Minimal spin Gauge theory

Purpose of a minimal theory is to achief most with least. Least may be for example the spacetime algebra. But the symmetric unitary groupSU(3) is not a part of any real Clifford algebra of 4-dimensional space, especially not of the algebraCl _1,3 of the Minkowski spacetime, nor of the algebraCl _3,1 in the opposite metric. Therefore we can ask how quantumchromodynamics enters into the theory. A first answer is that the groupSU(3) is an object of both the complexified algebras C ⊕Cl _1,3 and C⊕Cl _3,1. To show this we first define six color spaces which are spanned by conjugate triples of commuting base elements. These contain the six idempotent lattices that can be located inCl _3,1. Their images exist in both C⊕Cl _1,3 and C⊕Cl _3,1. Further in each color space there is defined an 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 artificialty of complexification can be avoided. The answer is yes. Defining the class of Clifford algebras with proper imaginary unit it turns out thatCl _1,3 andCl _3,1 do not belong to this class. ButCl _4,1 andCl _1,6 do. It is shown that in the latter algebra the whole color space Ansatz can be established and the generators ofSU(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.