Clifford Spaces Research Articles

Generalized relativistic transformations in Clifford spaces and their physical implications

A brief introduction of the Extended Relativity Theory in Clifford Spaces ([Formula: see text]-space) paves the way to the explicit construction of the generalized relativistic transformations of the Clifford multivector-valued coordinates in [Formula: see text]-spaces. The most general transformations furnish a full mixing of the grades of the multivector-valued coordinates. The transformations of the multivector-valued momenta follow leading to an invariant generalized mass [Formula: see text] in [Formula: see text]-spaces which differs from [Formula: see text]. No longer the proper mass appearing in the relativistic dispersion relation [Formula: see text] remains invariant under the generalized relativistic transformations. It is argued how this finding might shed some light into the cosmological constant problem, dark energy, and dark matter. We finalize with some concluding remarks about extending these transformations to phase spaces and about Born reciprocal relativity. An appendix is included with the most general (anti) commutators of the Clifford algebra multivector generators.

On Clifford multivector-valued actions, generalized Dirac equation and quantization of branes

We explore the construction of a generalized Dirac equation via the introduction of the notion of Clifford multivector-valued actions, which was inspired by the work of Kanatchikov [Kanatchikov IV. De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory. Rept Math Phys. 1999;43:157–170; Kanatchikov IV. De Donder-Weyl Hamiltonian formulation and precanonical quantization of Vielbein gravity. J Phys Conf Ser. 2013;442:012041] on the De Donder-Weyl formulation of field theory. Crucial in this construction is the evaluation of the exponentials of multivectors associated with Clifford (hypercomplex) analysis. Exact matrix solutions (instead of spinors) of the generalized Dirac equation in D = 2 , 3 space–time dimensions are found. This formalism can be extended to curved space–time backgrounds. We conclude by proposing a wave-functional equation governing the quantum dynamics of branes living in C-spaces (Clifford spaces), and which is based on the De Donder–Weyl Hamiltonian formulation of field theory.

The Clifford-Yang algebra, noncommutative Clifford phase spaces and the deformed quantum oscillator

We construct the novel Clifford-Yang algebra which is an extension of the Yang algebra in noncommutative phase spaces. The Clifford-Yang algebra allows us to write down the commutators of the [Formula: see text] polyvector-valued coordinates and momenta which are compatible with the Jacobi identities, the Weyl–Heisenberg algebra, and paves the way for a formulation of quantum mechanics (QM) in noncommutative Clifford spaces. We continue with a detailed study of the isotropic 3D quantum oscillator in noncommutative spaces and find the energy eigenvalues and eigenfunctions. These findings differ considerably from the ordinary quantum oscillator in commutative spaces. We find that QM in noncommutative spaces leads to very different solutions, eigenvalues, and uncertainty relations than ordinary QM in commutative spaces. The generalization of QM to noncommutative Clifford (phase) spaces is attained via the Clifford-Yang algebra. The operators are now given by the generalized angular momentum operators involving polyvector coordinates and momenta. The eigenfunctions (wave functions) are now more complicated functions of the polyvector coordinates. We conclude with some important remarks.

Motion of Poly-vectors in Fractal Clifford Spaces

Fractal Clifford Spaces (FCS) may be considered as a challenging approach to the unification of micro-physics and macro-physics. Trajectories of these manifolds are described by different poly-vectors describing paths and their deviation equations for not only to test particles but also for charged, rotating and rotating charged objects. Due to relaxing the condition of differentiability for all extended objects in the manifold. In this approach, a specialized Bazanski Lagrangian will be discussed to describe the fractal version of extended objects in a fractal space-time.

Generalizations of Schwarzschild and (Anti) de Sitter Metrics in Clifford Spaces

After a very brief introduction to generalized gravity in Clifford spaces (C-spaces), generalized metric solutions to the C-space gravitational field equations are found, and inspired from the (Anti) de Sitter metric solutions to Einstein’s field equations with a cosmological constant in ordinary spacetimes. C-space analogs of static spherically symmetric metrics solutions are constructed. Concluding remarks are devoted to a thorough discussion about Areal metrics, Kawaguchi–Finsler Geometry, Strings, and plausible novel physical implications of C-space Relativity theory.

A unified description of particles, strings and branes in Clifford spaces and p-brane/polyparticle duality

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

The Cosmological Constant from the Extended Theory of Gravitation in Clifford Spaces

The exploration of the novel physical consequences of the Extended Theory of Gravity in C-spaces (Clifford spaces) is continued. One of the most salient physical feature of the extended gravitational theory in C-spaces is that one can generate an effective stress energy tensor mimicking the effects of “dark” matter/energy. In particular, it is found that the presence of the cosmological constant, along with a plausible mechanism to explain its extremely small value and/or its cancellation, can be understood entirely from a purely Clifford algebraic and geometric perspective. For this reason we believe that this theory may have important consequences in Cosmology and further research in Gravitation and Particle Physics.

On Some Novel Consequences of Clifford Space Relativity Theory

Some of the novel physical consequences of the Extended Relativity Theory in C-spaces (Clifford spaces) are presented. In particular, generalized photon dispersion relations which allow for energy-dependent speeds of propagation while still retaining the Lorentz symmetry in ordinary spacetimes, while breaking the extended Lorentz symmetry in C-spaces. We analyze in further detail the extended Lorentz transformations in Clifford Space and their physical implications. Based on the notion of “extended events” one finds a very different physical explanation of the phenomenon of “relativity of locality” than the one described by the doubly special relativity framework. We finalize with a discussion of the modified dispersion relations, rainbow metrics and generalized uncertainty relations in C-spaces which are extensions of the stringy uncertainty relations.

Extended Lorentz Transformations in Clifford Space Relativity Theory

Some novel physical consequences of the Extended Relativity Theory in C-spaces (Clifford spaces) were explored recently. In particular, generalized photon dispersion relations allowed for energy-dependent speeds of propagation while still retaining the Lorentz symmetry in ordinary spacetimes, but breaking the extended Lorentz symmetry in C-spaces. In this work we analyze in further detail the extended Lorentz transformations in Clifford Space and their physical implications. Based on the notion of “extended events” one finds a very different physical explanation of the phenomenon of “relativity of locality” than the one described by the Doubly Special Relativity (DSR) framework. A generalized Weyl-Heisenberg algebra, involving polyvector-valued coordinates and momenta operators, furnishes a realization of an extended Poincare algebra in C-spaces. In addition to the Planck constant ħ, one finds that the commutator of the Clifford scalar components of the Weyl-Heisenberg algebra requires the introduction of a dimensionless parameter which is expressed in terms of the ratio of two length scales : the Planck and Hubble scales. We finalize by discussing the concept of “photons”, null intervals, effective temporal variables and the addition/subtraction laws of generalized velocities in C-space.

Novel Physical Consequences of the Extended Relativity in Clifford Spaces

Novel physical consequences of the Extended Relativity Theory in C-spaces (Clifford spaces) are explored. The latter theory provides a very different physical explanation of the phenomenon of “relativity of locality” than the one described by the Doubly Special Relativity (DSR) framework. Furthermore, an elegant nonlinear momentum-addition law is derived in order to tackle the “soccer-ball” problem in DSR. Neither derivation in C-spaces requires a curved momentum space nor a deformation of the Lorentz algebra. While the constant (energy-independent) speed of photon propagation is always compatible with the generalized photon dispersion relations in C-spaces, another important consequence is that these generalized photon dispersion relations allow also for energy-dependent speeds of propagation while still retaining the Lorentz symmetry in ordinary spacetimes, while breaking the extended Lorentz symmetry in C-spaces. This does not occur in DSR nor in other approaches, like the presence of quantum spacetime foam. We conclude with some comments on the quantization program and the key role that quantum Clifford-Hopf algebras might have in the future developments since the latter q-Clifford algebras naturally contain the κ-deformed Poincare algebras which are essential ingredients in the formulation of DSR.

The Extended Relativity Theory in Clifford Phase Spaces and Modifications of Gravity at the Planck/Hubble Scales

We extend the construction of Born’s Reciprocal Phase Space Relativity to the case of Clifford Spaces which involve the use of polyvectors and a lower/upper length scale. The generalized polyvector-valued velocity and acceleration/force boosts in Clifford Phase Spaces is presented and we find an explicit Clifford algebraic realization of the velocity and acceleration/force boosts in ordinary phase space. Finally, we provide a Clifford Phase-Space Gravitational Theory based in gauging the generalization of the Quaplectic group and invoking Born’s reciprocity principle between coordinates and momenta (maximal speed of light velocity and maximal force). The generalized gravitational vacuum field equations are explicitly displayed. We conclude with a brief discussion on the role of higher-order Finsler geometry in the construction of extended relativity theories with an upper and lower bound to the higher order accelerations (associated with the higher order tangent and cotangent spaces).

Lanczos-Lovelock and f(R) Gravity from Clifford Space Geometry

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(Rμν),… 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.

On Superluminal Particles and the Extended Relativity Theories

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 M2>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 mμ=105.7 MeV.

GENERALIZED GRAVITY IN CLIFFORD SPACES, VACUUM ENERGY AND GRAND UNIFICATION

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 Clifford Algebra Realization of Supersymmetry and its Polyvector Extension in Clifford Spaces

It is shown explicitly how to construct a novel (to our knowledge) realization of the Poincare superalgebra in 2D. These results can be extended to other dimensions and to (extended) superconformal and (anti) de Sitter superalgebras. There is a fundamental difference between the findings of this work with the other approaches to Supersymmetry (over the past four decades) using Grassmannian calculus and which is based on anti-commuting numbers. We provide an algebraic realization of the anticommutators and commutators of the 2D super-Poincare algebra in terms of the generators of the tensor product \({Cl_{1,1}(R) \otimes \mathcal{A}}\) of a two-dim Clifford algebra and an internal algebra A whose generators can be represented in terms of powers of a 3 × 3 matrix \({\mathcal{Q}}\) , such that \({\mathcal{Q}^3 = 0}\) . Our realization differs from the standard realization of superalgebras in terms of differential operators in Superspace involving Grassmannian (anti-commuting) coordinates θ α and bosonic coordinates x μ . We conclude in the final section with an analysis of how to construct Polyvector-valued extensions of supersymmetry in Clifford Spaces involving spinor-tensorial supercharge generators \({{{\mathcal {Q}}_{{\alpha}}^{\mu_1\mu_2\ldots\mu_n}}}\) and momentum polyvectors \({P_{\mu_1\mu_2\ldots\mu_n}}\) . Clifford-Superspace is an extension of Clifford-space and whose symmetry transformations are generalized polyvector-valued supersymmetries.

On n-ary algebras, branes and poly-vector gauge theories in noncommutative Clifford spaces

In this paper, poly-vector-valued gauge field theories in noncommutative Clifford spaces are presented. They are based on noncommutative (but associative) star products that require the use of the Baker–Campbell–Hausdorff formula. Using these star products allows the construction of actions for noncommutative p-branes (branes moving in noncommutative spaces). Noncommutative Clifford-space gravity as a poly-vector-valued gauge theory of twisted diffeomorphisms in Clifford spaces would require quantum Hopf algebraic deformations of Clifford algebras. We proceed with the study of n-ary algebras and find an important relationship among the n-ary commutators of the noncommuting spacetime coordinates [X1, X2, …, Xn] with the poly-vector-valued coordinates X123⋅⋅⋅n in noncommutative Clifford spaces given by [X1, X2, …, Xn] = n!X123⋅⋅⋅n. The large N limit of n-ary commutators of n hyper-matrices leads to Eguchi–Schild p-brane actions for p + 1 = n. A noncomutative n-ary • product of n functions is constructed which is a generalization of the binary star product * of two functions and is associated with the deformation quantization of n-ary structures and deformations of the Nambu–Poisson brackets.

THE CLIFFORD SPACE GEOMETRY BEHIND THE PIONEER AND FLYBY ANOMALIES

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.

THE CLIFFORD SPACE GEOMETRY OF CONFORMAL GRAVITY AND U(4) × U(4) YANG–MILLS UNIFICATION

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.

AN EXCEPTIONAL E8 GAUGE THEORY OF GRAVITY IN D = 8, CLIFFORD SPACES AND GRAND UNIFICATION

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.

On Chern–Simons (super) gravity, E8 Yang–Mills and polyvector-valued gauge theories in Clifford spaces

It is shown how the E8 Yang–Mills theory is a small sector of a Cl(16) algebra gauge theory and why the 11D Chern–Simons (super) gravity theory can be embedded into a Cl(11) algebra gauge theory. These results may shed some light into the origins behind the hidden E8 symmetry of 11D supergravity. To finalize, we explain how the Clifford algebra gauge theory (that contains the Chern–Simons gravity action in D=11, for example) can itself be embedded into a more fundamental polyvector-valued gauge theory in Clifford spaces involving tensorial coordinates xμ1μ2,xμ1μ2μ3,…,xμ1μ2…μD in addition to antisymmetric tensor gauge fields Aμ1μ2,Aμ1μ2μ3,…,Aμ1μ2…μD. The polyvector-valued supersymmetric extension of this polyvector valued bosonic gauge theory in Clifford spaces may reveal more important features of a Clifford-algebraic structure underlying M, F theory.