International Journal of Geometric Methods in Modern Physics

In this short review, we revisit inflation in F(R)-gravity. We find several F(R)-models for viable inflation by applying some reconstruction techniques. A special attention is payed in the reproduction of the last Planck satellite data. The possible generalizations of Starobinsky-like inflation are found and discussed. The early-time acceleration is analyzed in a higher derivative quantum gravitational model which mainly reduces to F(R)-gravity.

In this paper, we study the Ehlers' transformation (sometimes called gravitational duality rotation) for reciprocal static metrics. First, we introduce the concept of reciprocal metric. We prove a theorem which shows how we can construct a certain new static solution of Einstein field equations using a seed metric. Later, we investigate the family of stationary spacetimes of such reciprocal metrics. The key here is a theorem from Ehlers', which relates any static vacuum solution to a unique stationary metric. The stationary metric has a magnetic charge. The spacetime represents Newman-Unti-Tamburino (NUT) solutions. Since any stationary spacetime can be decomposed into a 1 + 3 time-space decomposition, Einstein field equations for any stationary spacetime can be written in the form of Maxwell's equations for gravitoelectromagnetic fields. Further, we show that this set of equations is invariant under reciprocal transformations. An additional point is that the NUT charge changes the sign. As an instructive example, by starting from the reciprocal Schwarzschild as a spherically symmetric solution and reciprocal Morgan–Morgan disk model as seed metrics we find their corresponding stationary spacetimes. Starting from any static seed metric, performing the reciprocal transformation and by applying an additional Ehlers' transformation we obtain a family of NUT spaces with negative NUT factor (reciprocal NUT factors).

In this paper, we derive the symplectic framework for field theories defined by higher order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher order system of partial differential equations to a constrained first-order one, the symplectic structures naturally arising in the dynamics of a first-order Lagrangian theory, and the importance of the Poincaré–Cartan form for variational problems, are all well-established facts. However, their adequate combination corresponding to higher order theories is missing in the literature. Here we obtain a consistent and truly finite-dimensional canonical formalism, as well as a higher order version of the Poincaré–Cartan form. In our exposition, the rigorous global proofs of the main results are always accompanied by their local coordinate descriptions, indispensable to work out practical examples.

In this paper, we will analyze the gravitational collapse in the framework of gravity's rainbow. We will demonstrate that the position of the horizon for a particle inside the black hole depends on the energy of that particle. It will also be observe that the position of the horizon for a particle falling radially into the black hole also depends on its energy. Thus, it is possible for a particle coming from outside to interact with a particle inside the black, and take some information outside the black hole. This is because for both these particles the position of horizon is different. So, even though the particle from inside the black hole is in its own horizon, it is not in the horizon of the particle coming from outside. Thus, we will demonstrate that in gravity's rainbow information can get out of a black hole.

In this paper, we have introduced a new notion of generalized Tanaka–Webster Reeb recurrent Ricci tensor of real hypersurfaces in complex two-plane Grassmannians G₂(ℂ^m+2). Next, we show a non-existence property for real hypersurfaces M in G₂(ℂ^m+2) with such a curvature condition.

An affine Kac–Moody algebra is a central extension of the Lie algebra of smooth mappings from S¹ to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S³ to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac–Moody algebras give it for two-dimensional conformal field theory.

The two-dimensional charge transport with parallel (in plane) magnetic field is considered from the physical and mathematical point of view. To this end, we start with the magnetic field parallel to the plane of charge transport, in sharp contrast to the configuration described by the theorems of Aharonov and Casher where the magnetic field is perpendicular. We explicitly show that the specific form of the arising equation enforces the respective field solution to fulfill the Majorana condition. Consequently, as soon any physical system is represented by this equation, the rise of fields with Majorana type behavior is immediately explained and predicted. In addition, there exists a quantized particular phase that removes the action of the vector potential producing interesting effects. Such new effects are able to explain due to the geometrical framework introduced, several phenomenological results recently obtained in the area of spintronics and quantum electronic devices. The quantum ring as spin filter is worked out in this framework and also the case of the quantum Hall effect.

In this paper, we will introduce a generalized soldering p-forms geometry, which can be the right framework to describe many new approaches and concepts in modern physics. Here we will treat some aspects of the theory of local cohomology in fields theory or more precisely the theory of soldering-form conservation laws in physics. We provide some illustrative applications, primarily taken from the Einstein equations of general theory of relativity and Yang–Mills theory. This theory can be considered to be a generalization of Noether's theory of conserved current to differential forms of any degree. An essential result of this, is that the conservation of the energy–momentum in general relativity, is linked to the fact that the vacuum field equations are equivalent to the integrability conditions of a first-order system of differential equations. We also apply the idea of the soldered space and the integrability conditions to the case of Yang–Mills theory. The mathematical framework, where these intuitive considerations would fit naturally, can be used to describe also the dynamics of changing manifolds.

Based on early theoretical work on tachyonic fermions we shall study a modification of the Gross–Neveu model in two dimensions. We shall see that the theory results to the dynamical generation of real and imaginary masses. These imaginary masses indicate the possibility that tachyonic solutions (or instabilities) could exist in the theory. The implications of a tachyonic neutrino coming from astrophysical sources are critically discussed. Moreover, we present a toy model that consists of an U(2, 2) invariant Dirac Lagrangian. This theory can have tachyonic masses as solutions. A natural mass splitting between the solutions is a natural outcome of the formalism.

In this paper a toy model of quantum topology is reviewed to study effects of matter and gauge fields on the topology fluctuations. In the model a collection of N one-dimensional manifolds is considered where a set of boundary conditions on states of Hilbert space specifies a set of all topologies perceived by quantum particle and probability of having a specific topology is determined by a partition function over all the topologies in the context of noncommutative spectral geometry. In general the topologies will be fuzzy with the exception of a particular case which is localized by imposing a specific boundary condition. Here fermions and bosons are added to the model. It is shown that in the presence of matter, the fuzziness of topology will be dependent on N, however for large N the dependence is removed similar to the case without matter. Also turning on a particular background gauge field can overcome the fuzziness of topology to reach a localized topology with classical interpretation. It can be seen that for large N more opportunities can be provided for choosing the background gauge field to localize the fuzzy topology.

In this paper, the bifurcation and attractor of the stochastic Rabinovich system with jump are discussed, and some new results for the system are presented. First, the sufficient condition and necessary condition for stochastic stability of the system are given. Second, the estimation of the global attractive set of system is obtained. The existence of random attractors of the stochastic Rabinovich system with jump is also discussed. Finally, stochastic bifurcation behavior for the system is analyzed. It is hoped that the investigation of this paper can help understanding the rich dynamic of the stochastic Rabinovich system and the true geometrical structure of the original amazing Rabinovich attractor.

A necessary condition for the emergence of chaos is given. It is well known that the emergence of chaos requires a positive exponent which entails diverging trajectories. Here we show that this is not enough. An additional necessary condition for the emergence of chaos in the region where the trajectory of the system goes through, is that the product of the maximal positive exponent times, the duration in which the system configuration point stays in the unstable region should exceed unity. We give a theoretical analysis justifying this result and a few examples. We stress that the criterion suggested involves only local exponents and is not concerned with asymptotic defined exponents.

We discuss the Minkowski stability problem in modified gravity by using dynamical system approach. The method to investigate dynamical stability of Minkowski space is proposed. This method was applied for some modified gravity theories, such as f(R) gravity, f(R)+αR□R gravity and scalar–tensor gravity models with non-minimal kinetic coupling. It was shown that in the case of f(R) gravity Minkowski solution is asymptotically stable in ghost-free (f′ > 0) and tachyon-free (f″ > 0) theories in expanding Universe with respect to isotropic and basic anisotropic perturbations. In the case of higher order gravity with αR□R correction conditions of Minkowski stability with respect to isotropic perturbations are significantly different: f′(0) < 0, f″(0) < 0 and 3f′(0) + f″(0)²/α > 0. And in the case of scalar-tensor gravity with non-minimal kinetic coupling Minkowski solution is asymptotically stable in expanding Universe with respect to isotropic perturbations of metric. Moreover, the developed method may be used for finding additional restrictions on parameters of different modified gravity theories.

We consider curvature-teleparallel F(R,T) gravity, where the gravitational Lagrangian density is given by an arbitrary function of the Ricci scalar R and the torsion scalar T. Using the Noether Symmetry Approach, we show that the functional form of the F(R,T) function can be determined by the presence of symmetries. Furthermore, we obtain exact solutions through the presence of conserved quantities and the reduction of cosmological dynamical system. Example of particular cosmological models are considered.

The variant of Fedosov construction based on fairly general fiberwise product in the Weyl bundle is studied. We analyze generalized star products of functions, of sections of endomorphisms bundle, and those generating deformed bimodule structure as introduced previously by Waldmann. Isomorphisms of generalized Fedosov algebras are considered and their relevance for deriving Seiberg–Witten map is described. The existence of the trace functional is established. Explicit expressions for star products and the trace functional are provided up to the second power of deformation parameter. The example of symmetric part of noncommutativity tensor is discussed as a case with possible field-theoretic application.

This paper considers the dependence of the Universe expansion rate in the era of radiation domination of the orbital angular momentum of photons filling the Universe. It is shown that the presence of a nonzero orbital angular momentum in photons leads to increase in the rate of expansion. As a consequence, the hypothesis is made that areas of the Universe, having photons with nonzero orbital angular momentum inside them, will cool faster. In these areas the elementary particles will form earlier, compared to the rest of the Universe. Therefore, these areas become the future centers of the primary material inhomogeneity.

Equivalence classes of gapped Hamiltonians compatible with given symmetry constraints, such as those underlying topological insulators, can be defined in many ways. For the non-chiral classes modeled by vector bundles over Brillouin tori, physically relevant equivalences include isomorphism, homotopy, and K-theory, which are inequivalent but closely related. We discuss an important subtlety which arises in the chiral Class AIII systems, where the winding number invariant is shown to be relative rather than absolute as is usually assumed. These issues are then analyzed and reconciled in the language of K-theory.

We consider the most general torsional completion of gravitation together with electrodynamics for the Dirac spinorial material fields, and we show that consistency arguments constrain torsion to be completely antisymmetric and the dynamics to be parity-invariant and described by actions that are either least-order derivative or renormalizable.

A method is presented which restricts the space of paths entering the path integral of quantum mechanics to subspaces of C^α, by only allowing paths which possess at least α derivatives. The method introduces two external parameters, and induces the appearance of a particular time scale ϵ_D such that for time intervals longer than ϵ_D the model behaves as usual quantum mechanics. However, for time scales smaller than ϵ_D, modifications to standard formulation of quantum theory occur. This restriction renders convergent some quantities which are usually divergent in the time-continuum limit ϵ → 0. We illustrate the model by computing several meaningful physical quantities such as the mean square velocity 〈v²〉, the canonical commutator, the Schrödinger equation and the energy levels of the harmonic oscillator. It is shown that an adequate choice of the parameters introduced makes the evolution unitary.

We define covariant Lie derivatives acting on vector-valued forms on Lie algebroids and study their properties. This allows us to obtain a concise formula for the Frölicher–Nijenhuis bracket on Lie algebroids.

Following the line of the history, if by one side the electromagnetic theory was consolidated on the 19th century, the emergence of the special and the general relativity theories on the 20th century opened possibilities of further developments, with the search for the unification of the gravitation and the electromagnetism on a single unified theory. Some attempts to the geometrization of the electromagnetism emerged in this context, where these first models resided strictly on a classical basis. Posteriorly, they were followed by more complete and embracing quantum field theories. The present work reconsiders the classical viewpoint, with the purpose of showing that at first-order of approximation the electromagnetism constitutes a geometric structure aside other phenomena as gravitation, and that magnetic monopoles do not exist at least up to this order of approximation. Even though being limited, the model is consistent and offers the possibility of an experimental test of validity.