The $C\ell(8)$ algebra of three fermion generations with spin and full internal symmetries

Supergroups are defined in the framework of $\dZ_2$-graded Clifford algebras over the fields of real and complex numbers, respectively. It is shown that cyclic structures of complex and real supergroups are defined by Brauer-Wall groups related with the modulo 2 and modulo 8 periodicities of the complex and real Clifford algebras. Particle (fermionic and bosonic) representations of a universal covering (spinor group $\spin_+(1,3)$) of the proper orthochronous Lorentz group ar...

The aim of this paper is to find generating sets of commuting involutions and use them to explicitly construct minimal representations of Clifford algebras $Cl(n)_{p,q}$. By results of [HL] and [LW], we know the dimension of such minimal representations, which is linked to the maximal number of commuting involutions in the algebra, dependent only on $p$ and $q$. We provide an algorithm to construct these generating sets of involutions explicitly for all Clifford algebras $Cl(...

To describe external and internal attributes of fundamental fermions, a theory of multi-spinor fields is developed on an algebra, a {\it triplet algebra}, which consists of all the triple-direct-products of Dirac \gamma-matrices. The triplet algebra is decomposed into the product of two subalgebras, an external algebra and an internal algebra, which are exclusively related with external and internal characteristic of the multi-spinor field named {\it triplet fields}. All elem...

The Cartan's equations definig simple spinors (renamed pure by C. Chevalley) are interpreted as equations of motion in momentum spaces, in a constructive approach in which at each step the dimesions of spinor space are doubled while those momentum space increased by two. The construction is possible only in the frame of geometry of simple or pure spinors, which imposes contraint equations on spinors with more than four components, and the momentum spaces result compact, isomo...

This paper is the second one of a series of three and it is the continuation of math-ph/0412074. We review some properties of the algebraic spinors in Cl(3,0) and Cl(0,3) and how Weyl, Pauli and Dirac spinors are constructed in Cl(3,0) (and Cl(0,3) in the case of Weyl spinors. A plane wave solution for the Dirac equation is obtained, and the Dirac equation is written in terms of Weyl spinors, and alternatively, in terms of Pauli spinors. Finally the covariant and contravarian...

The internal degrees of freedom of fermions are in the spin-charge-family theory described by the Clifford algebra objects, which are superposition of an odd number of $\gamma^a$'s. Arranged into irreducible representations of "eigenvectors" of the Cartan subalgebra of the Lorentz algebra $S^{ab}$ $(= \frac{i}{2} \gamma^a \gamma^b|_{a \ne b})$ these objects form $2^{\frac{d}{2}-1}$ families with $2^{\frac{d}{2}-1}$ family members each. Family members of each family offer the ...

It is shown that the generators of Clifford algebras behave as creation and annihilation operators for fermions and bosons. They can create extended objects, such as strings and branes, and can induce curved metric of our spacetime. At a fixed point, we consider the Clifford algebra $Cl(8)$ of the 8-dimensional phase space, and show that one quarter of the basis elements of $Cl(8)$ can represent all known particles of the first generation of the Standard model, whereas the ot...

Despite its tremendous success, the Standard Model of particle physics does not explain why the weak interaction breaks chiral symmetry. Various unified theories got us closer to an answer, but too often the explanation consists of labeling the $\operatorname{SU}(2)_w$ singlet representations as right-handed, and the doublet ones as left handed. This by itself does not ensure a chiral preference, because chirality itself, arising in the Dirac spinors, is not a property of the...

In the current version of QCD the quarks are described by ordinary Dirac fields, organized in the following internal symmetry multiplets: the $SU(3)$ colour, the $SU(2)$ flavour, and broken $SU(3)$ providing the family triplets. \noindent In this paper we argue that internal and external (i.e. space-time) symmetries are entangled at least in the colour sector in order to introduce the spinorial quark fields in a way providing all the internal quark's degrees of freedom which ...

It is a well known fact from the group theory that irreducible tensor representations of classical groups are suitably characterized by irreducible representations of the symmetric groups. However, due to their different nature, vector and spinor representations are only connected and not united in such description. Clifford algebras are an ideal tool with which to describe symmetries of multi-particle systems since they contain spinor and vector representations within the ...