Dirac and Maxwell Systems in Split Octonions

The known equivalence of 8-dimensional chiral spinors and vectors, also referred to as triality, is discussed for (4 + 4)-space. Split octonionic representation of SO(4, 4) and Spin(4, 4) groups and the trilinear invariant form are explicitly written and compared with Clifford algebraic matrix representation. It is noted that the complete algebra of split octonionic basis units can be recovered from the Moufang and Malcev relations for the three vector-like elements. Lagrangians on split octonionic fields that generalize Dirac and Maxwell systems are constructed using group invariant forms. It is shown that corresponding equations are related to split octonionic analyticity conditions.

TAYLOR SERIES AND ORTHOGONALITY OF THE OCTONION ANALYTIC FUNCTIONS

The Taylor series of the Octonion analytic function is given. And the orthogonal formula for the Octonion analytic functions is also obtained.

AN IMPROVEMENT OF THE OCTONIONIC TAYLOR TYPE THEOREM

We prove that the octonionic polynomials Vl 1k……lk. are independent of the associative orders k. This improves the oetonionic Taylor type theorem.

THE ALL-ASSOCIATIVITY OF OCTONIONS AND ITS APPLICATIONS

Using an elementary method, we give a new proof of the all-associativity of octonions. As some applications, the known Taylor theorem is improved, and a new definition and new properties of octonionic determinant are also obtained.

The Transition from Pre-Octonionic to Octonionic Gravity and How It May Be Pertinent to a Re-Do of the HUP for Metric Tensors

The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “instants”, is avoided by having initial Octonionic geometry embedded in a larger, nonlinear “pilot model” (semi classical) embedding structure. The Penrose suggestion of recycled space time avoiding a “big crunch” is picked as the embedding structure, so as to avoid the “instants” of time issue. Getting Octionic gravity as embedded in a larger, Pilot theory embedding structure may restore Quantum Gravity to its rightful place in early cosmology without the complication of then afterwards “Schrodinger equation” states of the universe, and the transformation of Octonionic gravity to existing space-time is explored via its possible linkage to a new version of the HUP involving metric tensors. We conclude with how specific properties of Octonion numbers algebra influence the structure and behavior of the early-cosmology model. This last point is raised in Section 14, and is akin to a phase transition from Pre-Octonionic geometry, in pre-Planckian space-time, to Octonionic geometry in Planckian space-time. A simple phase transition is alluded to;making this clear is as simple as realizing that Pre-Octonionic is for Pre-Planckian Space-time and Octonionic is for Planckian Space-time. We state that the Standard Model of physics occurs during Planckian Space-time. We also argue that the Standard Model does not apply to Pre Planckian Space-time. This is commensurate with the Octonion number system NOT applying in pre-Planckian space-time, but applying in Plankian space-time. And the last line of Equation (54) gives a minimum time step in pre-Planckian space-time when we do NOT have the Standard Model of physics, or Octonionic Geometry.

Pre-Octonionic to Octonionic Gravity and Could Kinetic Energy Be Larger than Potential Energy before Inflation?

We look at early universe space-time which is characterized by a transition from Pre-Planckian to Planckian space-time. In doing so we also invoke the geometry of Octonionic non-commutative structure and when it breaks down. Doing so is also equivalent to a speculation given earlier by the author as to the kinetic energy of Pre-Planckian space-time being significantly larger than the Potential energy, which is the opposite of what happens after the onset of Inflation, with the assumption as to how this is justified given in a (Pre- Planckian) Hubble Parameter set as of Equation (16), and we close with a comparison of this proposal with string cosmology, as represented in the 2nd reference in this paper.

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-Laffier problem is discussed.

The Analyticity for the Product of Analytic Functions on Octonions and Its Applications

Given two left Oc-analytic functions f, g in some open set Ω of R8, we obtain some sufficient conditions for fg is also left Oc-analytic in Ω. Moreover, we prove?fλ that is a left Oc-analytic function for any constants λ∈Oc if and only if is a complex Stein-Weiss conjugate harmonic system. Some applications and connections with Cauchy-Kowalewski product are also considered.

Multidimensional Laplace Transforms over Quaternions, Octonions and Cayley-Dickson Algebras, Their Applications to PDE

Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.

Powers of Octonions

As it is known, Binomial expansion, De Moivre’s formula, and Euler’s formula are suitable methods for computing the powers of a complex number, but to compute the powers of an octonion number in easy way, we need to derive suitable formulas from these methods. In this paper, we present a novel way to compute the powers of an octonion number using formulas derived from the binomial expansion.

Projection of 5 Dimensions into Four Is When the Octonionic Structure Kicks in as an Emergent Gravity Phenomenon

The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “instants”, is avoided by having initial octonionic geometry embedded in a larger, nonlinear (semi-classical) embedding structure. We detail some of what the quantum HUP is, in terms of deterministic 5-dimensional geometry and show that the projection of 5 dimensions into four is when the octonionic structure kicks in as an emergent gravity phenomenon. The example of such is to consider what would happen if there was an aftermath to a presumed initial causal discontinuous structure, after math being the generation of millions of Planck mass black holes, which would in themselves generate emergent gravity.

The Paley-Wiener theorem in the non-commutative and non-associative octonions

The Paley-Wiener theorem in the non-commutative and non-associative octonion analytic function space is proved.

Octonion Analysis of Several Variables

The octonions are distinguished in the M-theory in which Universe is the usual Minkowski space R4 times a G2 manifold of very small diameter with G2 being the automorphism group of the octonions.The multidimensional octonion analysis is initiated in this article,which extends the theory of several complex variables,such as the Bochner–Martinelli formula,the theory of non-homogeneous Cauchy–Riemann equations,and the Hartogs principle,to the non-commutative and non-associative realm.

Three-line Theorem on the Octonions

The three-line theorem on the octonions is obtained, which generalizes the result of J. Peetre and P. Sj?lin from the associative Clifford algebra to non-associative octonion algebra.

Split Octonion Reformulation for Electromagnetic Chiral Media of Massive Dyons

In an explicit, unified, and covariant formulation of an octonion algebra, we study and generalize the electromagnetic chiral fields equations of massive dyons with the split octonionic representation. Starting with 2 × 2 Zorn's vector matrix realization of split-octonion and its dual Euclidean spaces, we represent the unified structure of split octonionic electric and magnetic induction vectors for chiral media. As such, in present paper, we describe the chiral parameter and pairing constants in terms of split octonionic matrix representation of Drude-Born-Fedorov constitutive relations. We have expressed a split octonionic electromagnetic field vector for chiral media, which exhibits the unified field structure of electric and magnetic chiral fields of dyons. The beauty of split octonionic representation of Zorn vector matrix realization is that, the every scalar and vector components have its own meaning in the generalized chiral electromagnetism of dyons. Correspondingly, we obtained the alternative form of generalized Proca–Maxwell's equations of massive dyons in chiral media. Furthermore, the continuity equations, Poynting theorem and wave propagation for generalized electromagnetic fields of chiral media of massive dyons are established by split octonionic form of Zorn vector matrix algebra.

Quaternion rings and octonion rings

In this paper, for rings R, we introduce complex rings C（R）, quaternion rings H（R）, and octonion rings O（R）, which are extension rings of R; R C（R） H（R） O（R）. Our main purpose of this paper is to show that if R is a Frobenius algebra, then these extension rings are Frobenius Mgebras and if R is a quasi-Frobenius ring, then C（R） and H（R） are quasi-Frobenius rings and, when Char（R）=2, O（R） is also a quasi-Frobenius ring.

Quantum Computing with Octonions

There are two schools of“measurement-only quantum computation”.The first(Phys.Rev.Lett.86(22),5188-5191(2001))using prepared entanglement(cluster states)and the second(Phys.Rev.Lett.101(1),010501(2008))using collections of anyons which,according to how they were produced,also have an entanglement pat-tern.We abstract the common principle behind both approaches and find the notion of a graph or even continuous family of equiangular projections.This notion is the leading character in the paper.The largest continuous family,in a sense made pre-cise in Corollary 4.2,is associated with the octonions and this example leads to a universal computational scheme.Adiabatic quantum computation also fits into this rubric as a limiting case:nearby projections are nearly equiangular,so as a gapped ground state space is slowly varied,the corrections to unitarity are small.

伏羲八卦与八元数

说明了19世纪西方人发明八元数的构造方法与我国古代邵雍(1011～1077)建立先天八卦的方法是一致的,并在先天八卦与八元数的基元之间建立了一种数学意义上的同构.

S^6上非复常Khler角曲面的结构方程

设M2是纯八元数ImO中的单位球面S6上非复常Ka¨hler角曲面.运用Cayley三角,选取M2在ImO上的一组特殊基底,通过这个基底得到M2在ImO中的结构方程,并给出这个结构方程的一个应用.

关于一类四元数及八元数方程解的显式表示

运用初等求根方法研究形如x2=x0(其中x,x0为四元数或八元数)的方程的解,获得其解的显式表示,并给出数值例子。