Search Results (10)

We investigate Snyder spacetime and its generalizations, including Yang and Snyder–de Sitter spaces, which constitute manifestly Lorentz-invariant noncommutative geometries. This work initiates a systematic study of gauge theory on such spaces in the semi-classical regime, formulated as Poisson gauge theory. As a first step, we construct the symplectic realizations of the relevant noncommutative spaces, a prerequisite for defining Poisson gauge transformations and field strengths. We present a general method for representing the Snyder algebra and its extensions in terms of canonical phase-space variables, enabling both the reproduction of known representations and the derivation of novel ones. These canonical constructions are employed to obtain explicit symplectic realizations for the Snyder–de Sitter space and to construct the deformed partial derivative which differentiates the underlying Poisson structure. Furthermore, we analyze the motion of freely falling particles in these backgrounds and comment on the geometry of the associated spaces. Full article

The Clifford–Weyl algebra ${𝒜}_q^{\pm }$, as a class of solvable polynomial algebras, combines the anti-commutation relations of Clifford algebras ${𝒜}_q^{+}$ with the differential operator structure of Weyl algebras ${𝒜}_q^{-}$. It exhibits rich algebraic and geometric properties. This paper employs Gröbner–Shirshov basis principles in concert with Poincaré–Birkhoff–Witt (PBW) basis methodology to delineate the iterated skew polynomial structures within ${𝒜}_q^{+}\mathrm{and}{𝒜}_q^{-}$. By constructing explicit PBW generators, we analyze the structural properties of both algebras and their modules using constructive methods. Furthermore, we prove that ${𝒜}_q^{+}\mathrm{and}{𝒜}_q^{-}$ are Auslander regular, Cohen–Macaulay, and Artin–Schelter regular. These results provide new tools for the representation theory in noncommutative geometry. Full article

In this review, we begin with a historical survey of some singular solutions in the theory of gravitation, as well as a very brief discussion of how black holes could physically form. Some possible scenarios which could perhaps eliminate these singularities are then reviewed and discussed. Due to the vastness of the field, its coverage is not exhaustive; instead, the concentration is on a small subset of topics such as possible quantum gravity effects, non-commutative geometry, and gravastars. A simple singularity theorem is also reviewed. Although parts of the manuscript assume some familiarity with relativistic gravitation or differential geometry, the aim is for the broad picture to be accessible to non-specialists of other physical sciences and mathematics. Full article

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. Full article

We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function, defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument’s evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups. Full article

In this work, using the noncommutative integration method of linear differential equations, we obtain a complete set of solutions to the Schrodinger equation for a quantum asymmetric top in Euler angles. It is shown that the noncommutative reduction of the Schrodinger equation leads to the Lame equation. The resulting set of solutions is determined by the Lame polynomials in a complex parameter, which is related to the geometry of the orbits of the coadjoint representation of the rotation group. The spectrum of an asymmetric top is obtained from the condition that the solutions are invariant with respect to a special irreducible $\lambda$-representation of the rotation group. Full article

Noncommutative geometry is an established potential candidate for including quantum phenomena in gravitation. We outlined the formalism of Hopf algebras and its connection to the algebra of infinitesimal diffeomorphisms. Using a Drinfeld twist, we deformed spacetime symmetries, algebra of vector fields and differential forms, leading to a formulation of noncommutative Einstein equations. We studied a concrete example of charged BTZ spacetime and deformations steaming from the so-called angular twist. The entropy of the noncommutative charged BTZ black hole was obtained using the brick-wall method. We used a charged scalar field as a probe and obtained its spectrum and density of states via WKB approximation. We provide the method used to calculate corrections to the Bekenstein–Hawking entropy in higher orders in WKB, but we present the final result in the lowest WKB order. The result is that, even in the lowest order in WKB, the entropy, in general, contains higher powers in ℏ, and it has logarithmic corrections and polynomials of logarithms of the black hole area. Full article

We study the known coherent states of a quantum harmonic oscillator from the standpoint of the originally developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schrödinger equation and on the orbit geometry of the coadjoint representation of Lie groups. We have shown that analogs of coherent states constructed by the noncommutative integration can be expressed in terms of the solution to a system of differential equations on the Lie group of the oscillatory Lie algebra. The solutions constructed are directly related to irreducible representation of the Lie algebra on the Hilbert space functions on the Lagrangian submanifold to the orbit of the coadjoint representation. Full article

A self-organizing joint system classical oscillator–random environment is considered within the framework of a complex probabilistic process that satisfies a Langevin-type stochastic differential equation. Various types of randomness generated by the environment are considered. In the limit of statistical equilibrium (SEq), second-order partial differential equations (PDE) are derived that describe the distribution of classical environmental fields. The mathematical expectation of the oscillator trajectory is constructed in the form of a functional-integral representation, which, in the SEq limit, is compactified into a two-dimensional integral representation with an integrand: the solution of the second-order complex PDE. It is proved that the complex PDE in the general case is reduced to two independent PDEs of the second order with spatially deviating arguments. The geometric and topological features of the two-dimensional subspace on which these equations arise are studied in detail. An algorithm for parallel modeling of the problem has been developed. Full article

The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n-forms ${\Omega }^n(X,M)$. Further restricting the notion of smoothness, we use our result to show that most k-algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where $k=\mathbb{C}$. Full article