Noncommutative Quantum Mechanics and Geometry from the Quantization of C-spaces

Abstract

Four years ago the Extended Scale Relativity (ESR) theory in C-spaces (Clifford manifolds) was proposed as the plausible physical foundations of string theory. In such theory the speed of light and the minimum Planck scale are the two universal invariants. All the dimensions of a C-space can be treated on equal footing by implementing the holographic principle associated with a nested family of p-loops of various dimensionalities. This is achieved by using polyvector valued coordinates in C-spaces that encode in one stroke points, lines, areas, volumes,.....We review the derivation of the minimal length/time string/brane uncertainty relations and the maximum Planck temperature thermodynamical uncertainty relation. The Weyl-Heisenberg algebra in C-spaces is constructed which $induces$ a Noncommutative Geometric structure in the $ X^A $ coordinates. Hence quantization in C-spaces involves in a natural fashion a Noncommutative Quantum Mechanics and Field Theory, rather than being postulated ad-hoc. A QFT in C-spaces may very likely involve (Braided Hopf) Quantum Clifford algebras and generalized Moyal-like star products associated with multisymplectic geometry.