Phenomenological QED model in the minimally complete Geometric Representation of Clifford Algebra - What is the Gauge Group?*

In geometric representation of Clifford algebra, Pauli matrices are the basis vectors of 3D space, Dirac those of 4D spacetime. Vacuum wavefunction is comprised of the Pauli algebra's 8 components - 1 scalar point, 3 vector lines (orientational degrees-of-freedom), 3 bivector areas, and 1 trivector volume. Interaction is modeled by the Clifford product, sum of dimension-reducing dot and increasing wedge products. The 8 wavefunction components one might take to be E8 root vectors, gauge group generators via the Clifford product.

The 4 fundamental constants (electric charge quantum e, Planck's angular momentum quantum h, speed of light c, and magnetic permeability of space μ) that define the QED coupling constant α permit assigning unique geometrically and topologically appropriate electric and magnetic flux quanta to each of the 8 vacuum wavefunction components. This removes the 3-fold degeneracy of vector and bivector, and increases dimensionality to 10D. Time emerges from interactions, via wedge products' dimension-increasing property. Taking quantum interaction impedances of the modes to be components of the 8 x 8 x 8 S-matrix makes it the gauge group as well. Do we call this E8 x E8 x E8?

In physical spacetime the S-matrix has 6 degrees-of-freedom, 3 each space and phase. Each orientation requires its own phase. Time is the integral of phase, collapses 6D phase space to 4D spacetime. Do we call the physical S-matrix E6, the largest possible unification group according to Georgi? Is Weinberg's preference for E8 x E8 compacted to E6 realized in this approach?