International Journal of Geometric Methods in Modern Physics

A new framework for perturbative quantum gravity is proposed following the geometry of non-holonomic distributions on (pseudo)Riemannian manifolds. There are considered such distributions and adap-ted connections, also completely defined by a metric structure, when gravitational models with infinite many couplings reduce to two-loop renormalizable effective actions. We use a key result from our partner work arXiv: 0902.0911 that the classical Einstein gravity theory can be reformulated equivalently as a non-holonomic gauge model in the bundle of affine/de Sitter frames on pseudo-Riemannian space–time. It is proven that (for a class of non-holonomic constraints and splitting of the Levi–Civita connection into a "renormalizable" distinguished connection, on a base background manifold, and a gauge-like distortion tensor, in total space) a non-holonomic differential renormalization procedure for quantum gravitational fields can be elaborated. Calculation labor is reduced to one- and two-loop levels and renormalization group equations for non-holonomic configurations.

In this work, we apply the Poincaré–Cartan formalism of Classical Field Theory to study the systems of balance equations (balance systems). We introduce the partial k-jet bundles of the configurational bundle π : Y → X and study their basic properties: partial Cartan structure, prolongation of vector fields, etc. A constitutive relation C of a balance system is realized as the mapping between a (partial) k-jet bundle and the extended dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. The invariant (variational) form of the balance system corresponding to a constitutive relation is studied. Special cases of balance systems — Lagrangian systems of order 1 with arbitrary sources and RET (Rational Extended Thermodynamics) systems are characterized in geometrical terms. The action of automorphisms of the bundle π on the constitutive mappings is studied and it is shown that the symmetry group of acts on the sheaf of solutions of balance system . A suitable version of Noether theorem for an action of a symmetry group is presented together with the special forms for semi-Lagrangian and RET balance systems. Examples of energy momentum and gauge symmetries balance laws are provided. At the end, we introduce the secondary balance laws for a balance system and classify these laws for the Cattaneo heat propagation system.

A (n + 1)-dimensional Einstein–Gauss–Bonnet (EGB) model is considered. For diagonal cosmological-type metrics, the equations of motion are reduced to a set of Lagrange equations. The effective Lagrangian contains two "minisuperspace" metrics on ℝⁿ. The first one is the well-known 2-metric of pseudo-Euclidean signature and the second one is the Finslerian 4-metric that is proportional to n-dimensional Berwald–Moor 4-metric. When a "synchronous-like" time gauge is considered, the equations of motion are reduced to an autonomous system of first-order differential equations. For the case of the "pure" Gauss–Bonnet model, two exact solutions with power-law and exponential dependence of scale factors (with respect to "synchronous-like" variable) are obtained. (In the cosmological case, the power-law solution was considered earlier in papers of N. Deruelle, A. Toporensky, P. Tretyakov and S. Pavluchenko.) A generalization of the effective Lagrangian to the Lowelock case is conjectured. This hypothesis implies existence of exact solutions with power-law and exponential dependence of scale factors for the "pure" Lowelock model of mth order.

The Minkowski space–time quantum Clifford algebra structure associated with the conformal group and the Clifford–Hopf alternative κ-deformed quantum Poincaré algebra is investigated in the Atiyah–Bott–Shapiro mod 8 theorem context. The resulting algebra is equivalent to the deformed anti-de Sitter algebra , when the associated Clifford–Hopf algebra is taken into account, together with the associated quantum Clifford algebra and a (not braided) deformation of the periodicity Atiyah–Bott–Shapiro theorem.

We analyze Ricci flow (normalized/unnormalized) of product manifolds — unwarped as well as warped, through a study of generic examples. First, we investigate such flows for the unwarped scenario with manifolds of the type 𝕊ⁿ × 𝕊^m, 𝕊ⁿ × ℍ^m, ℍ^m × ℍⁿ and also, similar multiple products. We are able to single out generic features such as singularity formation, isotropization at particular values of the flow parameter and evolution characteristics. Subsequently, motivated by warped braneworlds and extra dimensions, we look at Ricci flows of warped space–times. Here, we are able to find analytic solutions for a special case by variable separation. For others, we numerically solve the equations and draw certain useful inferences about the evolution of the warp factor, the scalar curvature as well as the occurrence of singularities at finite values of the flow parameter. We also investigate the dependence of the singularities of the flow on the initial conditions. We expect our results to be useful in any physical/mathematical context where such product manifolds may arise.

In this paper, we study the relationship between closed-loop mechanical systems (CLMS) and higher-order constrained systems (HOCS). We show that second-order constraints, i.e. constraints in acceleration, can be used to construct a CLMS, where the control law is given by the related constraint force. As an example we show that, by means of appropriate constraints of this kind, stabilization can be achieved for the inertia wheel pendulum. We also consider the inverse problem, that is to say: can every CLMS be constructed from a HOCS, i.e. can every control law be seen as the constraint force related to a given set of constraints? We answer this question affirmatively, reflecting a deep connection between feedback systems and systems with constraints. Also, we study when these constraints can be integrated to constraints in velocities. We prove that this is possible for CLMSs constructed by means of linear methods and also by some global ones, as the kinetic shaping method with symmetry.

We shall here propose a class of relativistic theories of gravitation, based on a foundational paper of Ehlers Pirani and Schild (EPS). All "extended theories of gravitation" (also known as f(R) theories) in Palatini formalism are shown to belong to this class. In a forthcoming paper, we shall show that this class of theories contains other more general examples. EPS framework helps in the interpretation and solution of these models that however have exotic behaviors even compared to f(R) theories.

We shall present and analyze two examples of extended theories of gravitation in Palatini formalism with matter that couples to the connection. This will show that the class of Further Extended Theories of Gravitation introduced in [1] does not trivially reduce to f(R) models. It will also produce an example of theory that on-shell endows space–time with a non-trivial Weyl geometry where the connection is not induced by the metric structure (though it is compatible with it in the sense of Ehlers–Pirani–Schild; see [2]).

From points of view of physics, fractional operators represent a vital role for describing intermediate processes and critical phenomena in physics. Subsequently, fractional Action-like Variational Approach in the sense of Riemann–Liouville fractional integral has lately gained significance in exploring non-conservative dynamical systems found in classical and quantum field theories. Within the same framework, fractional Dirac operators are introduced and the fractional spectral action principle is constructed and some interesting consequences are discussed. In particular, we show that the fractional spectral triplet action is complexified and the disturbing huge cosmological term may be eliminated. The generalization of the problem in view of the generalized fractional integration operators, namely the Erdelyi–Kober fractional integral, is discussed as well.