C. Castro and M. Pavsic
The Extended Relativity Theory
in Clifford Spaces
Abstract
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.
|
