On N-ary Algebras, Polyvector Gauge Theories in Noncommutative Clifford Spaces and Deformation Quantization

Polyvector-valued gauge field theories in noncommutative Clifford spaces are presented. The noncommutative binary star products are associative and require the use of the Baker-Campbell-Hausdorff formula. An important relationship among the n-ary commutators of noncommuting spacetime coordinates [X, X, ......, X] and the poly-vector valued coordinates X in noncommutative Clifford spaces is explicitly derived and is given by [X, X, ......, X] = n! X. It is argued how the large N limit of n-ary commutators of n hyper-matrices Xi1i2....in leads to Eguchi-Schild p-brane actions when p+1 = n. A noncomutative n-ary generalized star product of functions is provided which is associated with the deformation quantization of n-ary structures. Finally, brief comments are made about the mapping of the Nambu-Heisenberg n-ary commutation relations of linear operators into the deformed Nambu-Poisson brackets of their corresponding symbols. 1 Gauge Theories in Noncommutative Clifford Spaces 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. C-space Relativity naturally incorporates the ideas of an invariant length (Planck scale), maximal acceleration, non-commuting coordinates, supersymmetry, holography, higher derivative