The Extended Relativity Theory in Clifford Spaces

A brief review of some of the most important features of the Extended Relativity theory in Clifford-spaces ($C$-spaces) is presented whose " point" coordinates are noncommuting Clifford-valued quantities and which incorporate the lines, areas, volumes,.... degrees of freedom associated with the collective particle, string, membrane,... dynamics of $p$-loops (closed p-branes) living in target $D$-dimensional spacetime backgrounds. $C$-space Relativity naturally incorporates the ideas of an invariant length (Planck scale), maximal acceleration, noncommuting coordinates, supersymmetry, holography, higher derivative gravity with torsion and variable dimensions/signatures that allows to study the dynamics of all (closed) p-branes, for all values of $ p $, on a unified footing. It resolves the ordering ambiguities in QFT and the problem of time in Cosmology. A discussion of the maximal-acceleration Relativity principle in phase-spaces follows along with the study of the invariance group of symmetry transformations in phase-space that allows to show why Planck areas are invariant under acceleration-boosts transformations and which seems to suggest that a maximal-string tension principle may be operating in Nature. We continue by pointing out how the relativity of signatures of the underlying $n$-dimensional spacetime results from taking different $n$-dimensional slices through $C$-space. The conformal group emerges as a natural subgroup of the Clifford group and Relativity in $C$-spaces involves natural scale changes in the sizes of physical objects without the introduction of forces nor Weyl's gauge field of dilations. We finalize by constructing the generalization of Maxwell theory of Electrodynamics of point charges to a theory in $C$-spaces that involves extended charges coupled to antisymmetric tensor fields of arbitrary rank. In the concluding remarks we outline briefly the current promising research programs and their plausible connections with $C$-space Relativity.