International Journal of Geometric Methods in Modern Physics

A compact, self-contained approach to gravitation, based on the local Poincaré gauge invariance, is proposed. Starting from the general invariance principle, we discuss the global and the local Poincaré invariance developing the spinor, vector and tetrad formalisms. These tools allow to construct the curvature, torsion and metric tensors by the Fock–Ivanenko covariant derivative. The resulting Einstein–Cartan theory describes a space endowed with non-vanishing curvature and torsion while the gravitational field equations are similar to the Yang–Mills equations of motion with the torsion tensor playing the role of the Yang–Mills field strength.

Fractional calculus of variations and in particular the Fractional Action-Like Variational Approach has recently gained importance in studying nonconservative and weak decaying dynamical systems. Until now, in high-energy physics including cosmology and quantum field theories the derivative and integral operators have only been used in integer steps. In this work, we develop the fractional Lagrangian formulation of General Relativity based on the Cresson's fractional differential operators that generalize the differential operators of conventional Einstein's General Relativity but that reduces to the standard formalism in the integer limit. Our main aim is to build the multi-dimensional setting for the Einstein's field equations. The presence of a complex number in the Cresson's fractional derivatives forces the field equations to be complex and even noncommutative. An immediate consequence of this is that all gravitational and curvature fields are complexified and as a result may be decomposed into real part and complex parts. We have also indications that the fractional Einstein's field equations in the real space behave differently from those in the complex plane. Furthermore, we show how the generalization of the fractional Cresson's fractional derivative to the directional derivative at an arbitrary orientation θ leads to a spinor-like gravitational field equation in the sense of Penrose.

A nonassociative generalization of the principal fiber bundles with a smooth loop mapping on the fiber is presented. Our approach allows us to construct a new kind of gauge theories that involve higher "nonassociative" symmetries.

We prove that a four-dimensional Lorentzian manifold that is curvature homogeneous of order 3, or CH₃ for short, is necessarily locally homogeneous. We also exhibit and classify four-dimensional Lorentzian, CH₂ manifolds that are not homogeneous. The resulting metrics belong to the class of null electromagnetic radiation, type N solutions on an anti-de Sitter background. These findings prove that the four-dimensional Lorentzian Singer number k_1,3 = 3, falsifying some recent conjectures [1]. We also prove that invariant classification for these proper CH₂ solutions requires ∇⁽⁷⁾ R, and that these are the unique metrics requiring the seventh order.

We briefly review the most relevant aspects of complete integrability for classical systems and identify those aspects which should be present in a definition of quantum integrability.

We show that a naive extension of classical concepts to the quantum framework would not work because all infinite dimensional Hilbert spaces are unitarilly isomorphic and, as a consequence, it would not be easy to define degrees of freedom. We argue that a geometrical formulation of quantum mechanics might provide a way out.

We present a variational formulation of electrodynamics using de Rham even and odd differential forms. Our formulation relies on a variational principle more complete than the Hamilton principle and thus leads to field equations with external sources and permits the derivation of the constitutive relations. We interpret a domain in space-time as an odd de Rham 4-current. This permits a treatment of different types of boundary problems in an unified way. In particular we obtain a smooth transition to the infinitesimal version by using a current with a one point support.