Transposition in Clifford Algebra: SU(3) from Reorientation Invariance

Abstract

Recoding base elements in a spacetime algebra is an act of cognition. But at the same time this act refers to the process of nature. That is, the internal in-teractions with their standard symmetries reconstruct the orientation of spacetime. This can best be represented in the Clifford algebra CC3,1 of the Minkowski space-time. Recoding is carried out by the involutive automorphism of transposition. The set of transpositions of erzeugende Einheiten (primitive idempotents), as Hermann Weyl called them, generates a finite group: the reorientation group of the Clifford algebra. Invariance of physics laws with respect to recoding is not a mere matter of computing, but one of physics. One is able to derive multiplets of strong in-teracting matter from the recoding invariance of CC3,1 alone. So the SU(3) flavor symmetry essentially turns out to be a spacetime group. The original quark multi-plet independently found by Gell-Mann and Zweig is reconstructed from Clifford algebraic eigenvalue equations of isospin, hypercharge, charge, baryon number and flavors treated as geometric operators. Proofs are given by constructing six possible commutative color spinor spaces Chχ, or color tetrads, in the noncommutative ge-ometry of the Clifford algebra CC3,1. Calculations are carried out with CLIFFORD, Maple V package for Clifford algebra computations. Color spinor spaces are isomor-phic with the quaternary ring 4 R = R ⊕ R ⊕ R ⊕ R. Thus, the differential (Dirac) operator takes a very handsome form and equations of motion can be handled easily. Surprisingly, elements of CC3,1 representing generators of SU(3) bring forth (1) the well-known grade-preserving transformations of the Lorentz group together with (2) the heterodimensional Lorentz transformations, as Jose Vargas denoted them: Lorentz transformations of inhomogeneous differential forms. That is, trigonal tetra-hedral rotations do not preserve the grade of a multivector but instead, they permute the base elements of the color tetrad having grades 0, 1, 2 and 3. In the present model the elements of each color space are exploited to reconstruct the flavor SU(3) such that each single commutative space contains three flavors and one color. Clearly, the six color spaces do not commute, and color rotations act in the noncommutative ge-ometry. To give you a picture: Euclidean space with its reorientation group, i.e., the This work is an outgrowth of a paper presented at the