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Placing a Dirac--Schrödinger operator along the orbit of a flow on a compact manifoldM defines an R-equivariant spectral triple over the algebra of smooth functions on M. We study some of the properties of these triples, with special attention to their zeta functions. These zeta functions are defined for Re(s) > 1 by Trace(fpH-8), where fp is the uniformly continuous function on the real line obtained by restricting the continuous or smooth function f on M to the orbit of a point p ∈ M, and H = -∂²/∂x² + x² is the harmonic oscillator. The meromorphic continuation property and pole structure of these zeta functions are related to ergodic time averages in dynamics. In the case of the periodic flow on the circle, one obtains a spectral triple over the smooth irration torus A∞h ⊂ Ah already studied by Lesch and Moscovici. We strengthen a result of these authors, showing that the zeta function Trace(aH-8) extends meromorphically to C for any element a of the C*-algebra Ah. Another variant of our construction yields a spectral cycle for Ah ⊗ A1/h and a spectral triple over a suitable subalgebra with the meromorphic continuation property if h satisfies a Diophantine condition. The class of this cycle defines a fundamental class in the sense that it determines a KK-duality between Ah and A1/h. We employ the local index theorem of Connes and Moscovici in order to elaborate an index theorem of Connes for certain classes of differential operators on the line and compute the intersection form on K-theory induced by the fundamental class. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

It has been claimed that wormholes are just as good a prediction of Einstein’s theory as black holes, but they are subject to severe restrictions from quantum field theory. The purpose of this paper is to show that the claim can be substantiated in spite of these restrictions.

We investigate a quantum mechanical harmonic oscillator based on the extended Snyder model. This realization of the Snyder model is constructed as a quantum phase space generated by D spatial coordinates and D (D − 1) / 2 tensorial degrees of freedom, together with their conjugated momenta. The coordinates obey nontrivial commutation relations and generate a noncommutative geometry, which admits nicer properties than the usual realization of the model, in particular giving rise to an associative star product. The spectrum of the harmonic oscillator is studied through the introduction of creation and annihilation operators. Some physical consequences of the introduction of the additional degrees of freedom are discussed. [ABSTRACT FROM AUTHOR]

This paper deals with some questions regarding the notion of integral in the framework of Connes' noncommutative geometry. First, we present a purely spectral theoretic construction of Connes' integral. This answers a question of Alain Connes. We also deal with the compatibility of Dixmier traces with Lebesgue's integral. This answers another question of Alain Connes. We further clarify the relationship of Connes' integration with Weyl's laws for compact operators and Birman-Solomyak's perturbation theory. We also give a "soft proof" of Birman-Solomyak's Weyl's law for negative order pseudodifferential operators on closed manifold. This Weyl's law yields a stronger form of Connes' trace theorem. Finally, we explain the relationship between Connes' integral and semiclassicalWeyl's law for Schrödinger operators, including for (fractional) Schrödinger operators on Euclidean spaces and on noncommutative manifolds. We thus get a neat links between noncommutative geometry and semiclassical analysis. [ABSTRACT FROM AUTHOR]

When gravity is quantum, the point structure of space-time should be replaced by a non-commutative geometry. This is true even for quantum gravity in the infra-red. Using the octonions as space-time coordinates, we construct pre-spacetime, pre-quantum Lagrangian dynamics. We show that the symmetries of this non-commutative space unify the standard model of particle physics with SU(2)R chiral gravity. The algebra of the octonionic space yields spinor states which can be identified with three generations of quarks and leptons. The geometry of the space implies quantisation of electric charge, and leads to a theoretical derivation of the mysterious mass ratios of quarks and the charged leptons. Quantum gravity is quantisation not only of the gravitational field, but also of the point structure of space-time.

The Arens-Michael functor in noncommutative geometry is an analogue of the analytification functor in algebraic geometry: out of the ring of "algebraic functions" on a noncommutative affine scheme, it constructs the ring of "holomorphic functions" on it when viewed as a noncommutative complex analytic space. In this paper, we explicitly compute the Arens-Michael envelopes of the Jordan plane and the quantum enveloping algebra Uq(sl)2)) of sl(2) for |q| = 1. [ABSTRACT FROM AUTHOR]

We take a fresh look at the geometrization of logic using the recently developed tools of "quantum Riemannian geometry" applied in the digital case over the field F2 = {0, 1}, extending de Morgan duality to this context of differential forms and connections. The 1-forms correspond to graphs and the exterior derivative of a subset amounts to the arrows that cross between the set and its complement. The line graph 0 - 1 - 2 has a non-flat but Ricci flat quantum Riemannian geometry. The previously known four quantum geometries on the triangle graph, of which one is curved, are revisited in terms of left-invariant differentials, as are the quantum geometries on the dual Hopf algebra, the group algebra of Z3. For the square, we find a moduli of four quantum Riemannian geometries, all flat, while for an n-gon with n > 4 we find a unique one, again flat. We also propose an extension of de Morgan duality to general algebras and differentials over F2. [ABSTRACT FROM AUTHOR]

Noncommutative geometry, an offshoot of string theory, replaces point-like particles with smeared objects. These local effects have led to wormhole solutions in a semiclassical setting, but it has also been claimed that the noncommutative effects can be implemented by modifying only the energy-momentum tensor in the Einstein field equations, while leaving the Einstein tensor unchanged. The implication is that noncommutative-geometry wormholes could be macroscopic. The purpose of this paper is to confirm this conclusion in a simpler and more concrete manner by showing that the throat radius can indeed be macroscopic. This result can be readily explained by considering the noncommutative-geometry background to be a fundamental property and the macroscopic wormhole spacetime to be emergent.

Based on the perturbation expansion, we compute the noncommutative corrections to the minimal surface areas of the pure AdS spacetime and Schwarzschild-AdS black hole, where the noncommutative background is suitably constructed in terms of the Poincaré coordinate system. In particular, we find a reasonable tetrad with subtlety, which not only matches the metrics of the pure AdS spacetime and Schwarzschild-AdS black hole in the commutative case, but also makes the corrections real rather than complex in the noncommutative case. For the pure AdS spacetime, the nocommutative effect is only a logarithmic term, while for the Schwarzschild-AdS black hole, it contains a logarithmic contribution plus both a mass term and a noncommutative parameter related term. Furthermore, we show that the holographic entanglement entropy with noncommutativity obeys a relation which is similar to the first law of thermodynamics in the pure AdS spacetime. [ABSTRACT FROM AUTHOR]

In the setting of non-type II 1 representations, we propose a definition of deformed Fredholm module [ D T | D T | - 1 , · ] T for a modular spectral triple T , where D T is the deformed Dirac operator. D T is assumed to be invertible for the sake of simplicity, and its domain is an “essential” operator system E T . According to such a definition, we obtain [ D T | D T | - 1 , · ] T = | D T | - 1 d T (·) + d T (·) | D T | - 1 , where d T is the deformed derivation associated to D T . Since the “quantum differential” 1 / | D T | appears in a symmetric position, such a definition of Fredholm module differs from the usual one even in the undeformed case, that is in the tracial case. Therefore, it seems to be more suitable for the investigation of noncommutative manifolds in which the nontrivial modular structure might play a crucial role. We show that all models in Fidaleo and Suriano (J Funct Anal 275:1484–1531, 2018) of non-type II 1 representations of noncommutative 2-tori indeed provide modular spectral triples, and in addition deformed Fredholm modules according to the definition proposed in the present paper. Since the detailed knowledge of the spectrum of the Dirac operator plays a fundamental role in spectral geometry, we provide a characterisation of eigenvalues and eigenvectors of the deformed Dirac operator D T in terms of the periodic solutions of a particular class of eigenvalue Hill equations. [ABSTRACT FROM AUTHOR]

A fuzzy geometry is a certain type of spectral triple whose Dirac operator crucially turns out to be a finite matrix. This notion incorporates familiar examples like fuzzy spheres and fuzzy tori. In the framework of random noncommutative geometry, we use Barrett's characterization of Dirac operators of fuzzy geometries in order to systematically compute the spectral action S.D/ D Tr f .D/ for 2n-dimensional fuzzy geometries. In contrast to the original Chamseddine-- Connes spectral action, we take a polynomial f with f .x/ ! 1 as jxj ! 1 in order to obtain a well-defined path integral that can be stated as a random matrix model with action of the type S.D/ D N tr F CPi tr Ai tr Bi, being F, Ai and Bi noncommutative polynomials in 22n are obtained via chord diagrams and satisfy: independence of N; self-adjointness of the main polynomial F (modulo cyclic reordering of each monomial); also up to cyclicity, either self-adjointness or anti-self-adjointness of Ai and Bi simultaneously, for fixed i . Collectively, this favors a free probabilistic perspective for the large-N limit we elaborate on. [ABSTRACT FROM AUTHOR]

We investigate a quantum mechanical harmonic oscillator based on the extended Snyder model. This realization of the Snyder model is constructed as a quantum phase space generated by D spatial coordinates and D(D−1)/2 tensorial degrees of freedom, together with their conjugated momenta. The coordinates obey nontrivial commutation relations and generate a noncommutative geometry, which admits nicer properties than the usual realization of the model, in particular giving rise to an associative star product. The spectrum of the harmonic oscillator is studied through the introduction of creation and annihilation operators. Some physical consequences of the introduction of the additional degrees of freedom are discussed.

When gravity is quantum, the point structure of space-time should be replaced by a non-commutative geometry. This is true even for quantum gravity in the infra-red. Using the octonions as space-time coordinates, we construct pre-spacetime, pre-quantum Lagrangian dynamics. We show that the symmetries of this non-commutative space unify the standard model of particle physics with S U (2) R chiral gravity. The algebra of the octonionic space yields spinor states which can be identified with three generations of quarks and leptons. The geometry of the space implies quantisation of electric charge, and leads to a theoretical derivation of the mysterious mass ratios of quarks and the charged leptons. Quantum gravity is quantisation not only of the gravitational field, but also of the point structure of space-time. [ABSTRACT FROM AUTHOR]

Over the last couple of decades, there have been direct and indirect evidences for massive compact objects than their conventional counterparts. A couple of such examples are super-Chandrasekhar white dwarfs and massive neutron stars. The observations of more than a dozen peculiar over-luminous type Ia supernovae predict their origins from super-Chandrasekhar white dwarf progenitors. On the other hand, recent gravitational wave detection and some pulsar observations provide arguments for massive neutron stars, lying in the famous mass-gap between lowest astrophysical black hole and conventional highest neutron star masses. We show that the idea of a squashed fuzzy sphere, which brings in noncommutative geometry, can self-consistently explain either of the massive objects as if they are actually fuzzy or squashed fuzzy spheres. Noncommutative geometry is a branch of quantum gravity. If the above proposal is correct, it will provide observational evidences for noncommutativity. [ABSTRACT FROM AUTHOR]

We extend the scope of noncommutative geometry by generalizing the construction of the noncommutative algebra of a quotient space to situations in which one is no longer dealing with an equivalence relation. For these so-called tolerance relations, passing to the associated equivalence relation looses crucial information as is clear from the examples such as coarse graining in physics or the relation d(x, y) < ε on a metric space. Fortunately, thanks to the formalism of operator systems such an extension is possible and provides new invariants, such as the C ∗-envelope and the propagation number. After a thorough investigation of the structure of the (non-unital) operator systems associated to tolerance relations, we analyze the corresponding state spaces. In particular, we determine the pure state space associated to the operator system for the relation d(x, y) < ε on a path metric measure space. [ABSTRACT FROM AUTHOR]

In the present paper, within the framework of the Deformed Schrödinger equation, using the generalized Bopp's shift method and standard perturbation theory, we have obtained the energy eigenvalues of a newly proposed improved trigonometric Rosen-Morse potential model (ITRMP) for the hydrogenic atoms and the Heavy Light Mesons (HLM) QQ(Q = b; c) and we also applied the results obtained in the production of the modified massspectra of the HLM in the three-dimensional nonrelativistic noncommutative phase space (3D-NCPS) symmetries. The potential is a superposition of the trigonometric Rosen-Morse potential and new central terms appear as a result of the effects of noncommutativity properties of space and phase on the TRMP model. The obtained energy eigenvalues and modified mass spectra appeared as a function of the discreet atomic quantum numbers (j; n; l; s and m), infinitesimal parameters (Θ, λ and τ) and (η, ¯λ and ¯τ) which are induced by (spacespace) and (phase-phase), in addition to, the dimensional parameters (μ, b, d) of the studied potential. Furthermore, we have shown that the corresponding new Hamiltonian operator is the sum of the ordinary Hamiltonian operator of the TRMP model and three operators, the first one is the modified spin-orbit interaction, the second is the modified Zeeman operator while the third part is the rotational part for the previous hydrogenic and the HLM systems. [ABSTRACT FROM AUTHOR]

We discuss various aspects of the noncommutative geometry of a smooth subalgebra of the Toeplitz algebra. In particular, we study the structure of derivations on this subalgebra. [ABSTRACT FROM AUTHOR]

We generalize the notion of spectral triple with reality structure to spectral triples with multitwisted real structure, the class of which is closed under the tensor product composition. In particular, we introduce a multitwisted first-order condition (characterizing the Dirac operators as an analogue of first-order differential operator). This provides a unified description of the known examples, which include rescaled triples with the conformal factor from the commutant of the algebra and (on the algebraic level) triples on quantum disc and on quantum cone, that satisfy twisted first-order condition of Brzeziński et al. (2016, 2019), as well as asymmetric tori, non-scalar conformal rescaling and noncommutative circle bundles. In order to deal with them, we allow twists that do not implement automorphisms of the algebra of spectral triple. [ABSTRACT FROM AUTHOR]

We show how the twisting of spectral triples induces a transition from a Euclidean to a Lorentzian noncommutative geometry at the level of the fermionic action. More specifically, we compute the fermionic action for the twisting of a closed Euclidean manifold, then that of a two-sheet Euclidean manifold, and finally the twisting of the spectral triple of electrodynamics in Euclidean signature. We obtain the Weyl and the Dirac equations in Lorentzian signature (and in the temporal gauge). The twisted fermionic action is then shown to be invariant under an action of the Lorentz group. This permits us to interpret the field of 1-form that parametrises the twisted fluctuation of a manifold as the (dual) of the energy-momentum 4-vector. [ABSTRACT FROM AUTHOR]

We present a broad selection of results on endomorphisms and automorphisms of the Cuntz algebras On that have been obtained in the last decades. A wide variety of open problems is also included.

Whether an algebraic or a geometric or a phenomenological prescription is applied, the first fundamental form is unambiguously related to the modeling of the curved spacetime. Accordingly, we assume that the possible quantization of the first fundamental form could be proposed. For precise accurate measurement of the first fundamental form ds2=gμνdxμdxν, the author derived a quantum-induced revision of the fundamental tensor. To this end, the four-dimensional Riemann manifold is extended to the eight-dimensional Finsler manifold, in which the quadratic restriction on the length measure is relaxed, especially in the relativistic regime; the minimum measurable length could be imposed ad hoc on the Finsler structure. The present script introduces an approach to quantize the fundamental tensor and first fundamental form. Based on gravitized quantum mechanics, the resulting relativistic generalized uncertainty principle (RGUP) is directly imposed on the Finsler structure, F(x^0μ,p^0ν), which is obviously homogeneous to one degree in p^0μ. The momentum of a test particle with mass m¯=m/mp with mp is the Planck mass. This unambiguously results in the quantized first fundamental form ds˜2=[1+(1+2βp^0ρp^0ρ)m¯2(|x¨|/A)2]gμνdx^μdx^ν, where x¨ is the proper spacelike four-acceleration, A is the maximal proper acceleration, and β is the RGUP parameter. We conclude that an additional source of curvature associated with the mass m¯, whose test particle is accelerated at |x¨|, apparently emerges. Thereby, quantizations of the fundamental tensor and first fundamental form are feasible.

Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these ‘lattice spacing’ weights do not have to be independent of the direction of the arrow. We use this greater freedom to give a quantum geometric interpretation of discrete Markov processes with transition probabilities as arrow weights, namely taking the diffusion form ∂+f = (−Δθ + q − p)f for the graph Laplacian Δθ, potential functions q, p built from the probabilities, and finite difference ∂+ in the time direction. Motivated by this new point of view, we introduce a ‘discrete Schrödinger process’ as ∂+ψ = ı(−Δ + V )ψ for the Laplacian associated to a bimodule connection such that the discrete evolution is unitary. We solve this explicitly for the 2-state graph, finding a 1-parameter family of such connections and an induced ‘generalised Markov process’ for f = |ψ|2 in which there is an additional source current built from ψ. We also mention our recent work on the quantum geometry of logic in ‘digital’ form over the field F2 = {0, 1}, including de Morgan duality and its possible generalisations.

Based on the perturbation expansion, we compute the noncommutative corrections to the minimal surface areas of the pure AdS spacetime and Schwarzschild-AdS black hole, where the noncommutative background is suitably constructed in terms of the Poincaré coordinate system. In particular, we find a reasonable tetrad with subtlety, which not only matches the metrics of the pure AdS spacetime and Schwarzschild-AdS black hole in the commutative case, but also makes the corrections real rather than complex in the noncommutative case. For the pure AdS spacetime, the nocommutative effect is only a logarithmic term, while for the Schwarzschild-AdS black hole, it contains a logarithmic contribution plus both a mass term and a noncommutative parameter related term. Furthermore, we show that the holographic entanglement entropy with noncommutativity obeys a relation which is similar to the first law of thermodynamics in the pure AdS spacetime.

Over the last couple of decades, there have been direct and indirect evidences for massive compact objects than their conventional counterparts. A couple of such examples are super-Chandrasekhar white dwarfs and massive neutron stars. The observations of more than a dozen peculiar over-luminous type Ia supernovae predict their origins from super-Chandrasekhar white dwarf progenitors. On the other hand, recent gravitational wave detection and some pulsar observations provide arguments for massive neutron stars, lying in the famous mass-gap between lowest astrophysical black hole and conventional highest neutron star masses. We show that the idea of a squashed fuzzy sphere, which brings in noncommutative geometry, can self-consistently explain either of the massive objects as if they are actually fuzzy or squashed fuzzy spheres. Noncommutative geometry is a branch of quantum gravity. If the above proposal is correct, it will provide observational evidences for noncommutativity.

The strategy adopted in the original Morris-Thorne wormhole was to retain complete control over the geometry at the expense of certain engineering considerations. The purpose of this paper is to obtain several complete wormhole solutions by assuming a noncommutative-geometry background with a concomitant isotropic pressure condition. This condition allows us to consider a cosmological setting with a perfect-fluid equation of state. An extended form of the equation generalizes the first solution and subsequently leads to the generalized Chaplygin gas model and hence to a third solution. The solutions obtained extend several previous results. This paper also reiterates the need for a noncommutative-geometry background to account for the enormous radial tension that is a characteristic of Morris-Thorne wormholes.

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 present here what we consider a new picture of quantum mechanics with the position and momentum observables as coordinates of the usual quantum phase space of a single particle, which also serves as the model of the physical space. To minimize the mathematics involved, we stick here to the Hilbert space picture of the phase space. We argue that a quantum observable should be seen as taking noncommutative values each of which can equivalently be represented by an infinite number of real numbers. The six noncommutative values of the position and momentum observables hence serve as an alternative system of coordinates for the Hilbert space. The values can, at least in principle, be experimentally determined. This can be seen as a complete resolution of the Einstein–Bohr debate that Einstein would probably be happy with. What we have is a solid noncommutative geometrical picture of the physical space, or spacetime, beyond the Newtonian and Einsteinian framework that is sure to be relevant to Nature directly coupled with the idea that each physical quantity should better be seen as having a value beyond what can be represented by a number. We finish by sketching some implications of the results for the physics and mathematics of quantum spacetime in general.

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.

I discuss some issues related to the noncommutative spaces κ and its angular variant ρ-Minkowski with particular emphasis on the role of observers.

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]

We construct a noncommutative analog of additive functional equations on discrete quantum semigroups and show that this noncommutative functional equation has Hyers-Ulam stability on amenable discrete quantum semigroups. The discrete quantum semigroups that we consider in this paper are in the sense of van Daele, and the amenability is in the sense of Bèdos-Murphy-Tuset. Our main result generalizes a famous and old result due to Forti on the Hyers-Ulam stability of additive functional equations on amenable classical discrete semigroups.

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]

Continuing the study of the Moyal Higher Spin Yang–Mills theory started in our previous paper we provide a detailed discussion of matter coupling and the corresponding tree-level amplitudes. We also start the investigation of the spectrum by expanding the master fields in terms of ordinary spacetime fields. We note that the spectrum can be consistent with unitarity while still preserving Lorentz covariance, albeit not in the usual way, but by employing an infinite-dimensional unitary representation of the Lorentz group.

Using a fully-fledged formulation of gauge field theory deformed by the spacetime noncommutativity, we study its impact on relic neutrino direct detection, as proposed recently by the PTOLEMY experiment. The noncommutative background tends to influence the propagating neutrinos by providing them with a tree-level vector-like coupling to photons, enabling thus otherwise sterile right-handed (RH) neutrinos to be thermally produced in the early universe. Such a new component in the universe's background radiation has been switched today to the almost fully active sea of non-relativistic neutrinos, exerting consequently some impact on the capture on tritium at PTOLEMY. The peculiarities of our nonperturbative approach tend to reflect in the cosmology as well, upon the appearances of the coupling temperature, above which RH neutrinos stay permanently decoupled from thermal environment. This entails the maximal scale of noncommutativity as well, being of order of 10−4MPl, above which there is no impact whatsoever on the capture rates at PTOLEMY. The latter represents an exceptional upper bound on the scale of noncommutativity coming from phenomenology.

Noncommutative or ‘quantum’ differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics) but also differential forms, bundles and Riemannian structures at this level. The data for the algebra quantisation is a classical Poisson bracket while the data for quantum differential forms is a Poisson-compatible connection. We give an introduction to our recent result whereby further classical data such as classical bundles, metrics etc. all become quantised in a canonical ‘functorial’ way at least to 1st order in deformation theory. The theory imposes compatibility conditions between the classical Riemannian and Poisson structures as well as new physics such as typical nonassociativity of the differential structure at 2nd order. We develop in detail the case of ℂℙn where the commutation relations have the canonical form [wi, w̄j] = iλδij similar to the proposal of Penrose for quantum twistor space. Our work provides a canonical but ultimately nonassociative differential calculus on this algebra and quantises the metric and Levi-Civita connection at lowest order in λ.

In the present contribution, the Hirota-Satsuma coupled KdV hierarchy on noncommutative phase-space is investigated using the noncommutative extension of Lax pair generating technique. In particular, the explicit representation of its associated Lax pair operators is constructed. It is shown that the obtained results for phase-space noncommutativity case reduce back to the standard commutative case when θ goes to 1/2 . [ABSTRACT FROM AUTHOR]

Abstract We equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four dimensional space. [ABSTRACT FROM AUTHOR]

In this paper, we studied the string theory in its original form of Nambu-Goto. We found that the calculation of the variation of the Nambu-Goto action with respect to the metric of the word sheet leads to a nonlinear action which is not strictly equivalent to the Polyakov action. Then the equation of motion corresponding to this action is nonlinear. This was the reason why we decided to solve the nonlinear Nambu-Goto equation of motion directly. This has been done for a string interacting with a constant, antisymmetric tensor field B in the Nambu-Goto formulation, and we derive the noncommutativity among the string end points coordinates. [ABSTRACT FROM AUTHOR]

In this paper, we review some recent developments of compact quantum groups that arise as θ-deformations of compact Lie groups of rank at least two. A θ-deformation is merely a 2-cocycle deformation using an action of a torus of dimension higher than 2. Using the formula (Lemma 5.3) developed in [11], we derive the noncommutative 7-sphere in the sense of Connes and Landi [3] as the fixed-point subalgebra. [ABSTRACT FROM AUTHOR]