Mathematical Tools to Understand the Field Theories of the Standard Model and Beyond

Since Isaac Newton the understanding of the physical world is more and more complex. The Euclidean space of three dimensions , independent of time is replaced in Enstein’s vision by the Lorentzian space-time at first, then by four dimensions manifold to unify space and matter. String theorists add to space more dimensions to make their theory consistent. Complex topological invariants which characterize different kind of spaces are developed. Space is discretized at the quantum scale in the loop quantum gravity theory. A non-commutative and spectral geometry is defined from the theory of operator algebra by Alain Connes. In this review, our goal is to enumerate different approaches implementing algebra and topology in order to understand the standard model of particles and beyond

Generalized Holographic Principle, Gauge Invariance and the Emergence of Gravity à la Wilczek

We show that a generalized version of the holographic principle can be derived from the Hamiltonian description of information flow within a quantum system that maintains a separable state. We then show that this generalized holographic principle entails a general principle of gauge invariance. When this is realized in an ambient Lorentzian space-time, gauge invariance under the Poincaré group is immediately achieved. We apply this pathway to retrieve the action of gravity. The latter is cast à la Wilczek through a similar formulation derived by MacDowell and Mansouri, which involves the representation theory of the Lie groups SO(3,2) and SO(4,1).

Chen-like inequalities on null hypersurfaces with closed rigging of a Lorentzian manifold

We study null hypersurfaces of a Lorentzian manifold with a closed rigging for the hypersurface. We derive inequalities involving Ricci tensors, scalar curvature, squared mean curvatures for a null hypersurface with a closed rigging of a Lorentzian space form and for a screen homothetic null hypersurface of a Lorentzian manifold. We also establish a generalized Chen–Ricci inequality for a screen homothetic null hypersurface of a Lorentzian manifold with a closed rigging for the hypersurface.

Spherical kinematics in 3-dimensional generalized space

In this study, we obtained generalized Cayley formula, Rodrigues equation and Euler parameters of an orthogonal matrix in 3-dimensional generalized space [Formula: see text]. It is shown that unit generalized quaternion, which is defined by the generalized Euler parameters, corresponds to a rotation in [Formula: see text] space.We found that the rotation in matrix equation forms using matrix form of the generalized quaternion product. Besides, in [Formula: see text] space, we obtained the rotations determined by the unit quaternions and unit split quaternions, which are special cases of generalized quaternions for [Formula: see text] in 3-dimensional Eulidean space [Formula: see text] in 3-dimensional Lorentzian space [Formula: see text] respectively.

A new approach to the bienergy and biangle of a moving particle lying in a surface of lorentzian space

In this research, we study bienergy and biangles of moving particles lying on the surface of Lorentzian 3-space by using their energy and angle values. We present the geometrical characterization of bienergy of the particle in Darboux vector fields depending on surface. We also give the relationship between bienergy of the surface curve and bienergy of the elastic surface curve. We conclude the paper by providing bienergy-curve graphics for different cases.

Slant Curves in Contact Lorentzian Manifolds with CR Structures

In this paper, we first find the properties of the generalized Tanaka–Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the ∇ ^ -geodesic is a magnetic curve (for ∇) along slant curves. Finally, we prove that when c ≤ 0 , there does not exist a non-geodesic slant Frenet curve satisfying the ∇ ^ -Jacobi equations for the ∇ ^ -geodesic vector fields in M. Thus, we construct the explicit parametric equations of pseudo-Hermitian pseudo-helices in Lorentzian space forms M 1 3 ( H ^ ) for H ^ = 2 c > 0 .

Wavefronts of traveling trajectories of geometric particles

Confining the traveling trajectory of a tachyon to the two-dimensional Lorentzian space forms, we describe the trajectory as a spacelike front in these Lorentzian space forms. Introducing the differential geometry of singular curves in Lorentzian space forms, that is, the hyperbolic space and de Sitter space, and applying the Legendrian duality theorems, we establish the moving frame along the front, whereby the definitions of the evolutes of spacelike fronts in Lorentzian space forms are presented and the geometric properties of these evolutes are investigated in detail. It is shown that these evolutes can be interpreted as wavefronts under the viewpoint of Legendrian singularity theory.

Tetrads in SU(N) Yang–Mills geometrodynamics

The discovery of the [Formula: see text] symmetry was fundamental as to establishing an ordering principle in particle physics. We already studied how to couple the [Formula: see text] symmetry to the gravitational field in four-dimensional curved Lorentzian space–times. The multiplets of equal quantum numbers are translated through natural elements in Riemannian geometry into local multiplets of equal gravitational field. As quark physics developed since in the 1970s, it was necessary to incorporate new symmetries to the models, that ensued in the incorporation of new quantum numbers like charm, for example, charm is an additive quantum number like isospin [Formula: see text] and hypercharge [Formula: see text] and the standard [Formula: see text] diagrams were extended onto another third axis. Then, instead of the fundamental triplet, we have a quartet [Formula: see text] as the smallest representation of the symmetry group, leading to the introduction of [Formula: see text] as the new group of symmetries. In this paper, we will not restrict ourselves exclusively to the symmetry group [Formula: see text] and we will set out to analyze the coupling of the [Formula: see text] symmetry to the gravitational field. To this end, new tetrads will be introduced as we did for the [Formula: see text] case. These tetrads have outstanding properties that enable these constructions. New theorems will be proved regarding the isomorphic nature of these local symmetry gauge groups and tensor products of groups of local tetrad transformations. This is a paper about grand field unification in four-dimensional curved Lorentzian space–times.