**Authors:** Carlos Castro

We construct the Extended Relativity Theory in Born-Clifford-Phase spaces with an
upper R and lower length λ scales (infrared/ultraviolet cutoff ). The invariance symmetry
leads naturally to the real Clifford algebra Cl(2, 6,R) and complexified Clifford
Cl_{C}(4) algebra related to Twistors. A unified theory of all Noncommutative branes in
Clifford-spaces is developed based on the Moyal-Yang star product deformation quantization
whose deformation parameter involves the lower/upper scale (ℏλ/R). Previous
work led us to show from first principles why the observed value of the vacuum
energy density (cosmological constant ) is given by a geometric mean relationship
ρ ~ L^{-2}_{Planck}R^{-2} = L^{-4}_{P}(L_{Planck}/R)^{2} ~ 10^{-122}M^{4}_{Planck},
and can be obtained when the infrared
scale R is set to be of the order of the present value of the Hubble radius. We
proceed with an extensive review of Smith's 8D model based on the Clifford algebra
Cl(1,7) that reproduces at low energies the physics of the Standard Model and Gravity,
including the derivation of all the coupling constants, particle masses, mixing angles,
....with high precision. Geometric actions are presented like the Clifford-Space extension
of Maxwell's Electrodynamics, and Brandt's action related to the 8D spacetime tangentbundle
involving coordinates and velocities ( Finsler geometries ). Finally we outline
the reasons why a Clifford-Space Geometric Unification of all forces is a very reasonable
avenue to consider and propose an Einstein-Hilbert type action in Clifford-Phase spaces
(associated with the 8D Phase space) as a Unified Field theory action candidate that
should reproduce the physics of the Standard Model plus Gravity in the low energy limit.

**Comments:** 61 pages, This article appeared in Foundations of Physics vol 35, no.6 (2005) 971.

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[v1] 28 Aug 2009

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