Cartan's Soldered Spaces and Conservation Laws in Physics

Modern Physics Letters A, 2000

The gauge symmetry inherent in the concept of manifold has been discussed. Within the scope of this symmetry the linear connection or displacement field can be considered as a natural gauge field on the manifold. The gauge-invariant equations for the displacement field have been derived. It has been shown that the energy–momentum tensor of this field conserves and hence the displacement field can be treated as one that transports energy and gravitates. To show the existence of the solutions of the field equations, we have derived the general form of the displacement field in Minkowski space–time which is invariant under rotation and space and time inversion. With this ansatz we found spherically-symmetric solutions of the equations in question.

2008

Using a nonlocal field transformation for the gauge field known as Cho-Faddeev-Niemi-Shabanov decomposition as well as ideas taken from generalized integrability, we derive a new family of infinitely many conserved currents in the self-dual sector of SU (2) Yang-Mills theory. These currents may be related to the area preserving diffeomorphisms on the reduced target space. The calculations are performed in a completely covariant manner and, therefore, can be applied to the self-dual equations in any space-time dimension with arbitrary signature.

Classical and Quantum Gravity, 1992

The conservation laws are examined from the transformation properties of the Lagrangian. The energymomentum complex obtained has mixed indices, T", whereas a symmetric quantity V" is required for the definition of angular momentum. Such a symmetric quantity has been constructed by Landau and Lifshitz. In the course of examining the relationship between these quantities, two hierarchies of complexes T'(")""and g"(")t"" are constructed. Under linear coordinate transformations the former are tensor densities of weight (I+1) and the latter of weight (++2).For I=0 these reduce to the canonical T""and the Landau-Lifshitz K"', respectively. By requiring the energy-momentum complex to generate the coordinate transformations, and the total energy and momentum to form a free vector, one can identify the canonical complex T"' as the appropriate quantity to describe the energy and momentum of the field plus matter. Similarly, by requiring the total angular momentum to behave as a free antisymmetric tensor, one can construct, in the usual manner, an appropriate quantity from T& 1)"".The angular momentum complex so defined differs from that proposed by Landau and Lifshitz as well as from an independent construction by Bergmann and Thomson.

2006

In this paper using a Clifford bundle formalism we examine (a): the strong conditions for existence of conservation laws involving only the energy-momentum and angular momentum of the matter fields on a general Riemann-Cartan spacetime and also in the particular cases of Lorentzian and teleparallel spacetimes and (b): the conditions for the existence of conservation laws of energy-momentum and angular momentum for the matter and gravitational fields when this latter concept can be rigorously defined. We examine in details some misleading and even erroneous and often quoted statements concerning the issues of the conservation laws in General Relativity and Riemann-Cartan (including the particular case of the teleparallel one) theories.

Annals of Physics, 2013

Gravity is commonly thought of as one of the four force fields in nature. However, in standard formulations its mathematical structure is rather different from the Yang-Mills fields of particle physics that govern the electromagnetic, weak, and strong interactions. This paper explores this dissonance with particular focus on how gravity couples to matter from the perspective of the Cartan-geometric formulation of gravity. There the gravitational field is represented by a pair of variables: 1) a 'contact vector' V A which is geometrically visualized as the contact point between the spacetime manifold and a model spacetime being 'rolled' on top of it, and 2) a gauge connection A AB µ , here taken to be valued in the Lie algebra of SO(2, 3) or SO(1, 4), which mathematically determines how much the model spacetime is rotated when rolled. By insisting on two principles, the gauge principle and polynomial simplicity, we shall show how one can reformulate matter field actions in a way that is harmonious with Cartan's geometric construction. This yields a formulation of all matter fields in terms of first order partial differential equations. We show in detail how the standard second order formulation can be recovered. Furthermore, the energy-momentum and spin-density three-forms are naturally combined into a single object here denoted the spin-energy-momentum three-form. Finally, we highlight a peculiarity in the mathematical structure of our first-order formulation of Yang-Mills fields. This suggests a way to unify a U (1) gauge field with gravity into a SO(1, 5)-valued gauge field using a natural generalization of Cartan geometry in which the larger symmetry group is spontaneously broken down to SO(1, 3) × U (1). The coupling of this unified theory to matter fields and possible extensions to non-Abelian gauge fields are left as open questions.

International Journal of Geometric Methods in Modern Physics

In this paper, we explore some new off-shell and on-shell conserved quantities for a scalar field in Minkowski space, using integrability condition. The off-shell conserved tensors are related to the kinematics of the field, while a linear combination of the off-shell and the on-shell conserved tensors ends up with the energy–momentum tensor for the scalar field. In the curved background, using Ricci and Bianchi identities, scalar–tensor theory of gravity, that resembles somewhat with Brans–Dicke-type field equations, emerges without requiring the principle of equivalence. Further, starting from the curvature scalar and using these identities, the field equations for modified gravity (Einstein–Hilbert action in the presence of higher-order terms) follow.

International Journal of Modern Physics D, 2016

This review is dedicated to some modern applications of the remarkable paper written in 1918 by E. Noether. On a single paper, Noether discovered the crucial relation between symmetries and conserved charges as well as the impact of gauge symmetries on the equations of motion. Almost a century has gone since the publication of this work and its applications have permeated modern physics. Our focus will be on some examples that have appeared recently in the literature. This review aims at students, not researchers. The main three topics discussed are (i) global symmetries and conserved charges (ii) local symmetries and gauge structure of a theory (iii) boundary conditions and algebra of asymptotic symmetries. All three topics are discussed through examples.

Physical Review D, 2012

Despite the fact that the integral form of the equations of classical electrodynamics is well known, the same is not true for non-abelian gauge theories. The aim of the present paper is threefold. First, we present the integral form of the classical Yang-Mills equations in the presence of sources, and then use it to solve the long standing problem of constructing conserved charges, for any field configuration, which are invariant under general gauge transformations and not only under transformations that go to a constant at spatial infinity. The construction is based on concepts in loop spaces and on a generalization of the non-abelian Stokes theorem for two-form connections. The third goal of the paper is to present the integral form of the self dual Yangs-Mills equations, and calculate the conserved charges associated to them. The charges are explicitly evaluated for the cases of monopoles, dyons, instantons and merons, and we show that in many cases those charges must be quantized. Our results are important in the understanding of global properties of non-abelian gauge theories. 1

International Journal of Modern Physics A, 1995

We investigate the possible form of local translation-invariant conservation laws associated with the relativistic field equations [Formula: see text] for a multicomponent field ϕ. Under the assumptions that (i) the vi' s can be expressed as linear combinations of partial derivatives ∂wj / ∂ϕk of a set of functions wj (ϕ), (ii) the space of functions spanned by the wj' s is closed under partial derivations, and (iii) the fields ϕ take values in a simply connected space, the local conservation laws can be transformed either to the form [Formula: see text] (where [Formula: see text] and [Formula: see text] are homogeneous polynomials in the variables [Formula: see text]), or to the parity-transformed version of this expression, [Formula: see text].

arXiv (Cornell University), 2022

The conventional Rosenfeld-Bergmann-Dirac constrained Hamiltonian algorithm applied to Einstein-Yang-Mills theory is shown to be equivalent to a local gauge theoretic extension of Cartan's invariant integral approach to classical mechanics. In addition, the Hamiltonian generators of Legendreprojectable spacetime diffeomorphism and gauge symmetries are derived directly as vanishing Noether charges. This leads directly to their interpretation as delivering the correct symmetry variations of both configuration and momentum field variables.