On the Unification of Interactions by Clifford Algebra

Abstract

A theory based on the concept of 16-dimensional Clifford space C, a manifold whose tangent space is Clifford algebra, is investigated. The elements of the space C are oriented r-volumes, r = 0, 1, 2, 3, associated with extended objects such as strings and branes. Although the latter objects form an infinite dimensional configuration space, they can be sampled in terms of a finite dimensional subspace, namely, the Clifford space. The connection and the curvature of C describe what, from the point of view of 4-dimensional spacetime, appear as gravitation and Yang-Mills gauge fields and field strengths. This is demonstrated on the case of a Clifford algebra valued field Ψ(X) which depends on position X in C and satisfies the generalized Dirac equation. Mathematics Subject Classification (2000). Primary 15A66, 83E30; Secondary 11E88, 83E15, 81R99.

Probing Clifford Algebras Through Spin Groups: A Standard Model Perspective

“…The emergence of the Pati-Salam group resembles aspects found in Kaluza-Klein theories and unification models. The connection explored in this article between a real Clifford algebra and its Complex Clifford algebra through the Spin groups and general Witt decomposition will aid in understanding how an operator based Clifford action [33][34][35][36] for example can be translated into its bivector field theory representation form [37][38][39].This will allow the exploration of extra dimensional aspects of Cl(6, 0) and Cl(4, 0) as a bivector field theory. Understanding the implication that our Cl(1, 4) left and right projection operator has on Cl(1, 4) and whether one can find a dual ladder description by compactifying the additional dimension through its complex structure are worth studying.…”

“…We can express a spinor as Ψ = ψ α Ω α where Ω α represent the vacuum ideal and its conjugate. This can be useful in the study of Spin gauge fields as described in references [35,36]. In our last sections we discovered that the correct gauge symmetry over R is unique for a given D dimensional space we are interested in.…”

“…The algebra at hand can be questioned given that color gauge symmetry does not exhibit chiral behavior, not allowing one to compute a projection operator such as Eq. (36). From a represenation perspective there still lies the questions of having leptons in our particle spectrum when considering pure SU (3) gauge theory.…”

“…Clifford algebras provide the structure and versatility to probe ties between Spin groups and their relation to the Standard Model, leading to a deeper grasp pertaining to the intimate constructs relating geometry, symmetries and fundamental interactions. It will provide the pathway to probe a complex Clifford spinor action as those explored in [35,36], and having a foundational understanding of how one might translate the theory into its bivector counterpart applying field theoretical techniques appropriate to Clifford algebras [37][38][39], in addition to studying the connection between gauge groups and complex algebras.…”