Three fermion generations with two unbroken gauge symmetries from the complex sedenions

Abstract

We show that three generations of leptons and quarks with unbroken Standard Model gauge symmetry SU (3) c × U (1) em can be described using the algebra of complexified sedenions C ⊗ S. A primitive idempotent is constructed by selecting a special direction, and the action of this projector on the basis of C ⊗ S can be used to uniquely split the algebra into three complex octonion subalgebras C ⊗ O. These subalgebras all share a common quaternionic subalgebra. The left adjoint actions of the 8 C-dimensional C ⊗ O subalgebras on themselves generates three copies of the Clifford algebra C (6). It was previously shown that the minimal left ideals of C (6) describe a single generation of fermions with unbroken SU (3) c × U (1) em gauge symmetry. Extending this construction from C ⊗ O to C ⊗ S naturally leads to a description of exactly three generations.

Sedenions, the Clifford algebra $\mathbb{C}l(8)$, and three fermion generations

“…These representations of the minimal left ideals are invariant under the electrocolor symmetry SU(3) c ⊗U(1) em , with each basis state in the ideals transforming as a specific lepton or quark as indicated by their suggestively labeled complex coefficients. One way of extending this model from a single generation to exactly three generations is to go beyond the division algebras, and consider the next Cayley-Dickson algebra after the octonions [10]. This is the algebra of sedenions S. The sedenions S are spanned by the identity 1 = e 0 and seven anti-commuting square roots of minus one e i satisfying the multiplication table given in Appecndix A of [10].…”

“…One way of extending this model from a single generation to exactly three generations is to go beyond the division algebras, and consider the next Cayley-Dickson algebra after the octonions [10]. This is the algebra of sedenions S. The sedenions S are spanned by the identity 1 = e 0 and seven anti-commuting square roots of minus one e i satisfying the multiplication table given in Appecndix A of [10]. This larger algebra also generates a larger left adjoint algebra (C ⊗ S) L ∼ = C (8).…”

“…One prominent open question in the division algebra program is how to extend the many results from a single generation to exactly three generations. In this short paper some recent results are reviewed which show that by going beyond the division algebras, and including the Cayley-Dickson algebra of sedenions S, it is possible to extend the one generation results of [2] to exactly three generations in a very natural way [10]. Some ongoing research and speculations relating to this approach are then discussed.…”

Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics

“…In case of the K3 surface, one has the intersection form containing the matrix E 8 ⊕E 8 which corresponds to two copies of octonions O × O. Here, there is a link to the recent work [8] of complex sedions. Now, every element of O is related to a surface (unique up to homotopy).…”

“…, O(4), O(8), one can construct the real numbers R, the complex numbers C and the quaternions H, respectively. Interestingly, the triality of the O(8) group reflects the symmetry of the three quaternionic units e i = −iσ i , where σ i are the Pauli matrices.…”