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

Division algebras have demonstrated their utility in studying non-associative algebras and their connection to the Standard Model through complex Clifford algebras. This article focuses on exploring the connection between these complex Clifford algebras and their corresponding real Clifford algebras providing insight into geometric properties of bivector gauge symmetries. We first generate gauge symmetries in the complex Clifford algebra through a general Witt decomposition. ...

This is a short review of the algebraic properties of Clifford algebras and spinors. Their use in the description of fundamental physics (elementary particles) is also summarized. Lecture given at the ICCA7 conference, Toulouse (23/05/2005)

We investigate a model in which spinors are considered as being embedded within the Clifford algebra that operates on them. In Minkowski space $M_{1,3}$, we have four independent 4-component spinors, each living in a different minimal left ideal of $Cl(1,3)$. We show that under space inversion, a spinor of one left ideal transforms into a spinor of another left ideal. This brings novel insight to the role of chirality in weak interactions. We demonstrate the latter role by co...

In this note, we speculate about the fundamental role being played by the $SO(8)$ group representations displaying the triality structure that necessarily arise in models constructed under the free fermionic methodology as being remnants of the higher-dimensional triality algebra tri$(\mathbb{O}) = \mathfrak{so}(8)$.

In this paper, theory and construction of spinor representations of real Clifford algebras $\cl_{p,q}$ in minimal left ideals are reviewed. Connection with a general theory of semisimple rings is shown. The actual computations can be found in, for example, [2].

We consider aspects of the noncommutative approach to the standard model based on the spectral action principle. We show that as a consequence of the incorporation of the Clifford structures in the formalism, the spectral action contains an extended scalar sector, with respect to the minimal Standard Model. This may have interesting phenomenological consequences. Some of these new scalar fields carry both weak isospin and colour indexes. We calculate the new terms in spectral...

This paper is meant to be an informative introduction to spinor representations of Clifford algebras. In this paper we will have a look at Clifford algebras and the octonion algebra. We begin the paper looking at the quaternion algebra $\mathbb{H}$ and basic properties that relate Clifford algebras and the well know Pin and Spin groups. We then will look at generalized spinor representations of Clifford algebras, along with many examples. We conclude the paper looking at the ...

We prove in this paper that the elliptic $R$--matrix of the eight vertex free fermion model is the intertwiner $R$--matrix of a quantum deformed Clifford--Hopf algebra. This algebra is constructed by affinization of a quantum Hopf deformation of the Clifford algebra.

It is shown how the old Cartan's conjecture on the fundamental role of the geometry of simple (or pure) spinors, as bilinearly underlying euclidean geometry, may be extended also to quantum mechanics of fermions (in first quantization), however in compact momentum spaces, bilinearly constructed with spinors, with signatures unambiguously resulting from the construction, up to sixteen component Majorana-Weyl spinors associated with the real Clifford algebra $\Cl(1,9)$, where, ...

This article presents the description of the internal spaces of fermion and boson fields in $d$-dimensional spaces, with the odd and even "basis vectors" which are the superposition of odd and even products of operators $\gamma^a$. While the Clifford odd "basis vectors" manifest properties of fermion fields, appearing in families, the Clifford even "basis vectors" demonstrate properties of the corresponding gauge fields. In $d\ge (13+1)$ the corresponding creation operators m...