Polyvector-valued Gauge Field Theories and Quantum Mechanics in Noncommutative Clifford Spaces

The basic ideas and results behind polyvector-valued gauge field theories and Quantum Mechanics in Noncommutative Clifford spaces are presented. The star products are noncommutative but associative up to second order only. The construction of Noncommutative Clifford-space gravity as polyvector-valued gauge theories of twisted diffeomorphisms in Clifford-spaces would require quantum Hopf algebraic deformations of Clifford algebras. Clifford algebras are deeply related and essential tools in many aspects in Physics. The Extended Relativity theory in Clifford-spaces ( C-spaces ) is a natural extension of the ordinary Relativity theory [3] whose generalized polyvectorvalued coordinates are Clifford-valued quantities which incorporate lines, areas, volumes, hyper-volumes.... degrees of freedom associated with the collective particle, string, membrane, p-brane,... dynamics of p-loops (closed p-branes) in D-dimensional target spacetime backgrounds. It was recently shown [1] how an unification of Conformal Gravity and a U(4) × U(4) Yang-Mills theory in four dimensions could be attained from a Clifford Gauge Field Theory in C-spaces (Clifford spaces) based on the (complex) Clifford Cl(4, C) algebra underlying a complexified four dimensional spacetime (8 real dimensions). Clifford-space tensorial-gauge fields generalizations of Yang-Mills theories allows to predict the existence of new particles (bosons, fermions) and tensor-gauge fields of higher-spins in the 10 TeV regime [2]. Tensorial Generalized Yang-Mills in C-spaces (Clifford spaces) based on poly-vector valued (anti-symmetric tensor fields) gauge fields AM (X) and field strengths FMN (X) have been studied in [2], [3] where X = XMΓ is a C-space polyvector valued coordinate X = σ 1 + xμ γ + xμ1μ2 γ μ1 ∧ γ2 + xμ1μ2μ3 γ1 ∧ γ2 ∧ γ3 + ...... +