity models far above the Planck scale for nonvanishing θ_{μν} electriclike components, and up to 6.9×10^{-2} Planck scales if θ_{0i}=0.

The AdS𝜃2/CFT1 correspondence and noncommutative geometry II: Noncommutative quantum black holes

In this paper, we present the construction of noncommutative [Formula: see text] black hole and its four-dimensional Yang–Mills IKKT-type matrix model which includes two competing Myers term one responsible for the condensation of pure [Formula: see text] and the other one responsible for the condensation of the dilaton field. It is argued that the phase diagram of this matrix model features three phases: (1) a gravitational phase ([Formula: see text] black hole), (2) a geometric phase ([Formula: see text] background) and (3) a Yang–Mills phase. The Hawking process is therefore seen as an exotic line of discontinuous transitions between the gravitational and geometrical phases. Alternatively, a noncommutative nonlinear sigma model describing the transition of the dilaton field between the gravitational and geometrical phases is also constructed.

Diffeomorphisms in Momentum Space: Physical Implications of Different Choices of Momentum Coordinates in the Galilean Snyder Model

It has been pointed out that different choices of momenta can be associated to the same noncommutative spacetime model. The question of whether these momentum spaces, related by diffeomorphisms, produce the same physical predictions is still debated. In this work, we focus our attention on a few different momentum spaces that can be associated to the Galilean Snyder noncommutative spacetime model and show that they produce different predictions for the energy spectrum of the harmonic oscillator.

Spectral action and the electroweak \u03b8-terms for the Standard Model without fermion doubling

We compute the leading terms of the spectral action for a noncommutative geometry model that has no fermion doubling. The spectral triple describing it, which is chiral and allows for CP-symmetry breaking, has the Dirac operator that is not of the product type. Using Wick rotation we derive explicitly the Lagrangian of the model from the spectral action for a flat metric, demonstrating the appearance of the topological θ-terms for the electroweak gauge fields.

Associative realizations of κ -deformed extended Snyder model

Usually, the realizations of the noncommutative Snyder model lead to a nonassociative star product. However, it has been shown that this problem can be avoided by adding to the spacetime coordinates new tensorial degrees of freedom. The model so obtained, called extended Snyder model, can be subject to a $\kappa$-deformation, giving rise to a unification of the Snyder and the $\kappa$-Poincar\'e algebras in the formalism of extended spacetime. In this paper we review this construction and consider the generic realizations of the $\kappa$-deformed extended Snyder model, calculating the associated star product, coproduct and twist in a perturbative setting. We also introduce a representation of the Lorentz algebra in the extended space and speculate on possible interpretations of the tensorial degrees of freedom.

Fuzzy dark matter black holes and droplets

We consider the possibility of having Dark Matter (DM) black holes motivated by the Einasto density profile. This generalizes both the noncommutative mini black hole model and allows DM to enter as the matter constituent which makes up the black hole. We show that it is possible to construct a black hole solution for each value of the Einasto index and for different values of the mass parameter, provided that the we work with the energy–momentum tensor of an anisotropic fluid. In particular, we achieve that by first considering the equation of state (EOS) p_r=-rho . It turns out that the corresponding black hole solution exhibits a horizon structure similar to that of a Reissner–Nordström black hole and the central singularity is replaced by a regular de Sitter core. We also show that if the previous EOS is replaced by a nonlocal one, it is possible to construct a self-gravitating fuzzy DM droplet but also in this case, the radial pressure is negative. Finally, we contemplate scenarios of different dark matter black holes with moderate mass values which could have formed in galaxies. In particular, we probe the possibility whether such black holes could also be the central galactic objects.

Four‐Dimensional Gravity on a Covariant Noncommutative Space (II)

AbstractBased on the construction of the 4‐dim noncommutative gravity model described in our previous work, first, a more extended description of the covariant noncommutative space (fuzzy 4‐dim de Sitter space), which accommodates the gravity model, is presented and then the corresponding field equations, which are obtained after variation of the previously proposed action, are extracted. Also, a spontaneous breaking of the initial symmetry is performed, this time induced by the introduction of an auxiliary scalar field, and its implications in the reduced theory, which is produced after considering the commutative limit, are examined.

Complete noncommutativity in a cosmological model with radiation

In order to try explaining the present accelerated expansion of the universe, we consider the most complete noncommutativity, of a certain type, in a Friedmann–Robertson–Walker cosmological model, coupled to a perfect fluid. We use the ADM formalism in order to write the gravitational Hamiltonian of the model and the Schutz’s formalism in order to write the perfect fluid Hamiltonian. The noncommutativity is introduced by four nontrivial Poisson brackets between all geometrical as well as matter variables of the model. Each nontrivial Poisson bracket is associated with a noncommutative parameter. We recover the description in terms of commutative variables by introducing four variables transformations that depend on the noncommutative parameters. Using those variables transformations, we rewrite the total noncommutative Hamiltonian of the model in terms of commutative variables. From the resulting Hamiltonian, we obtain the scale factor dynamical equations for a generic perfect fluid. In order to solve these equations, we restrict our attention to a model where the perfect fluid is radiation. The solutions depend on six parameters: the four noncommutative parameters, a parameter associated with the fluid energy C, and the curvature parameter k. They also depend on the initial conditions of the model variables. We compare the noncommutative solutions to the corresponding commutative ones and determine how the former ones differ from the latter ones. The comparison shows that the noncommutative model is very useful for describing the accelerated expansion of the universe. We also obtain estimates for one of the noncommutative parameters.

Noncommutative Homological Mirror Functor

We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of their results in terms of mirror symmetry and symplectic geometry. As an interesting application, we construct an explicit global deformation quantization of an affine del Pezzo surface as a noncommutative mirror to an elliptic orbifold.

Anisotropic constant-roll inflation with noncommutative model and swampland conjectures

In this paper, we study a constant-roll inflationary model in the presence of a noncommutative parameter with a homogeneous scalar field minimally coupled to gravity. The specific noncommutative inflation conditions proposed new consequences. On the other hand, we use anisotropic conditions and find new anisotropic constant-roll solutions with respect to noncommutative parameter. Also, we will plot some figures with respect to the specific values of the corresponding parameter and the swampland criteria which is raised from the exact potential obtained from the constant-roll condition. Finally, different of figures lead us to analyze the corresponding results and also show the effect of above mentioned parameter on the inflationary model.