Narrow Results

Using Clifford algebraic methods we describe how to generalize Maxwell theory of Electrodynamics associated with ordinary point-charges to a generalized Maxwell theory in Clifford spaces involving extended charges and p-forms of arbitrary rank. Clifford algebras contain the appropriate algebraic-geometric features to implement the principle of polydimensional transformations (branes of different dimensionality are rotated into each other) that could possibly lead to a proper formulation of string and M theory.

Representations of SO(4, 2) are constructed using 4×4 and 2×2 matrices with elements in ℍ' ⊗ ℂ and the known isomorphism between the conformal group and SO(4, 2) is written explicitly in terms of the 4×4 representation. The Clifford algebra structure of SO(4, 2) is briefly discussed in this language, as is its relationship to other groups of physical interest.

We give an algebraic formulation based on Clifford algebras and algebraic spinors for quantum information. In this context, logic gates and concepts such as chirality, charge conjugation, parity and time reversal are introduced and explored in connection with states of qubits. Supersymmetry and M-superalgebra are also analyzed with our formalism. Specifically we use extensively the algebras $C{l}_{3,0}$ and $C{l}_{1,3}$ as well as tensor products of Clifford algebras.

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.

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.

It is rigorously shown how the extended relativity theory in Clifford spaces (C-spaces) can explain the variable radial dependence a_p(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 R_H 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.

Motivated by evolution equations on Clifford algebras and illustrated with the n-particle fermion algebra, a theory of invertible left- and right-Appell systems is developed for Clifford algebras of an arbitrary quadratic form. This work extends and clarifies the authors' earlier work on Clifford Appell systems, operator calculus, and operator homology/cohomology. A direct connection is also shown between blade factorization algorithms and the construction of Appell systems in these algebras.

We construct Lagrange interpolating polynomials for a set of points and values belonging to the algebra of real quaternions ℍ ≃ ℝ_0,2, or to the real Clifford algebra ℝ_0,3. In the quaternionic case, the approach by means of Lagrange polynomials is new, and gives a complete solution of the interpolation problem. In the case of ℝ_0,3, such a problem is dealt with here for the first time. Elements of the recent theory of slice regular functions are used. Leaving apart the classical cases ℝ_0,0 ≃ ℝ, ℝ_0,1 ≃ ℂ and the trivial case ℝ_1,0 ≃ ℝ⊕ℝ, the interpolation problem on Clifford algebras ℝ_p,q with (p,q) ≠ (0,2), (0,3) seems to have some intrinsic difficulties.

ℤ₂-gradings of Clifford algebras are reviewed and we shall be concerned with an α-grading based on the structure of inner automorphisms, which is closely related to the spacetime splitting, if we consider the standard conjugation map automorphism by an arbitrary, but fixed, splitting vector. After briefly sketching the orthogonal and parallel components of products of differential forms, where we introduce the parallel [orthogonal] part as the space [time] component, we provide a detailed exposition of the Dirac operator splitting and we show how the differential operator parallel and orthogonal components are related to the Lie derivative along the splitting vector and the angular momentum splitting bivector. We also introduce multivectorial-induced α-gradings and present the Dirac equation in terms of the spacetime splitting, where the Dirac spinor field is shown to be a direct sum of two quaternions. We point out some possible physical applications of the formalism developed.

Some properties of the Clifford algebras and are presented, and three isomorphisms between the Dirac–Clifford algebra and are exhibited, in order to construct conformal maps and twistors, using the paravector model of spacetime. The isomorphism between the twistor space inner product isometry group SU(2,2) and the group $pin₊(2,4) is also investigated, in the light of a suitable isomorphism between and . After reviewing the conformal spacetime structure, conformal maps are described in Minkowski spacetime as the twisted adjoint representation of $pin₊(2,4), acting on paravectors. Twistors are then presented via the paravector model of Clifford algebras and related to conformal maps in the Clifford algebra over the Lorentzian ℝ^4,1 spacetime. We construct twistors in Minkowski spacetime as algebraic spinors associated with the Dirac–Clifford algebra using one lower spacetime dimension than standard Clifford algebra formulations, since for this purpose, the Clifford algebra over ℝ^4,1 is also used to describe conformal maps, instead of ℝ^2,4. Our formalism sheds some new light on the use of the paravector model and generalizations.

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.

We continue to study the Chern–Simons E₈ Gauge theory of Gravity developed by the author which is a unified field theory (at the Planck scale) of a Lanczos–Lovelock Gravitational theory with a E₈ Generalized Yang–Mills (GYM) field theory, and is defined in the 15D boundary of a 16D bulk space. The Exceptional E₈ Geometry of the 256-dim slice of the 256 × 256-dimensional flat Clifford (16) space is explicitly constructed based on a spin connection , that gauges the generalized Lorentz transformations in the tangent space of the 256-dim curved slice, and the 256 × 256 components of the vielbein field , that gauge the nonabelian translations. Thus, in one-scoop, the vielbein encodes all of the 248 (nonabelian) E₈ generators and 8 additional (abelian) translations associated with the vectorial parts of the generators of the diagonal subalgebra [Cl(8) ⊗ Cl(8)]_diag ⊂ Cl(16). The generalized curvature, Ricci tensor, Ricci scalar, torsion, torsion vector and the Einstein–Hilbert–Cartan action is constructed. A preliminary analysis of how to construct a Clifford Superspace (that is far richer than ordinary superspace) based on orthogonal and symplectic Clifford algebras is presented. Finally, it is shown how an E₈ ordinary Yang–Mills in 8D, after a sequence of symmetry breaking processes E₈ → E₇ → E₆ → SO(8, 2), and performing a Kaluza–Klein–Batakis compactification on CP², involving a nontrivial torsion, leads to a (Conformal) Gravity and Yang–Mills theory based on the Standard Model in 4D. The conclusion is devoted to explaining how Conformal (super) Gravity and (super) Yang–Mills theory in any dimension can be embedded into a (super) Clifford-algebra-valued gauge field theory.

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.

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.

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(R_μν), … 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.

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 2^D dimensions. The most physically relevant conclusion, besides the presence of torsion in the vacuum, is the contribution of the higher grade metric components g^{μ₁μ₂ν₁ν₂}, g^{μ₁μ₂μ₃ν₁ν₂ν₃},… of the C-space metric to dark energy/dark matter.

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 ρ_obs ~ 10^-120(M_P)⁴ (M_P 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 Cl(3, 1)-algebra-valued scalar field Φ in four dimensions.

It is described how the Extended Relativity Theory in $C$-spaces (Clifford spaces) allows a unified formulation of point particles, strings, membranes and $p$-branes, moving in ordinary target spacetime backgrounds, within the description of a single polyparticle moving in $C$-spaces. The degrees of freedom of the latter are provided by Clifford polyvector-valued coordinates (antisymmetric tensorial coordinates). A correspondence between the $p$-brane ($p$-loop) “Schrödinger-like” equations of Ansoldi–Aurilia–Spallucci and the polyparticle wave equation in $C$-spaces is found via the polyparticle/$p$-brane correspondence. This correspondence might provide another unexplored avenue to quantize $p$-branes (a notoriously difficult and unsolved problem) from the more straightforward quantization of the polyparticle in $C$-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 $p$-brane coordinates via the evaluation of $n$-ary commutators.

A Moyal deformation of a Clifford $\mathrm{\text{Cl}}(3,1)$ Gauge Theory of (Conformal) Gravity is performed for canonical noncommutativity (constant ${\mathrm{\Theta }}^{\mu \nu }$ parameters). In the very special case when one imposes certain constraints on the fields, there are $no$ first-order contributions in the ${\mathrm{\Theta }}^{\mu \nu }$ parameters to the Moyal deformations of Clifford gauge theories of gravity. However, when one does $not$ impose constraints on the fields, there are first-order contributions in ${\mathrm{\Theta }}^{\mu \nu }$ 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 $U(2,2),\mathrm{\text{SO}}(4,2),\mathrm{\text{SO}}(2,3)$ can be expressed in terms of the Clifford algebra generators this does $not$ 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 ${\mathrm{\Theta }}^{\mu \nu }$. Finally, we provide a mechanism which furnishes a plausible cancellation of the huge vacuum energy density.

Using the toroidal compactification of string theory on $n$-dimensional tori, ${𝕋}^n$, we investigate dyonic objects in arbitrary dimensions. First, we present a class of dyonic black solutions formed by two different D-branes using a correspondence between toroidal cycles and objects possessing both magnetic and electric charges, belonging to $\text{U}{(1)}_e^{2^{n-1}}\times \text{U}{(1)}_m^{2^{n-1}}$ dyonic gauge symmetry. This symmetry could be associated with electrically charged magnetic monopole solutions in stringy model buildings of the standard model (SM) extensions. Then, we consider in some detail such black hole classes obtained from even-dimensional toroidal compactifications, and we find that they are linked to $Cl(n)$ Clifford algebras using the vee product. It is believed that this analysis could be extended to dyonic objects which can be obtained from local Calabi–Yau manifold compactifications.