Gauge invariant method for maximum simplification of the field strength in non-Abelian Yang-Mills theories

A new local gauge invariant method is introduced in order to maximally simplify the expression for a SU(2) non-Abelian field strength. The new tetrads introduced in previous works are going to play a fundamental role in the algorithm presented in this manuscript. Three new local gauge invariant objects are going to guide us through the process of making a field strength block diagonal. The process is also covariant. Any non-trivial isospace field strength projection will become block diagonal through this algorithm. Along with the local gauge invariant method already developed in order to diagonalize the stress-energy tensor, we have with this new local gauge invariant method to maximally simplify the field strength, a new gauge invariant method to classify Yang-Mills field theories.Comment: 17 pages, no figures. arXiv admin note: substantial text overlap with arXiv:gr-qc/060204

Second proof for SU(2) isomorphic to LB1 X LB1 X LB1 theorem

In this note we present a second independent proof for the theorem introduced previously that establishes an isomorphism between SU(2) and LB1 X LB1 X LB1. Since the local groups LB1 and LB2 are isomorphic, it was also previously proved a similar result for LB2 X LB2 X LB2. We are going to reverse the three sets of tetrads that are going to be used in order to prove this new version. Instead of choosing three SU(2) different tetrads keeping the electromagnetic tetrad the same for all three sets of SU(2) tetrads, we are going to keep fixed the SU(2) tetrad, that is, we are going to pick just one SU(2) tetrad but choose three different arbitrary but fixed electromagnetic tetrads in order to gauge locally the only local SU(2) tetrad involved in our new version of this theorem. The same result will be obtained via an alternative but equivalent way.Comment: A new and straightforward version of the theorem presented in "Tetrads in Yang-Mills geometrodynamics

Dynamical symmetry breaking in non-Abelian geometrodynamics

We are going to analyze through a first order perturbative formulation the local loss of symmetry when a source of non-Abelian Yang-Mills and gravitational fields interacts with an external agent that perturbes the original geometry associated to the source. Then, as the symmetry in Abelian and non-Abelian field structures in four-dimensional Lorentzian spacetimes is displayed through the existence of local orthogonal planes of symmetry that we previously called blades one and two, the loss of symmetry will be manifested by the tilting of these planes under the influence of the external agent. It was found already that there is an algorithm to block diagonalize the Yang-Mills field strength isospace projections in a local gauge invariant way. Independently, it was also found an algorithm to diagonalize the Yang-Mills stress-energy tensor in a gauge invariant way. Using these results and perturbative analysis from a previous manuscript dealing with the Abelian case, we are going to demonstrate how to develop an algorithm for constructing local energy-momentum conserved currents inside both local orthogonal planes. As the interaction proceeds, the planes are going to tilt perturbatively, and in this strict sense the original local symmetries will be lost. But we will prove that the new blades at the same point will correspond after the tilting generated by perturbation, to new symmetries, with associated new local currents, both on each new local planes of symmetry. Old symmetries will be broken, new will arise. There will be a local symmetry evolution in the non-Abelian case as well.Comment: arXiv admin note: substantial text overlap with arXiv:1306.0602, arXiv:1306.2174, arXiv:gr-qc/060204

Dynamical symmetry breaking in geometrodynamics

We are going to analyze through a first order perturbative formulation the local loss of symmetry when a source of electromagnetic and gravitational field interacts with an agent that perturbes the original geometry associated to the source. As the symmetry in Abelian or even non-Abelian field structures in four dimensional Lorentzian spacetimes is displayed through the existence of local planes of symmetry that we previously called blades one and two, the loss of symmetry will be manifested by the tilting of these planes under the influence of the external agent. In this strict sense the original local symmetry will be lost. But we will prove that the new blades at the same point will correspond after the tilting generated by perturbation to a new symmetry. The point of this note is to prove that the geometrical manifestation of local gauge symmetries is dynamic. The local original symmetries will be lost, nonetheless new symmetries will arise. There is a dynamic evolution of local symmetries.Comment: Several modifications and corrections have been implemented in sections (3) and (7

The equivalence between local inertial frames and electromagnetic gauge in Einstein-Maxwell theories

We are going to prove that locally the inertial frames and gauge states of the electromagnetic field are equivalent. This proof will be valid for Einstein-Maxwell theories in four-dimensional Lorentzian spacetimes. Use will be made of theorems proved in a previous manuscript. These theorems state that locally the group of electromagnetic gauge transformations is isomorphic to the local Lorentz transformations of a special set of tetrad vectors. The tetrad that locally and covariantly diagonalizes any non-null electromagnetic stress-energy tensor. Two isomorphisms, one for each plane defined locally by two separate sets of two vectors each. In particular, we are going to use the plane defined by the timelike and one spacelike vector, plane or blade one. These results will be extended to any tetrad that results in a local Lorentz transformation of the special tetrad that locally and covariantly diagonalizes the stress-energy tensor.Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1306.2174, arXiv:1306.0602, arXiv:1306.5784, arXiv:1306.400

Integrability of dual coactions on Fell bundle C*-algebras

We study integrability for coactions of locally compact groups. For abelian groups, this corresponds to integrability of the associated action of the Pontrjagin dual group. The theory of integrable group actions has been previously studied by Ruy Exel, Ralf Meyer and Marc Rieffel. Our goal is to study the close relationship between integrable group coactions and Fell bundles. As a main result, we prove that dual coactions on C*-algebras of Fell bundles are integrable, generalizing results by Ruy Exel for abelian groups.Comment: 24 page

Tetrads in Yang-Mills geometrodynamics

A new set of tetrads is introduced within the framework of SU(2) X U(1) Yang-Mills field theories in four dimensional Lorentz curved spacetimes. Each one of these tetrads diagonalizes separately and explicitly each term of the Yang-Mills stress-energy tensor. Therefore, three pairs of planes also known as blades, can be defined, and make up the underlying geometrical structure, at each point. These tetrad vectors are gauge dependent on one hand, and also in their definition, there is an additional inherent freedom in the choice of two vector fields. In order to get rid of the gauge dependence, another set of tetrads is defined, such that the only choice we have to make is for the two vector fields. A particular choice is made for these two vector fields such that they are gauge dependent, but the transformation properties of these tetrads are analogous to those already known for curved spacetimes where only electromagnetic fields are present. This analogy allows to establish group isomorphisms between the local gauge group SU(2), and the tensor product of the groups of local Lorentz tetrad transformations, either on blade one or blade two. These theorems show explicitly that the local internal groups of transformations are isomorphic to local spacetime groups of transformations. As an example of application of these new tetrads, we exhibit three new gauge invariant objects, and using these objects we show how to diagonalize the Yang-Mills stress-energy tensor in a gauge invariant way.Comment: This new version has a new abstract and the introduction has been modified. There is also a new section, applications. This new section has two subsections: Gauge invariants and Diagonalization of the stress-energy tensor. The conclusion section has also been substantially modified. There is a new appendi

Generalized fixed point algebras for coactions of locally compact quantum groups

We extend the construction of generalized fixed point algebras to the setting of locally compact quantum groups - in the sense of Kustermans and Vaes - following the treatment of Marc Rieffel, Ruy Exel and Ralf Meyer in the group case. We mainly follow Meyer's approach analyzing the constructions in the realm of equivariant Hilbert modules. We generalize the notion of continuous square-integrability, which is exactly what one needs in order to define generalized fixed point algebras. As in the group case, we prove that there is a correspondence between continuously square-integrable Hilbert modules over an equivariant C*-algebra B and Hilbert modules over the reduced crossed product of B by the underlying quantum group. The generalized fixed point algebra always appears as the algebra of compact operators of the associated Hilbert module over the reduced crossed product.Comment: 40 pages; article based on author's doctoral dissertation. Final version to appear in M\"unster Journal of Mathematic

Euler observers in geometrodynamics

Euler observers are a fundamental tool for the study of spacetime evolution. Cauchy surfaces are evolved through the use of hypersurface orthogonal fields and their relationship to coordinate observers, that enable the use of already developed algorithms. In geometrodynamics new tetrad vectors have been introduced with outstanding simplifying properties. We are going to use these already introduced tetrad vectors in the case where we consider a curved four dimensional Lorentzian spacetime with the presence of electromagnetic fields. These Einstein-Maxwell geometries will provide the new tetrad that we are going to use in order to develop an algorithm to produce Cauchy evolution with additional simplifying properties.Comment: 11 pages. arXiv admin note: substantial text overlap with arXiv:1306.0602, arXiv:1306.217

Weakly proper group actions, Mansfield's imprimitivity and twisted Landstad duality

Using the theory of weakly proper actions of locally compact groups recently developed by the authors, we give a unified proof of both reduced and maximal versions of Mansfield's Imprimitivity Theorem and obtain a general version of Landstad's Duality Theorem for twisted group coactions. As one application, we obtain the stabilization trick for arbitrary twisted coactions, showing that every twisted coaction is Morita equivalent to an inflated coaction.Comment: 29 pages. Revised version, to appear in Trans. Amer. Math. So