The aim of this paper is to create a large geometrical background on the dual 1-jet space J^{1*}(T,M) for a multi-time Hamiltonian approach of the electromagnetic and gravitational physical fields. Our geometric-physical construction is achieved starting only from a given quadratic Hamiltonian function of polymomenta H, which naturally produces a canonical nonlinear connection N, a canonical Cartan N-linear connection CΓ(N) and their corresponding local distinguished (d-) torsions and curvatures. In such a context, we construct some geometrical electromagnetic-like and gravitational-like field theories which are characterized by some natural geometrical Maxwell-like and Einstein-like equations. Some abstract and geometrical conservation laws for the multi-time Hamiltonian gravitational physical field are also given.
In this paper, we analyze the stability of the real-valued Maxwell-Bloch equations with a control that depends on state variables quadratically. We also investigate the topological properties of the energy-Casimir map, as well as the existence of periodic orbits and explicitly construct the heteroclinic orbits.
We prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the time-harmonic Maxwell equations in certain anisotropic settings. We give a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold, and a uniqueness result for Maxwell equations in Euclidean space with admissible matrix coefficients. The proofs are based on a new Fourier analytic construction of complex geometrical optics solutions on admissible manifolds, and involve a proper notion of uniqueness for such solutions.
In this paper we first show that any coupled system consisting of a gravitational plus a free electromagnetic field can be described geometrically in the sense that both Maxwell equations and Einstein equation having as source term the energy-momentum of the electromagnetic field can be derived from a geometrical Lagrangian proportional to the scalar curvature R of a particular kind of Riemann-Cartan spacetime structure, where those fields are identified as geometrical objects of the structure. We show moreover that the contorsion tensor of the particular Riemann-Cartan spacetime structure of …
A general affine connection has enough degrees of freedom to describe the classical gravitational and electromagnetic fields in the metric-affine formulation of gravity. The gravitational field is represented in the Lagrangian by the symmetric part of the Ricci tensor, while the classical electromagnetic field is represented geometrically by the tensor of homothetic curvature. We introduce matter as the four-velocity field subject to the kinematical constraint in which the Lagrange multiplier represents the energy density. A coupling between the four-velocity and the trace of the nonmetricity …
In this paper we show how a gravitational field generated by a given energy-momentum distribution (for all realistic cases) can be represented by distinct geometrical structures (Lorentzian, teleparallel and non null nonmetricity spacetimes) or that we even can dispense all those geometrical structures and simply represent the gravitational field as a field in the Faraday's sense living in Minkowski spacetime. The explicit Lagrangian density for this theory is given and the field equations (which are Maxwell's like equations) are shown to be equivalent to Einstein's equations. Some examples ar…
In this manuscript, we present an alternative method for calculating null geodesics in General Static Isotropic Metrics in General Relativity and Extended Theories of Gravity. By applying a conformal transformation, we are able to consider an analogue gravity model, where curvature is encoded in the dielectric and magnetic properties of a medium. In other words, we pass from curved to flat space-times, where instead of the Einstein field equations, the Maxwell equations are solved. Within this geometrical background, the photon geodesics are calculated. Then, given different black hole and wor…
The concept of gauge theories progressively emerged during the 20th century while the building of the Standard Model (SM) of particle physics. This work focusses on some aspects of gauge theories. First, a modern interpretation of Maxwell's equations in terms of differential forms motivates a geometric definition of gauge theories as a theory of connections on principle bundles. After presenting some historical facts about the notion of gauge in particle physics, the geometrical construction is used to study the electroweak theory and the SM. Though the SM has not been defaulted by any experim…
We show that the first-order D=4, N=1 pure supergravity lagrangian four-form can be obtained geometrically as a quadratic expression in the curvatures of the Maxwell superalgebra. This is achieved by noticing that the relative coefficient between the two terms of the Lagrangian that makes the action locally supersymmetric also determines trivial field equations for the gauge fields associated with the extra generators of the Maxwell superalgebra. Along the way, a convenient geometric procedure to check the local supersymmetry of a class of lagrangians is developed.
This article is concerned with the time-harmonic electromagnetic (EM) scattering from a generic inhomogeneous medium. It is shown that if there is a right corner on the support of the medium, then it scatters every pair of incident EM fields, excluding a possible class of EM fields which are of very particular forms. That is, for every pair of admissible incident EM fields, the corresponding scattered wave fields associated to the medium scatterer cannot be identically vanishing outside the support of the medium. Indeed, we achieve the corner scattering result by establishing a stronger result…
Electromagnetism: an intrinsic approach to Hadamard's method of descent
Sapienza Università di Roma（罗马第一大学）；INFN, Sezione di Bologna（博洛尼亚意大利国家核物理研究所）；Università di Bologna（博洛尼亚大学）；INFN, Sezione di Bari（巴里意大利国家核物理研究所）；Università di Bari（巴里大学）；INFN, Sezione di Napoli（那不勒斯意大利国家核物理研究所）；Università di Napoli Federico II（那不勒斯费德里科二世大学）
An integrated approach to Lie derivatives of spinors, spinor connections and the gravitational field is presented, in the context of a previously proposed, partly original formulation of a theory of Einstein-Carta-Maxwell-Dirac fields based on "minimal geometric data": all the needed underlying structure is geometrically constructed from the unique assumption of a complex vector field $S\to M$ with 2-dimensional fibers. The Lie derivatives of objects of all considered types, with respect to a vector field $X:M\to TM$, are well-defined without making any special assumption about $X$, and fulfil…
It has been shown that, for all dimensions and signatures, the most general first-order linear symmetry operators for the Dirac equation including interaction with Maxwell field in curved background are given in terms of Killing-Yano (KY) forms. As a general gauge invariant condition it is found that among all KY-forms of the underlying (pseudo) Riemannian manifold, only those which Clifford commute with the Maxwell field take part in the symmetry operator. It is also proved that associated with each KY-form taking part in the symmetry operator, one can define a quadratic function of velocitie…
We present the manifestly covariant Lagrangian of a massless polarized particle, that implies all dynamic and algebraic equations as the conditions of extreme of this variational problem. The model allows for minimal interaction with a gravitational field, leading to the equations, coinciding with Maxwell equations in the geometrical optics approximation. The model allows also a wide class of non minimal interactions, which suggests an alternative way to study the electromagnetic radiation beyond the leading order of geometrical optics. As a specific example, we construct a curvature-dependent…
Many important theories in modern physics can be stated using differential geometry. Symplectic geometry is the natural framework to deal with autonomous Hamiltonian mechanics. This admits several generalizations for nonautonomous systems, both regular and singular. Some of these extensions are the subject of this thesis. Recently there has been a growing interest in studying dissipative mechanical systems from a geometric perspective using contact geometry. In this thesis we review what has been done in this topic and go deeper, studying symmetries and dissipated quantities of contact systems…
In the Relativistic Quantum Geometry (RQG) formalism recently introduced, was explored the possibility that the variation of the tensor metric can be done in a Weylian integrable manifold using a geometric displacement, from a Riemannian to a Weylian integrable manifold, described by the dynamics of an auxiliary geometrical scalar field $θ$, in order that the Einstein tensor (and the Einstein equations) can be represented on a Weyl-like manifold. In this framework we study jointly the dynamics of electromagnetic fields produced by quantum complex vector fields, which describes charges without …
