Minimal spin Gauge theory

Abstract

Purpose of a minimal theory is to achief most with least. Least may be for example the spacetime algebra. But the symmetric unitary group SU(3) is nota part of any real Clifford algebra of 4-dimensional space, especially not of the algebra C11,3 of the Minkowski spacetime, nor of the algebra Cl3,~ in the opposite metric. Therefore we can ask how quantumchromodynamics enters into the theory. A ¡ answer is that the group SU (3) is an object of both the complexified algebras C | Cll,3 and C|To show this we first define six color spaces which ate spanned by conjugate triples of commuting base elements. These contain the six idempotent lattices that can be located in Cl3,l. Their images exist in both (J | Cll,3 and C | Cl3,1. Further in each color space there is defined ah octahedral orientation stabilizer group which fixates one lepton and color rotates the states in its quark family. Thus quantum numbers of strong interacting fields such as isospin, charge, hypercharge and color turn out as geometric properties. Next we ask if the axtificialty of complexification can be avoided. The answer is yes. Defining the class of Clifford algebras with proper imaginary unit it turns out that Cll,s and Cls.~ do not belong to this class. But C14,1 and C11,6 do. It is shown that in the latter algebra the whole color space Ansatz can be established and the generators of SU(3) represented most naturally and without complexification. That the proposed theory becomes a physically true statement requires that there exists a non rank preserving freedom of motion within the constituents of primitive idempotents, that is, transpositions among conjugate triples in color space.

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

“…These matrices are diagonal 8 × 8 block matrices due to the isomorphism Cℓ + (8) ∼ = Cℓ(7) ∼ = C(8) ⊕ C(8). This means that the left and right adjoint actions of Cℓ( 8) are a straightforward direct sum extension 13 of the Cℓ(6) left and right adjoint actions in [1],…”

“…) 12 The action is simply left matrix multiplication by ρ(J 3 ). 13 Apart from the action of ρ(J i ) which are not direct sum extensions.…”

“…In the 1970s the octonions were related to the color symmetries of a generation of quarks [10,11]. More recently there has been a revived interest in using division algebras to construct a theoretical basis for the SM gauge groups and the observed particle spectrum [12,13,14,15,16,17,18,19,20,21,22,23]. For example, in [17], the algebra Cℓ (6), that also forms the basis of [1], is derived from the left adjoint actions of the complex octonions C ⊗ O on themselves.…”