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A Moyal deformation of a Clifford Gauge Theory of (Conformal) Gravity is performed for canonical noncommutativity (constant parameters). In the very special case when one imposes certain constraints on the fields, there are first-order contributions in the parameters to the Moyal deformations of Clifford gauge theories of gravity. However, when one does impose constraints on the fields, there are first-order contributions in to the Moyal deformations in variance with the previous results obtained by other authors and based on different gauge groups. Despite that the generators of can be expressed in terms of the Clifford algebra generators this does imply that these algebras are isomorphic to the Clifford algebra. Therefore one should not expect identical results to those obtained by other authors. In particular, there are Moyal deformations of the Einstein-Hilbert gravitational action with a cosmological constant to first-order in . Finally, we provide a mechanism which furnishes a plausible cancellation of the huge vacuum energy density. [ABSTRACT FROM AUTHOR]

It is described how the Extended Relativity Theory in -spaces (Clifford spaces) allows a unified formulation of point particles, strings, membranes and -branes, moving in ordinary target spacetime backgrounds, within the description of a single polyparticle moving in -spaces. The degrees of freedom of the latter are provided by Clifford polyvector-valued coordinates (antisymmetric tensorial coordinates). A correspondence between the -brane (-loop) 'Schrödinger-like' equations of Ansoldi-Aurilia-Spallucci and the polyparticle wave equation in -spaces is found via the polyparticle/-brane correspondence. This correspondence might provide another unexplored avenue to quantize -branes (a notoriously difficult and unsolved problem) from the more straightforward quantization of the polyparticle in -spaces, even in the presence of external interactions. We conclude with comments about the compositeness nature of the polyvector-valued coordinate operators in terms of ordinary -brane coordinates via the evaluation of -ary commutators. [ABSTRACT FROM AUTHOR]

A Clifford-gravity-based model is exploited to build a generalized action (beyond the current ones used in the literature) and arrive at relevant numerical results which are consistent with the presently-observed de Sitter accelerating expansion of the universe driven by a very small vacuum energy density ρobs ~ 10-120(MP)4 (MP is the Planck mass) and provide promising dark energy/matter candidates in terms of the 16 scalars corresponding to the degrees of freedom associated with a (3, 1)-algebra-valued scalar field Φ in four dimensions. [ABSTRACT FROM AUTHOR]

A Clifford Cl(5, C) unified gauge field theory formulation of conformal gravity and U(4) × U(4) × U(4) Yang-Mills in 4D, is reviewed along with its implications for the Pati-Salam (PS) group SU(4) × SU(2)L × SU(2)R, and trinification grand unified theory models of three fermion generations based on the group SU(3)C × SU(3)L × SU(3)R. We proceed with a brief review of a unification program of 4D gravity and SU(3) × SU(2) × U(1) Yang-Mills emerging from 8D pure quaternionic gravity. A realization of E8 in terms of the Cl(16) = Cl(8) ⊗ Cl(8) generators follows, as a preamble to F. Smith's E8 and Cl(16) = Cl(8) ⊗ Cl(8) unification model in 8D. The study of chiral fermions and instanton backgrounds in CP2 and CP3 related to the problem of obtaining three fermion generations is thoroughly studied. We continue with the evaluation of the coupling constants and particle masses based on the geometry of bounded complex homogeneous domains and geometric probability theory. An analysis of neutrino masses, Cabbibo-Kobayashi-Maskawa quark-mixing matrix parameters, and neutrino-mixing matrix parameters follows. We finalize with some concluding remarks about other proposals for the unification of gravity and the Standard Model, like string, M, and F theories and noncommutative and nonassociative geometry. [ABSTRACT FROM AUTHOR]

We continue with the study of Clifford-space (C-space) gravity and analyze further the C-space generalized gravitational field equations which are obtained from a variational principle based on the generalization of the Einstein-Hilbert-Cartan action. One of the main features is that the C-space connection requires torsion in order to have consistency with the Clifford algebraic structure associated with the curved C-space basis generators. Hence no spin matter is required to induce torsion since it already exists in the vacuum. The field equations in C-spaces associated to a Clifford algebra in D dimensions are not equivalent to the ordinary gravitational equations with torsion in higher 2D dimensions. The most physically relevant conclusion, besides the presence of torsion in the vacuum, is the contribution of the higher grade metric components gµ1µ2v1v2, gµ1µ2µ3v1v2v3,... of the C-space metric to dark energy/dark matter. [ABSTRACT FROM AUTHOR]

A rigorous construction of Clifford-space (C-space) gravity is presented which is compatible with the Clifford algebraic structure and permits the derivation of the expressions for the connections with torsion in C-spaces. The C-space generalized gravitational field equations are derived from a variational principle based on the extension of the Einstein-Hilbert-Cartan action. We continue by arguing how Lanczos-Lovelock-Cartan (LLC) higher curvature gravity with torsion can be embedded into gravity in C-spaces and suggest how this might also occur for extended gravitational theories based on f(R), f(Rμν), ... actions, for polynomial-valued functions. In essence, the LLC curvature tensors appear as Ricci-like traces of certain components of the C-space curvatures. Torsional gravity is related to higher-order corrections of the bosonic string-effective action. In the torsionless case, black-strings and black-brane metric solutions in higher dimensions D > 4 play an important role in finding specific examples of solutions to LL gravity. [ABSTRACT FROM AUTHOR]

A rigorous construction of Clifford-space Gravity is presented which is compatible with the Clifford algebraic structure and permits the derivation of the generalized connections in Clifford spaces ( C-space) in terms of derivatives of the C-space metric. We continue by arguing how Lanczos-Lovelock higher curvature gravity can be embedded into gravity in Clifford spaces and suggest how this might also occur for extended gravitational theories based on f( R), f( R),... actions, for polynomial-valued functions. Black-strings and black-brane metric solutions in higher dimensions D>4 play an important role in finding specific examples. [ABSTRACT FROM AUTHOR]

Superluminal particles are studied within the framework of the Extended Relativity theory in Clifford spaces ( C-spaces). In the simplest scenario, it is found that it is the contribution of the Clifford scalar component π of the poly-vector-valued momentum which is responsible for the superluminal behavior in ordinary spacetime due to the fact that the effective mass $\mathcal{M} = \sqrt{ M^{2} - \pi^{2} }$ is imaginary (tachyonic). However, from the point of view of C-space, there is no superluminal (tachyonic) behavior because the true physical mass still obeys M>0. Therefore, there are no violations of the Clifford-extended Lorentz invariance and the extended Relativity principle in C-spaces. It is also explained why the charged muons (leptons) are subluminal while its chargeless neutrinos may admit superluminal propagation. A Born's Reciprocal Relativity theory in Phase Spaces leads to modified dispersion relations involving both coordinates and momenta, and whose truncations furnish Lorentz-violating dispersion relations which appear in Finsler Geometry, rainbow-metrics models and Double (deformed) Special Relativity. These models also admit superluminal particles. A numerical analysis based on the recent OPERA experimental findings on alleged superluminal muon neutrinos is made. For the average muon neutrino energy of 17 GeV, we find a value for the magnitude $|\mathcal{M } | = 119.7\mbox{~MeV}$ that, coincidentally, is close to the mass of the muon m=105.7 MeV. [ABSTRACT FROM AUTHOR]

Polyvector-valued gauge field theories in Clifford spaces are used to construct a novel Cl(3, 2) gauge theory of gravity that furnishes modified curvature and torsion tensors leading to important modifications of the standard gravitational action with a cosmological constant. Vacuum solutions exist which allow a cancelation of the contributions of a very large cosmological constant term and the extra terms present in the modified field equations. Generalized gravitational actions in Clifford-spaces are provided and some of their physical implications are discussed. It is shown how the 16 fermions and their masses in each family can be accommodated within a Cl(4) gauge field theory. In particular, the Higgs fields admit a natural Clifford-space interpretation that differs from the one in the Chamseddine-Connes spectral action model of non-commutative geometry. We finalize with a discussion on the relationship with the Pati-Salam color-flavor model group SU(4)C × SU(4)F and its symmetry breaking patterns. An Appendix is included with useful Clifford algebraic relations. [ABSTRACT FROM AUTHOR]

It is rigorously shown how the extended relativity theory in Clifford spaces (C-spaces) can explain the variable radial dependence ap(r) of the Pioneer anomaly; its sign (pointing towards the Sun); why planets do not experience the anomalous acceleration and why the present day value of the Hubble scale RH appears. It is the curvature–spin coupling of the planetary motions that hold the key. The difference in the rate at which clocks tick in C-space translates into the C-space analog of Doppler shifts which may explain the anomalous redshifts in cosmology, where objects which are not that far apart from each other exhibit very different redshifts. We conclude by showing how the empirical formula for the flybys anomalies obtained by Anderson et al.17 can be derived within the framework of Clifford geometry. [ABSTRACT FROM AUTHOR]

It is shown how a conformal gravity and U(4) × U(4) Yang–Mills grand unification model in four dimensions can be attained from a Clifford gauge field theory in C-spaces (Clifford spaces) based on the (complex) Clifford Cl(4, C) algebra underlying a complexified four-dimensional space–time (eight real dimensions). Upon taking a real slice, and after symmetry breaking, it leads to ordinary gravity and the Standard Model in four real dimensions. A brief conclusion about the noncommutative star-product deformations of this Grand Unified Theory of gravity with the other forces of Nature is presented. [ABSTRACT FROM AUTHOR]

A candidate action for an Exceptional E8 gauge theory of gravity in 8D is constructed. It is obtained by recasting the E8 group as the semi-direct product of GL(8,R) with a deformed Weyl–Heisenberg group associated with canonical-conjugate pairs of vectorial and antisymmetric tensorial generators of rank two and three. Other actions are proposed, like the quarticE8 group-invariant action in 8D associated with the Chern–Simons E8 gauge theory defined on the 7-dim boundary of a 8D bulk. To finalize, it is shown how the E8 gauge theory of gravity can be embedded into a more general extended gravitational theory in Clifford spaces associated with the Cl(16) algebra and providing a solid geometrical program of a grand unification of gravity with Yang–Mills theories. The key question remains if this novel gravitational model based on gauging the E8 group may still be renormalizable without spoiling unitarity at the quantum level. [ABSTRACT FROM AUTHOR]

A novel Chern–Simons E8 gauge theory of gravity in D = 15 based on an octicE8 invariant expression in D = 16 (recently constructed by Cederwall and Palmkvist) is developed. A grand unification model of gravity with the other forces is very plausible within the framework of a supersymmetric extension (to incorporate spacetime fermions) of this Chern–Simons E8 gauge theory. We review the construction showing why the ordinary 11D Chern–Simons gravity theory (based on the Anti de Sitter group) can be embedded into a Clifford-algebra valued gauge theory and that an E8 Yang–Mills field theory is a small sector of a Clifford (16) algebra gauge theory. An E8 gauge bundle formulation was instrumental in understanding the topological part of the 11-dim M-theory partition function. The nature of this 11-dim E8 gauge theory remains unknown. We hope that the Chern–Simons E8 gauge theory of gravity in D = 15 advanced in this work may shed some light into solving this problem after a dimensional reduction. [ABSTRACT FROM AUTHOR]

Starting with a review of the Extended Relativity Theory in Clifford-Spaces, and the physical motivation behind this novel theory, we provide the generalization of the nonrelativistic supersymmetric point-particle action in Clifford-space backgrounds. The relativistic supersymmetric Clifford particle action is constructed that is invariant under generalized supersymmetric transformations of the Clifford-space background's polyvector-valued coordinates. To finalize, the Polyvector super-Poincaré and M, F theory superalgebras, in D = 11, 12 dimensions, respectively, are discussed followed by our final analysis of the novel Clifford-superspace realizations of generalized supersymmetries in Clifford spaces. [ABSTRACT FROM AUTHOR]