Higher Derivative Gravity and Torsion from the Geometry of C-spaces

Abstract

We start from a new theory (discussed earlier) in which the arena for physics is not spacetime, but its straightforward extension-the so called Clifford space ($C$-space), a manifold of points, lines, areas, etc..; physical quantities are Clifford algebra valued objects, called polyvectors. This provides a natural framework for description of supersymmetry, since spinors are just left or right minimal ideals of Clifford algebra. The geometry of curved $C$-space is investigated. It is shown that the curvature in $C$-space contains higher orders of the curvature in the underlying ordinary space. A $C$-space is parametrized not only by 1-vector coordinates $x^{\mu }$ but also by the 2-vector coordinates ${\sigma }^{\mu \nu }$, 3-vector coordinates ${\sigma }^{\mu \nu \rho }$, etc., called also {\it holographic coordinates}, since they describe the holographic projections of 1-lines, 2-loops, 3-loops, etc., onto the coordinate planes. A remarkable relation between the "area" derivative $\p/ \p σ^{μν}$ and the curvature and torsion is found: if a scalar valued quantity depends on the coordinates ${\sigma }^{\mu \nu }$ this indicates the presence of torsion, and if a vector valued quantity depends so, this implies non vanishing curvature. We argue that such a deeper understanding of the $C$-space geometry is a prerequisite for a further development of this new theory which in our opinion will lead us towards a natural and elegant formulation of $M$-theory.

19 pages; A section describing the main physical implications of C-space is added, and the rest of the text is modified accordingly

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Cited by in corpus (19)

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• Clifford Space as the Arena for Physics
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• How the Geometric Calculus Resolves the Ordering Ambiguity of Quantum Theory in Curved Space
• Finsler-Lagrange Geometries and Standard Theories in Physics: New Methods in Einstein and String Gravity
• On a Unified Theory of Generalized Branes Coupled to Gauge Fields, Including the Gravitational and Kalb-Ramond Fields
• Quantum Field Theories in Spaces with Neutral Signatures
• Nonholonomic Clifford Structures and Noncommutative Riemann--Finsler Geometry
• Clifford Space as a Generalization of Spacetime: Prospects for Unification in Physics
• An Extra Structure of Spacetime: A Space of Points, Areas and Volumes
• Spin Gauge Theory of Gravity in Clifford Space
• Noncommutative Quantum Mechanics and Geometry from the Quantization of C-spaces
• Beyond the Relativistic Point Particle: A Reciprocally Invariant System and its Generalisation
• Clifford Algebra of Spacetime and the Conformal Group
• Maximal-acceleration phase space relativity from Clifford algebras
• Clifford Algebra, Geometry and Physics
• On an Alternative Approach to the Relation between Bosons and Fermions: Employing Clifford Space