Octonionic Space Research Articles

Some properties of dark matter field in the complex octonion space

The paper aims to consider the electromagnetic adjoint-field in the complex octonion space as the dark matter field, describing some properties of the dark matter, especially the origin, particle category, existence region, force and so forth. Since Maxwell applied the algebra of quaternions to depict the electromagnetic theory, some scholars adopt the complex quaternion and octonion to study the physics property of electromagnetic and gravitational fields. In the paper, by means of the octonion operator, it is found that the gravitational field accompanies with one adjoint-field, whose property is partly similar to that of electromagnetic field. The electromagnetic field accompanies with another adjoint-field, whose feature is partly similar to that of gravitational field. As a result, the electromagnetic adjoint-field can be chosen as one candidate of the dark matter field. According to the electromagnetic adjoint-field, it is able to predict a few properties of the dark matter, for instance, the particle category, interaction intensity, interaction distance, existence region and so forth. The study reveals that the dark matter particle and the gravitational resource will be influenced by the gravitational strength and force. The dark matter field is capable of making a contribution to physics quantities of gravitational field, including the angular momentum, torque, energy, force and so on. Further, there may be comparatively more chances to discover the dark matter in some regions with the ultrastrong field strength, surrounding the neutral star, white dwarf, galactic nucleus, black hole, astrophysical jet and so on.

Inhomogeneous Cauchy-Riemann equation in Octonion space

The integral representation of differentiable functions in Octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the Cousin problem analogue of Mittag-Laffler problem is discussed.

Dynamic of astrophysical jets in the complex octonion space

The paper aims to consider the strength gradient force as the dynamic of astrophysical jets, explaining the movement phenomena of astrophysical jets. J. C. Maxwell applied the quaternion analysis to describe the electromagnetic theory. This encourages others to adopt the complex quaternion and octonion to depict the electromagnetic and gravitational theories. In the complex octonion space, it is capable of deducing the field potential, field strength, field source, angular momentum, torque, force and so forth. As one component of the force, the strength gradient force relates to the gradient of the norm of field strength only, and is independent of not only the direction of field strength but also the mass and electric charge for the test particle. When the strength gradient force is considered as the thrust of the astrophysical jets, one can deduce some movement features of astrophysical jets, including the bipolarity, matter ingredient, precession, symmetric distribution, emitting, collimation, stability, continuing acceleration and so forth. The above results reveal that the strength gradient force is able to be applied to explain the main mechanical features of astrophysical jets, and is the competitive candidate of the dynamic of astrophysical jets.

Octonionic matrix representation and electromagnetism

Keeping in mind the important role of octonion algebra, we have obtained the electromagnetic field equations of dyons with an octonionic 8×8 matrix representation. In this paper, we consider the eight — dimensional octonionic space as a combination of two (external and internal) four-dimensional spaces for the existence of magnetic monopoles (dyons) in a higher-dimensional formalism. As such, we describe the octonion wave equations in terms of eight components from the 8 × 8 matrix representation. The octonion forms of the generalized potential, fields and current source of dyons in terms of 8 × 8 matrix are discussed in a consistent manner. Thus, we have obtained the generalized Dirac-Maxwell equations of dyons from an 8×8 matrix representation of the octonion wave equations in a compact and consistent manner. The generalized Dirac-Maxwell equations are fully symmetric Maxwell equations and allow for the possibility of magnetic charges and currents, analogous to electric charges and currents. Accordingly, we have obtained the octonionic Dirac wave equations in an external field from the matrix representation of the octonion-valued potentials of dyons.

Angular momentum and torque described with the complex octonion

The paper aims to adopt the complex octonion to formulate the angular momentum, torque, and force etc in the electromagnetic and gravitational fields. Applying the octonionic representation enables one single definition of angular momentum (or torque, force) to combine some physics contents, which were considered to be independent of each other in the past. J. C. Maxwell used simultaneously two methods, the vector terminology and quaternion analysis, to depict the electromagnetic theory. It motivates the paper to introduce the quaternion space into the field theory, describing the physical feature of electromagnetic and gravitational fields. The spaces of electromagnetic field and of gravitational field can be chosen as the quaternion spaces, while the coordinate component of quaternion space is able to be the complex number. The quaternion space of electromagnetic field is independent of that of gravitational field. These two quaternion spaces may compose one octonion space. Contrarily, one octonion space can be separated into two subspaces, the quaternion space and S-quaternion space. In the quaternion space, it is able to infer the field potential, field strength, field source, angular momentum, torque, and force etc in the gravitational field. In the S-quaternion space, it is capable of deducing the field potential, field strength, field source, current continuity equation, and electric (or magnetic) dipolar moment etc in the electromagnetic field. The results reveal that the quaternion space is appropriate to describe the gravitational features, including the torque, force, and mass continuity equation etc. The S-quaternion space is proper to depict the electromagnetic features, including the dipolar moment and current continuity equation etc. In case the field strength is weak enough, the force and the continuity equation etc can be respectively reduced to that in the classical field theory.

Field Equations in the Complex Quaternion Spaces

The paper aims to adopt the complex quaternion and octonion to formulate the field equations for electromagnetic and gravitational fields. Applying the octonionic representation enables one single definition to combine some physics contents of two fields, which were considered to be independent of each other in the past. J. C. Maxwell applied simultaneously the vector terminology and the quaternion analysis to depict the electromagnetic theory. This method edified the paper to introduce the quaternion and octonion spaces into the field theory, in order to describe the physical feature of electromagnetic and gravitational fields, while their coordinates are able to be the complex number. The octonion space can be separated into two subspaces, the quaternion space and theS-quaternion space. In the quaternion space, it is able to infer the field potential, field strength, field source, field equations, and so forth, in the gravitational field. In theS-quaternion space, it is able to deduce the field potential, field strength, field source, and so forth, in the electromagnetic field. The results reveal that the quaternion space is appropriate to describe the gravitational features; meanwhile, theS-quaternion space is proper to depict the electromagnetic features.

Investigating Initial Conditions of the WdW Equation in Flat Space in a Transition from the Pre-Planckian Physics Era to the Electroweak Regime of Space-Time

This document is due to reviewing an article by Maydanyuk and Olkhovsky, of a Nova Science conpendium as of “The big bang, theory assumptions and Problems”, as of 2012, which uses the Wheeler De Witt equation as an evolution equation assuming a closed universe. Having the value of k, not as the closed universe, but nearly zero of a nearly flat universe, which leads to serious problems of interpretation of what initial conditions are. These problems of interpretations of initial conditions tie in with difficulties in using QM as an initial driver of inflation. And argue in favor of using a different procedure as far as forming a wave function of the universe initially. The author wishes to thank Abhay Ashtekar for his well thought out criticism but asserts that limitations in space-time geometry largely due to when is formed from semi classical reasoning, i.e. Maxwell’s equation involving a close boundary value regime between Octonionic geometry and flat space non Octonionic geometry is a datum which Abhay Ashekhar may wish to consider in his quantum bounce model and in loop quantum gravity in the future.

Looking at Graviton Properties, as Either Classical or QM, in Nature, via Alicki-Van Ryn Experimental Realization

Recently, the author read the Alicki-Van Ryn test as to behavior of photons in a test of violations of classicality. The same thing is propoosed via use of a spin two graviton, using typical spin 2 matrices. While the technology currently does not exist to perform such an analysis yet, the same sort of thought experiment is proposed in a way to allow for a first principle test of the either classical or quantum foundations of gravity. The reason for the present manuscript topic is due to a specific argument presented in a prior document as to how is formed from semiclassical reasoning. We referred to a procedure as to how to use Maxwell’s equations involving a closed boundary regime, in the boundary re- gime between Octonionic Geometry and quantum flat space. Conceivably, a similar argument could be made forgravi- tons, pending further investigations. Also the anlysis of if gravitons are constructed by a similar semiclassical argument is pending if gravitons as by the Alicki-Van Ryn test result in semiclassical and matrix observable eigenvalue behavior. This paper also indirectly raises the question of if Baysian statistics would be the optimal way to differentiate between and matrix observable eigenvalue behavior for reasons brought up in the conclusion.

八元数描述的电磁场方程组 Electromagnetic Field Equations Described with the Octonions

麦克斯韦首先同时采用矢量和四元数两种方法描述电磁场性质。这启发本文总结出三条基本假设，试图解决将八元数引进场论的问题。应用八元数可以描述电磁场的麦克斯韦方程组，从而部分程度实现采用八元数代数描述电磁场理论的努力目标。如果将八元数分解成为四元数和S-四元数两个部分，可以发现S-四元数适合于描述电磁场性质。这种方法在电磁场理论的以往研究中是不曾遇到过的。由八元数描述的电磁场的场源定义式可以分解出麦克斯韦方程组。同时电磁场势、场强定义式和麦克斯韦方程组均分别等同于采用矢量方法得到的结果，但位移电流方向和规范条件方程有所不同。研究结果揭示，应用八元数描述的电磁场理论可以涵盖经典电磁场理论中的大部分已有结论。采用S-四元数可以描述电磁场中的麦克斯韦方程组。 Inspired by J. C. Maxwell firstly using both the vector terminology and the algebra of quaternions to describe the property of electromagnetic fields, the paper summarizes 3 basic postulations to import the algebra of octonions into the field theory. In this case, the algebra of octonions can partly describe the electromagnetic theory by using the algebra of octonions to describe the Maxwell equations of electromagnetic field. The viewpoint of R. Descartes, M. Faraday, and A. Einstein etc claims that, the field is an irreducible element of physical description, while the space-time is only the extension of the field and does not claim existence on its own. According to the three basic postulations, the space-time extended from the electromagnetic field is different from the one extended from the gravitational field. They are quite similar but independent to each other. These two space-times, extended respectively from the electromagnetic field and the gravitational field, both can be considered as the quaternion space, and are perpendicular to each other so that they can combine together to become an octonion space. Therefore the octonion space can be used to describe the physical property of the electromagnetic field and the gravitational field. In order to maintain the dimensional homogeneity, the physical quantity of the electromagnetic field should be multiplied by the undetermined coefficient, when the two kinds of physical quantities exist in the same octonion formula. The undetermined coefficient can be determined by comparing with the classical theories of the electromagnetic field and the gravitational field. In math, the complex number can be divided into the real number and the imaginary number, while the imaginary number is the product of another real number and the imaginary unit. Just like the complex number, the octonion also can be divided into two components, the quaternion and the S-quaternion, while the S-quaternion is the product of another quaternion and the ‘fourth imaginary unit’. The ‘fourth imaginary unit’ is independent to the ‘three imaginary unit’ in the quaternion. After several years studies the author finds out that the quaternion is suitable for describing the property of the gravitational field, while the S-quaternion is suitable for describing the property of electromagnetic field. Comparing the physi-cal quantities describing by the algebra of octonions and the vector terminology can also prove this conclu-sion. This method is never been used before in all the theoretical studies of electromagnetic fields. And may be that can partly explain why it is not completely successful before to describe the electromagnetic field theory directly by the quaternion (rather than the S-quaternion). In the paper, the definition of field source described by the octonions can deduce Maxwell equation in the electromagnetic field. The Maxwell equa-tions, the electromagnetic potential, and the field strength definition, deduced by the field source definition of electromagnetic describe by octonions, are respectively identical with that deduced by the vector terminology in classical electromagnetic theory, but the direction of displacement current and the gauge equation are not. The study claims that the electromagnetic field theory described by the algebra of octonions can cover most of the existing conclusions of classical electromagnetic field theory. And the Maxwell equations in the elec-tromagnetic field can be described by S-quaternion.

Cauchy integrals on Lipschitz surfaces in octonionic space

With the newly developed octonion analytic function theory, we confirm the octonionic analogue of the Calderón's conjecture. As application, we obtain the Plemelj formula in octonionic space.

Quaternion-Octonion Analyticity for Abelian and Non-Abelian Gauge Theories of Dyons

Einstein-Schrödinger (ES) non-symmetric theory has been extended to accommodate the Abelian and non-Abelian gauge theories of dyons in terms of the quaternion-octonion metric realization. Corresponding covariant derivatives for complex, quaternion and octonion spaces in internal gauge groups are shown to describe the consistent field equations and generalized Dirac equation of dyons. It is also shown that quaternion and octonion representations extend the so-called unified theory of gravitation and electromagnetism to the Yang-Mill’s fields leading to two SU(2) gauge theories of internal spaces due to the presence of electric and magnetic charges on dyons.

On the noncommutative and nonassociative geometry of octonionic space time, modified dispersion relations and grand unification

The octonionic geometry (gravity) developed long ago by Oliveira and Marques, J. Math. Phys. 26, 3131 (1985) is extended to noncommutative and nonassociative space time coordinates associated with octonionic-valued coordinates and momenta. The octonionic metric Gμν already encompasses the ordinary space time metric gμν, in addition to the Maxwell U(1) and SU(2) Yang-Mills fields such that it implements the Kaluza-Klein Grand unification program without introducing extra space time dimensions. The color group SU(3) is a subgroup of the exceptional G2 group which is the automorphism group of the octonion algebra. It is shown that the flux of the SU(2) Yang-Mills field strength Fμν through the area-momentum Σμν in the internal isospin space yields corrections O(1∕MPlanck2) to the energy-momentum dispersion relations without violating Lorentz invariance as it occurs with Hopf algebraic deformations of the Poincare algebra. The known octonionic realizations of the Clifford Cl(8), Cl(4) algebras should permit the construction of octonionic string actions that should have a correspondence with ordinary string actions for strings moving in a curved Clifford-space target background associated with a Cl(3, 1) algebra.

The Dirac equation in a non-Riemannian manifold III: An analysis using the algebra of quaternions and octonions

The geometrical properties of a flat tangent space-time local to the manifold of the Einstein–Schrödinger nonsymmetric theory to which an internal octonionic space is attached, is developed here. As an application of the theory, an octonionic Dirac equation for a spin-1/2 particle is also obtained, where is now used an octonionic-like gauge field. It is shown that the (quaternionic) nonsymmetric Yang–Mills theory can be easily recovered and from there, the usual gauge theory on a curved space.

Geometrical properties of an internal local octonionic space in curved space-time.

A geometrical treatment on a flat tangent space local to a generalized complex, quaternionic, and octonionic space-time is constructed. It is shown that it is possible to find an Einstein-Maxwell-Yang-Mills correspondence in this generalized (Minkowskian) tangent space.

An extension of quaternionic metrics to octonions

A treatment of a non-Riemannian geometry including internal complex, quaternionic, and octonionic space is made. Then, an interpretation of this geometry for the nonsymmetric theory of Einstein–Schrödinger, and for the unified theory of Borchsenius is showed. Finally, field equations in the extended octonionic geometry of space-time are obtained through a minimal action principle.

Higher-dimensional Riemannian geometry and quaternion and octonion spaces

An eight-dimensional Riemannian geometry is shown to be the basis of a nonsymmetric theory of gravitation. A hyperbolic complex structure is imposed and the group structure is GL(8,R)→GL(4,R)⊗GL(4,R)⊇GL(4,R). Octonion and quaternion division algebras are used to represent geometrical quantities and spinors. A Lagrangian is constructed that is related to supersymmetry and supergravity theories. The group structure for a hyperbolic octonion scheme is GL(8,qH)→GL(4,OH)≂GL(4,qH) ⊗GL(4,qH)⊇GL(4,qH), while a simpler scheme based on hyperbolic quaternions is GL(8,C)→GL(4,qH)≂GL(4,C)⊗GL(4,C)⊇GL(4,C).

Algebraic Gauge Theory of Quarks and Leptons

The quaternionic Weinberg· Salam theory developed in the previous paper is combined with an alternative gauge theory based on local quaternions, which is naturally extended to SU(3) octonionic gauge theory, an algebraic version of QCD. The result is a reformulation of the standard theory of quarks and leptons with SU(3)X SU(2)LX SU(2)RX U(l), the family structure being incorporated with the hypercomplex algebra of octonions times quaternions. According to the superselection rule operative at present energies in the octonionic Hilbert space of GUnaydin, leptons and hadrons are classified into the associative coherent subspace, while quarks belong to the non·associative transverse component. This picture was originally proposed by GUnaydin and GUrsey, the present method offers QCD as a dynamics, leading to, say, Suura's equation for extended mesons.