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We study the Laplacian operator Δ ∂ ‾ associated to a Kähler structure (Ω (• , •) , κ) for the Heckenberger–Kolb differential calculus of the quantum quadrics O q (Q N) , which is to say, the irreducible quantum flag manifolds of types B n and D n. We show that the eigenvalues of Δ ∂ ‾ on zero forms tend to infinity and have finite multiplicity. [ABSTRACT FROM AUTHOR]

We introduce ringed spaces, referred to as para-manifolds, whose non-commutative nilpotent coordinates naturally describe parafermions at the classical level in a similar way as Grassmann variables describe usual fermions. Given a supermanifold X , we construct a family of para-manifolds X (p) for positive integers p , such that X (1) recovers the supermanifold itself. A differential analysis on para-manifolds is developed, which can be readily applied to model physical problems. Two classes of para-manifolds, respectively corresponding to X being affine superspaces and projective superspaces, are treated in detail as examples. Green's theory of parafermions is reformulated in terms of para-manifolds. [ABSTRACT FROM AUTHOR]

We consider operator systems associated to spectral truncations of tori. We show that their state spaces, when equipped with the Connes distance function, converge in the Gromov–Hausdorff sense to the space of all Borel probability measures on the torus equipped with the Monge–Kantorovich distance. A crucial role will be played by the relationship between Schur and Fourier multipliers. Along the way, we introduce the spectral Fejér kernel and show that it is a good kernel. This allows to make the estimates sufficient to prove the desired convergence of state spaces. We conclude with some structure analysis of the pertinent operator systems, including the C*-envelope and the propagation number, and with an observation about the dual operator system. [ABSTRACT FROM AUTHOR]

Based on the presentation of the Kauffman bracket skein algebra of the thickened torus given by the third author in previous work [4], Frohman and Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere. We present an algorithm to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves. We surmise that the algorithm has quasi-polynomial growth with respect to the number of crossings of a pair of curves. Further, we conjecture the existence of a positive basis for the algebra. [ABSTRACT FROM AUTHOR]

In this paper, within the framework of relativistic quantum mechanics and using the improved approximation scheme to the centrifugal term for any l states via Bopp's shift method and standard perturbation theory, we have obtained the modified energy eigenvalues of a newly proposed modified unequal vector and scalar Hellmann plus modified Kratzer potentials (DUVSHMK-Ps) for some diatomic N2, I2, CO, NO, O2 and HCl molecules. This study includes corrections of the first-order in noncommutativity parameters (Θ , σ). This potential is a superposition of the attractive Coulomb Yukawa potential plus the Kratzer potential and new central terms appear as a result of the effects of noncommutativity properties of space–space. The obtained energy eigenvalues appear as a function of noncommutativity parameters, the strength parameters (V 0 , S 0) and (V 1 , S 1) of the (scalar vector) Hellmann potential, the screening range parameter α , the dissociation energy of the vector, and scalar potential (D v , D s) , the equilibrium inter-nuclear distance r e in addition to the atomic quantum numbers (n , j , l , s , m). Furthermore, we obtained the corresponding modified energy of DUVSHMK-Ps in the symmetries of non-relativistic noncommutative quantum mechanics (NRNCQM). In both relativistic and non-relativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of RQM and NRQM. [ABSTRACT FROM AUTHOR]

A Gaussian approximation to the bosonic part of M-(atrix) theory with mass deformation is considered at large values of the dimension d. From the perspective of the gauge/gravity duality this action reproduces with great accuracy the stringy Hagedorn phase transition from a confinement (black string) phase to a deconfinement (black hole) phase whereas from the perspective of the matrix/geometry approach this action only captures a remnant of the geometric Yang–Mills-to-fuzzy-sphere phase where the fuzzy sphere solution is only manifested as a three-cut configuration termed the "baby fuzzy sphere" configuration. The Yang–Mills phase retains most of its characteristics with two exceptions: (i) the uniform distribution inside a solid ball suffers a crossover at very small values of the gauge coupling constant to a Wigner's semicircle law, and (ii) the uniform distribution at small values of the temperatures is nonexistent. [ABSTRACT FROM AUTHOR]

In this paper we construct a non‐commutative geometry over a configuration space of gauge connections and show that it gives rise to a candidate for an interacting, non‐perturbative quantum gauge theory coupled to a fermionic field on a curved background. The non‐commutative geometry is given by an infinite‐dimensional Bott‐Dirac type operator, whose square gives the Hamilton operator, and which interacts with an algebra generated by holonomy‐diffeomorphisms. The Bott‐Dirac operator and the associated Hilbert space relies on a metric on the configuration space of connections, which effectively works as a covariant ultra‐violet regulator. We show that the construction coincides with perturbative quantum field theory in a local limit. Questions concerning Lorentz invariance and the fermionic sector as well as the issue of existence are left open. In this paper a non‐commutative geometry over a configuration space of gauge connections is constructed and it is shown that it gives rise to a candidate for an interacting, non‐perturbative quantum gauge theory coupled to a fermionic field on a curved background. The noncommutative geometry is given by an infinite‐dimensional Bott‐Dirac type operator, whose square gives the Hamilton operator, and which interacts with an algebra generated by holonomy‐diffeomorphisms. The Bott‐Dirac operator and the associated Hilbert space relies on a metric on the configuration space of connections, which effectively works as a covariant ultra‐violet regulator. It is shown that the construction coincides with perturbative quantum field theory in a local limit. Questions concerning Lorentz invariance and the fermionic sector as well as the issue of existence are left open. [ABSTRACT FROM AUTHOR]

The indirect evidence for at least a dozen massive white dwarfs (WDs) violating the Chandrasekhar mass limit is considered to be one of the wonderful discoveries in astronomy for more than a decade. Researchers have already proposed a diverse amount of models to explain this astounding phenomenon. However, each of these models always carries some drawbacks. On the other hand, noncommutative geometry is one of the best replicas of quantum gravity, which is yet to be proved from observations. Madore introduced the idea of a fuzzy sphere to describe a formalism of noncommutative geometry. This paper shows that the idea of a squashed fuzzy sphere can self-consistently explain the super-Chandrasekhar limiting mass WDs. We further show that the length scale beyond which the noncommutativity is prominent is an emergent phenomenon, and there is no prerequisite for an ad hoc length scale. [ABSTRACT FROM AUTHOR]

We consider the orbits of test particles moving in the gravitational field of a noncommutative-inspired Einstein–Euler–Heisenberg black hole. Using the geometric metric, we determine the circular orbits followed by massless particles, comparing them with the circular photon orbits coming from the Plebanski pseudo-metric that takes into account the nonlinear nature of the Euler–Heisenberg electrodynamics. Using the impact parameter of the photon orbits, we define the shadow of the noncommutative-inspired black hole and discuss the constraints on the model by comparing its shadow with the prediction from General Relativity. [ABSTRACT FROM AUTHOR]

This study explores the Gaussian and the Lorentzian distributed spherically symmetric wormhole solutions in the f(τ,T) gravity. The basic idea of the Gaussian and Lorentzian noncommutative geometries emerges as the physically acceptable and substantial notion in quantum physics. This idea of the noncommutative geometries with both the Gaussian and Lorentzian distributions becomes more striking when wormhole geometries in the modified theories of gravity are discussed. Here we consider a linear model within f(τ,T) gravity to investigate traversable wormholes. In particular, we discuss the possible cases for the wormhole geometries using the Gaussian and the Lorentzian noncommutative distributions to obtain the exact shape function for them. By incorporating the particular values of the unknown parameters involved, we discuss different properties of the new wormhole geometries explored here. It is noted that the involved matter violates the weak energy condition for both the cases of the noncommutative geometries, whereas there is a possibility for a physically viable wormhole solution. By analyzing the equilibrium condition, it is found that the acquired solutions are stable. Furthermore, we provide the embedded diagrams for wormhole structures under Gaussian and Lorentzian noncommutative frameworks. Moreover, we present the critical analysis on an anisotropic pressure under the Gaussian and the Lorentzian distributions. This study explores the Gaussian and the Lorentzian distributed spherically symmetric wormhole solutions in the f(τ,T) gravity. The basic idea of the Gaussian and Lorentzian noncommutative geometries emerges as the physically acceptable and substantial notion in quantum physics. This idea of the noncommutative geometries with both the Gaussian and Lorentzian distributions becomes more striking when wormhole geometries in the modified theories of gravity are discussed. Here the authors consider a linear model within f(τ,T) gravity to investigate traversable wormholes. In particular, the possible cases for the wormhole geometries are considered using the Gaussian and the Lorentzian noncommutative distributions to obtain the exact shape function for them. By incorporating the particular values of the unknown parameters involved, there will be a discussiom of different properties of the new wormhole geometries explored here. It is noted that the involved matter violates the weak energy condition for both the cases of the noncommutative geometries, whereas there is a possibility for a physically viable wormhole solution. By analyzing the equilibrium condition, it is found that the acquired solutions are stable. Furthermore, the embedded diagrams for wormhole structures under Gaussian and Lorentzian noncommutative frameworks are provided. Moreover, the critical analysis on an anisotropic pressure under the Gaussian and the Lorentzian distributions will be presented. [ABSTRACT FROM AUTHOR]

In this work, we propose a model of noncommutative cosmology through the deformation of minisuperspace. We focus on an exponentially potential with a homogeneous scalar field minimally coupled to gravity in the spatially flat universe. To process, we use a particular case of noncommutativity by making a deformation of space coordinates only. Then, we compare results in both the commutative model and the noncommutative one. [ABSTRACT FROM AUTHOR]

We propose a general procedure to construct noncommutative deformations of an embedded submanifold M of R n determined by a set of smooth equations f a (x) = 0 . We use the framework of Drinfel’d twist deformation of differential geometry of Aschieri et al. (Class Quantum Gravity 23:1883, 2006); the commutative pointwise product is replaced by a (generally noncommutative) ⋆ -product determined by a Drinfel’d twist. The twists we employ are based on the Lie algebra Ξ t of vector fields that are tangent to all the submanifolds that are level sets of the f a (tangent infinitesimal diffeomorphisms); the twisted Cartan calculus is automatically equivariant under twisted Ξ t . We can consistently project a connection from the twisted R n to the twisted M if the twist is based on a suitable Lie subalgebra e ⊂ Ξ t . If we endow R n with a metric, then twisting and projecting to the normal and tangent vector fields commute, and we can project the Levi–Civita connection consistently to the twisted M, provided the twist is based on the Lie subalgebra k ⊂ e of the Killing vector fields of the metric; a twisted Gauss theorem follows, in particular. Twisted algebraic manifolds can be characterized in terms of generators and ⋆ -polynomial relations. We present in some detail twisted cylinders embedded in twisted Euclidean R 3 and twisted hyperboloids embedded in twisted Minkowski R 3 [these are twisted (anti-)de Sitter spaces d S 2 , A d S 2 ]. [ABSTRACT FROM AUTHOR]

Superheavy right-handed Majorana neutrinos are proposed as a promising candidate for dark matter, with dynamical axial torsion as the mediating agent. [ABSTRACT FROM AUTHOR]

Chandrasekhar made the startling discovery about nine decades back that the mass of compact object white dwarf has a limiting value once nuclear fusion reactions stop therein. This is the Chandrasekhar mass-limit, which is ∼ 1. 4 M ⊙ for a nonrotating non-magnetized white dwarf. On approaching this limiting mass, a white dwarf is believed to spark off with an explosion called type Ia supernova, which is considered to be a standard candle. However, observations of several over-luminous, peculiar type Ia supernovae indicate the Chandrasekhar mass-limit to be significantly larger. By considering noncommutativity among the components of position and momentum variables, hence uncertainty in their measurements, at the quantum scales, we show that the mass of white dwarfs could be significantly super-Chandrasekhar and thereby arrive at a new mass-limit ∼ 2. 6 M ⊙ , explaining a possible origin of over-luminous peculiar type Ia supernovae. The idea of noncommutativity, apart from the Heisenberg's uncertainty principle, is there for quite sometime, without any observational proof however. Our finding offers a plausible astrophysical evidence of noncommutativity, arguing for a possible second standard candle, which has many far-reaching implications. [ABSTRACT FROM AUTHOR]

The Snyder–de Sitter model is an extension of the Snyder model to a de Sitter background. It is called triply special relativity (TSR) because it is based on three fundamental parameters: speed of light, Planck mass and cosmological constant. In this paper, we study the three-dimensional DKP oscillator for spin-0 and spin-1 in the framework of Snyder–de Sitter algebra in momentum space. By using the technique of vector spherical harmonics the energy spectrum and the corresponding eigenfunctions are obtained for the both cases. [ABSTRACT FROM AUTHOR]

We study the quantum geometry of the fuzzy sphere defined as the angular momentum algebra [ x i , x j ] = 2 ı λ p ϵ ijk x k modulo setting ∑ i x i 2 to a constant, using a recently introduced 3D rotationally invariant differential structure. Metrics are given by symmetric 3 × 3 matrices g and we show that for each metric there is a unique quantum Levi-Civita connection with constant coefficients, with scalar curvature 1 2 (Tr (g 2) - 1 2 Tr (g) 2) / det (g) . As an application, we construct Euclidean quantum gravity on the fuzzy unit sphere. We also calculate the charge 1 monopole for the 3D differential structure. [ABSTRACT FROM AUTHOR]

In this work, we address the process of pair creation of scalar particles in (1 + 1) de Sitter space–time in presence of a constant electromagnetic field by applying the noncommutativity on the scalar field up to first-order in 𝜃. We calculate the density of particles created in the vacuum by the mean of the Bogoliubov transformations. In contrast to a previous result, we show that noncommutativity contributes to the pair creation process. We find that the noncommutativity plays the same role of chemical potential and gives an important interest for studies at high energies. [ABSTRACT FROM AUTHOR]

When minimal length uncertainty emerging from a generalized uncertainty principle (GUP) is thoughtfully implemented, it is of great interest to consider its impacts on gravitational Einstein field equations (gEFEs) and to try to assess consequential modifications in metric manifesting properties of quantum geometry due to quantum gravity. GUP takes into account the gravitational impacts on the noncommutation relations of length (distance) and momentum operators or time and energy operators and so on. On the other hand, gEFE relates classical geometry or general relativity gravity to the energy–momentum tensors, that is, proposing quantum equations of state. Despite the technical difficulties, we intend to insert GUP into the metric tensor so that the line element and the geodesic equation in flat and curved space are accordingly modified. The latter apparently encompasses acceleration, jerk, and snap (jounce) of a particle in the quasi‐quantized gravitational field. Finite higher orders of acceleration apparently manifest phenomena such as accelerating expansion and transitions between different radii of curvature and so on. [ABSTRACT FROM AUTHOR]

Based on quantum mechanical framework for the minimal length uncertainty, we demonstrate that the generalized uncertainty principle (GUP) parameter could be best constrained by recent gravitational waves observations on one hand. On the other hand, this suggests modified dispersion relations (MDRs) enabling an estimation for the difference between the group velocity of gravitons and that of photons. Utilizing features of the UV/IR correspondence and the obvious similarities between GUP (including non‐gravitating and gravitating impacts on Heisenberg uncertainty principle) and the discrepancy between the theoretical and the observed cosmological constant (apparently manifesting gravitational influences on the vacuum energy density), we suggest a possible solution for the cosmological constant problem. [ABSTRACT FROM AUTHOR]

We construct covariant q-deformed holomorphic structures for all finitely generated relative Hopf modules over the irreducible quantum flag manifolds endowed with their Heckenberger–Kolb calculi. In the classical limit, these reduce to modules of sections of holomorphic homogeneous vector bundles over irreducible flag manifolds. For the case of simple relative Hopf modules, we show that this covariant holomorphic structure is unique. This generalises earlier work of Majid, Khalkhali, Landi, and van Suijlekom for line modules of the Podleś sphere, and subsequent work of Khalkhali and Moatadelro for general quantum projective space. [ABSTRACT FROM AUTHOR]

The injective spectrum is a topological space associated to a ring R , which agrees with the Zariski spectrum when R is commutative noetherian. We consider injective spectra of right noetherian rings (and locally noetherian Grothendieck categories) and establish some basic topological results and a functoriality result, as well as links between the topology and the Krull dimension of the ring (in the sense of Gabriel and Rentschler). Finally, we use these results to compute a number of examples. [ABSTRACT FROM AUTHOR]

We construct canonical strongly bicovariant differential graded algebra structures on all four flavours of cross product Hopf algebras, namely double cross products A ↪ A ⋈ H ↩ H , double cross coproducts Image 1 , biproducts Image 2 and bicrossproducts Image 3 on the assumption that the factors have strongly bicovariant calculi Ω (A) , Ω (H) (or a braided version Ω (B)). We use super versions of each of the constructions. Moreover, the latter three quantum groups all coact canonically on one of their factors and we show that this coaction is differentiable. In the case of the Drinfeld double D (A , H) = A op ⋈ H (where A is dually paired to H), we show that its canonical actions on A , H are differentiable. Examples include a canonical Ω (C q [ G L 2 ⋉ C 2 ]) for the quantum group of affine transformations of the quantum plane and Ω (C λ [ Poin c 1 , 1 ]) for the bicrossproduct Poincaré quantum group in 2 dimensions. We also show that Ω (C q [ G L 2 ]) itself is uniquely determined by differentiability of the canonical coaction on the quantum plane and of the determinant subalgebra. [ABSTRACT FROM AUTHOR]

We review the main recent progresses in noncommutative space–time phenomenology in underground experiments. A popular model of noncommutative space–time is 𝜃 -Poincaré model, based on the Groenewold–Moyal plane algebra. This model predicts a violation of the spin-statistic theorem, in turn implying an energy and angular dependent violation of the Pauli exclusion principle. Pauli exclusion principle violating transitions in nuclear and atomic systems can be tested with very high accuracy in underground laboratory experiments such as DAMA/LIBRA and VIP(2). In this paper we derive that the 𝜃 -Poincaré model can be already ruled-out until the Planck scale, from nuclear transitions tests by DAMA/LIBRA experiment. [ABSTRACT FROM AUTHOR]

We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric. [ABSTRACT FROM AUTHOR]

We have recently proposed a new matrix dynamics at the Planck scale, building on the theory of trace dynamics. This is a Lagrangian dynamics in which the matrix degrees of freedom are made from Grassmann numbers, and the Lagrangian is trace of a matrix polynomial. Matrices made from even grade elements of the Grassmann algebra are called bosonic, and those made from odd grade elements are called fermionic. In the present article, we provide a basic definition of spin angular momentum in this matrix dynamics, and introduce a bosonic(fermionic) configuration variable conjugate to the spin of a boson(fermion). We then show that at energies below Planck scale, where the matrix dynamics reduces to quantum theory, fermions have half-integer spin (in multiples of Planck's constant), and bosons have integral spin. We also show that this definition of spin agrees with the conventional understanding of spin in relativistic quantum mechanics. Consequently, we obtain an elementary proof for the spin-statistics connection. [ABSTRACT FROM AUTHOR]

We discuss the corrections to the orbital period of a particle in a constant magnetic field, driven by the model of noncommutative geometry recently associated to a quantum clock. The effects are extremely small, but in principle detectable. [ABSTRACT FROM AUTHOR]

We review recent results obtained by using the spectral action applied to some simple physical models built in the noncommutative geometry framework. The first result, based on the generalization of the spectral action principle to the case of the fermionic action leads to the lowest order corrections that can possibly explain the huge difference between the electron and neutrino masses. In the second case, the extension of the family of Dirac operators for the product of finite geometry with the Riemannian manifold but without a product metric leads to a theory with two metrics that is similar to the models of bimetric gravity. [ABSTRACT FROM AUTHOR]

Using the methods of ordinary quantum mechanics, we study κ -Minkowski space as a quantum space described by noncommuting self-adjoint operators, following and enlarging T. Poulain and J.-C. Wallet, " κ -Poincaré invariant orientable field theories with at 1-loop: Scale-invariant couplings, preprint (2018), arXiv:1808.00350 [hep-th]. We see how the role of Fourier transforms is played in this case by Mellin transforms. We briefly discuss the role of transformations and observers. [ABSTRACT FROM AUTHOR]

This is a review of recent results regarding the application of Connes' noncommutative geometry to the Standard Model, and beyond. By twisting (in the sense of Connes-Moscovici) the spectral triple of the Standard Model, one does not only get an extra scalar field which stabilises the electroweak vacuum, but also an unexpected 1 -form field. By computing the fermionic action, we show how this field induces a transition from the Euclidean to the Lorentzian signature. Hints on a twisted version of the spectral action are also briefly mentioned. [ABSTRACT FROM AUTHOR]

We analyze a modified Dirac equation based on a noncommutative structure in phase space originating from a generalized uncertainty principle with a minimum length. The noncommutative structure induces generalized momenta and contributions to the energy levels of the standard Dirac equation. Applying techniques of perturbation theory, we find the lowest-order corrections to the energy levels and eigenfunctions of the Dirac equation in three dimensions for a spherically symmetric linear potential and for a square-well times triangular potential along one spatial dimension. We find that the corrections due to the noncommutative contributions may be of the same order as the relativistic ones, leading to an upper bound on the parameter fixing the minimum length induced by the generalized uncertainty principle. [ABSTRACT FROM AUTHOR]

We study covariant derivatives on a class of centered bimodules ℰ over an algebra 𝒜. We begin by identifying a 𝒵 (𝒜) -submodule 𝒳 (𝒜) which can be viewed as the analogue of vector fields in this context; 𝒳 (𝒜) is proven to be a Lie algebra. Connections on ℰ are in one-to-one correspondence with covariant derivatives on 𝒳 (𝒜). We recover the classical formulas of torsion and metric compatibility of a connection in the covariant derivative form. As a result, a Koszul formula for the Levi-Civita connection is also derived. [ABSTRACT FROM AUTHOR]

We have obtained the approximate analytical solutions of the nonrelativistic Hydrogen-like atoms such as ( He + , Li 2 + and Be +) and neutral atoms such as ( 2 2 Na , 12 C and 158 Au) atoms with a newly proposed generalized perturbed Yukawa potential with centrifugal barrier (GPYPCB) model using the generalized Bopp's shift method and standard perturbation theory in the symmetries of noncommutative three-dimensional real space phase (NC: 3D-RSP). By approximating the centrifugal term through the Greene–Aldrich approximation scheme, we have obtained the energy eigenvalues and generalized Hamiltonian operator for all orbital quantum numbers ℓ in the symmetries of NC: 3D-RSP. The potential is a superposition of the perturbed Yukawa potential and new terms proportional with (exp (− 2 a r) r 4 , exp (− 2 a r) r 3 , exp (− a r) r 3  and  exp (− a r) r 2 ) appear as a result of the effects of noncommutativity properties of space and phase on the perturbed Yukawa potential model. The obtained energy eigenvalues appear as functions of the generalized Gamma function, the discreet atomic quantum numbers (j , n , l , s  and  m) , two infinitesimal parameters (Θ , 𝜃 ¯) , which are induced by (position–position and phase–phase). In addition, the dimensional parameters (a , V 0 , V 1 , α) of perturbed Yukawa potential with centrifugal barrier model in NC: 3D-RSP. Furthermore, we have shown that the corresponding Hamiltonian operator in (NC: 3D-RSP) symmetries is the sum of the Hamiltonian operator of perturbed Yukawa potential model and the two operators are modified spin–orbit interaction and the modified Zeeman operator for the previous Hydrogenic and neutral atoms. [ABSTRACT FROM AUTHOR]

Noncommutative geometry, an offshoot of string theory, replaces point-like objects by smeared objects. The resulting uncertainty may cause a black hole to be observationally indistinguishable from a traversable wormhole, while the latter, in turn, may become observationally indistinguishable from a gravastar. The same noncommutative-geometry background allows the theoretical construction of thin-shell wormholes from gravastars and may even serve as a model for dark energy. [ABSTRACT FROM AUTHOR]

We consider a two-parameter family of Drinfeld twists generated from a simple Jordanian twist further twisted by 1-cochains. Twists from this family interpolate between two simple Jordanian twists. Relations between them are constructed and discussed. It is proved that there exists a one-parameter family of twists identical to a simple Jordanian twist. The twisted coalgebra, star product and coordinate realizations of the κ -Minkowski noncommutative space–time are presented. Real forms of Jordanian deformations are also discussed. The method of similarity transformations is applied to the Poincaré–Weyl Hopf algebra and two types of one-parameter families of dispersion relations are constructed. Mathematically equivalent deformations, that are related to nonlinear changes of symmetry generators and linked with similarity maps, may lead to differences in the description of physical phenomena. [ABSTRACT FROM AUTHOR]

We generalize the notions of shifted double Poisson and shifted double Lie-Rinehart structures, defined by Van den Bergh in [31, 32], to monoids in a symmetric monoidal abelian category. The main result is that an n-shifted double Lie-Rinehart structure on a pair (A;M) is equivalent to a non-shifted double Lie-Rinehart structure on the pair (A;M[-n]). [ABSTRACT FROM AUTHOR]

Using the gauged pseudo-Hermitian fuzzy Ginsparg–Wilson algebra, pseudo fuzzy Dirac and chirality operators on the fuzzy E A d S 2 have been studied. Also, the spectrum of the gauged pseudo fuzzy Dirac operator in the instanton sector has been studied. [ABSTRACT FROM AUTHOR]

We consider the 𝒫 𝒯 -symmetric quantum field theory on the noncommutative spacetime with angular twist and construct its pseudo-Hermitian interpretation. We explore the differences between internal and spatial parities in the context of the angular twist and for the latter we find new 𝒫 𝒯 -symmetric interactions that are nontrivial only for the noncommutative spacetime. We reproduce the same formula for the leading order T-matrix of the equivalent Hermitian model as the one obtained earlier for the quantum field theory on the commutative spacetime. This formula implies that the leading order scattering amplitude preserves the symmetries of the noncommutative geometry if they are not broken in the non-Hermitian formulation. [ABSTRACT FROM AUTHOR]

We examine the index data associated to twisted spectral triples and higher order spectral triples. In particular, we show that a Lipschitz regular twisted spectral triple can always be "logarithmically dampened" through functional calculus, to obtain an ordinary (i.e. untwisted) spectral triple. The same procedure turns higher order spectral triples into spectral triples. We provide examples of highly regular twisted spectral triples with nontrivial index data for which Moscovici's ansatz for a twisted local index formula is identically zero. [ABSTRACT FROM AUTHOR]

The paper builds the original foundations of a new operator theoretic setting for the study of quantum dynamics of non-perturbative aspects originated from Green's functions in Quantum Field Theory with strong couplings. [ABSTRACT FROM AUTHOR]

In the first article of this series, we have introduced the concept of an atom of space-time-matter (STM), which is described by the spectral action of noncommutative geometry, corresponding to a classical theory of gravity. In the present work, we use the Connes time parameter, along with the spectral action, to incorporate gravity into trace dynamics. We then derive the spectral equation of motion for the STM atom, which turns out to be the Dirac equation on a noncommutative space. [ABSTRACT FROM AUTHOR]

There ought to exist a reformulation of quantum theory which does not depend on classical time. To achieve such a reformulation, we introduce the concept of an atom of space-time-matter (STM). An STM atom is a classical noncommutative geometry (NCG), based on an asymmetric metric, and sourced by a closed string. Different such atoms interact via entanglement. The statistical thermodynamics of a large number of such atoms gives rise, at equilibrium, to a theory of quantum gravity. Far from equilibrium, where statistical fluctuations are large, the emergent theory reduces to classical general relativity. In this theory, classical black holes are far from equilibrium low entropy states, and their Hawking evaporation represents an attempt to return to the [maximum entropy] equilibrium quantum gravitational state. [ABSTRACT FROM AUTHOR]

We show that a class of braided Hopf algebras, which includes the braided S U q (2) of [P. Kasprzak, R. Meyer, S. Roy and S. L. Woronowicz, Braided quantum S U (2) groups, J. Noncommut. Geom. 10 (2016) 1161–1125], is obtained by twisting. We show further examples and demonstrate that twisting of bicovariant differential calculi gives braided bicovariant differential calculi. [ABSTRACT FROM AUTHOR]

In this paper, we provide five canonical and three Lie-algebraic twist-deformations of Carroll, para-Galilei and Static Hopf algebras. Particularly, in the case of Carroll Hopf structure, we get the same result with the use of contraction schemes of corresponding canonically and Lie-algebraically twisted Poincaré Hopf algebras as well. [ABSTRACT FROM AUTHOR]

In this contribution, we study the effects of space noncommutativity on the Landau system for a neutral particle with induced electric dipole moment subject to a linear confining potential. We analytically solve the Schrödinger equation and obtain the complete set of energy levels and the corresponding radial wave functions for the system in terms of the noncommutativity parameter 𝜃. [ABSTRACT FROM AUTHOR]

We review the approach to the Standard Model of particle interactions based on spectral noncommutative geometry. The paper is (nearly) self-contained and presents both the mathematical and phenomenological aspects. In particular, the bosonic spectral action and the fermionic action are discussed in detail, and how they lead to phenomenology. We also discuss the Euclidean versus Lorentz issues and how to go beyond the Standard Model in this framework. [ABSTRACT FROM AUTHOR]

This paper is the second part of a two-paper series whose aim is to give a detailed description of Connes' pseudodifferential calculus on noncommutative n -tori, n ≥ 2. We make use of the tools introduced in the 1st part to deal with the main properties of pseudodifferential operators on noncommutative tori of any dimension n ≥ 2. This includes the main results mentioned in [2, 5, 11]. We also obtain further results regarding action on Sobolev spaces, spectral theory of elliptic operators, and Schatten-class properties of pseudodifferential operators of negative order, including a trace-formula for pseudodifferential operators of order < − n. [ABSTRACT FROM AUTHOR]

This paper is the first part of a two-paper series whose aim is to give a thorough account on Connes' pseudodifferential calculus on noncommutative tori. This pseudodifferential calculus has been used in numerous recent papers, but a detailed description is still missing. In this paper, we focus on constructing an oscillating integral for noncommutative tori and laying down the main functional analysis ground for understanding Connes' pseudodifferential calculus. In particular, this allows us to give a precise explanation of the definition of pseudodifferential operators on noncommutative tori. More generally, this paper introduces the main technical tools that are used in the second part of the series to derive the main properties of these operators. [ABSTRACT FROM AUTHOR]

Fractal geometry is the study of sets which exhibit the same pattern at multiple scales. Developing tools to study these sets is of great interest. One step towards developing some of these tools is recognizing the duality between topological spaces and commutative C -algebras. When one lifts the commutativity axiom, one gets what are called noncommutative spaces and the study of noncommutative geometry. The tools built to study noncommutative spaces can in fact be used to study fractal sets. In what follows we will use the spectral triples of noncommutative geometry to describe various notions from fractal geometry. We focus on the fractal sets known as the harmonic Sierpiński gasket and the stretched Sierpiński gasket, and show that the spectral triples constructed in [7] and [23] can recover the standard self-affine measure in the case of the harmonic Sierpiński gasket and the Hausdorff dimension, geodesic metric, and Hausdorff measure in the case of the stretched Sierpiński gasket. [ABSTRACT FROM AUTHOR]