Starting with the usual definitions of octonions and split octonions in terms of zorn vector matrix realization, we have made an attempt to write the continuity equation and other wave equations of dyons in split octonions. Hamiltons equations in a nonassociative quantum theory. The known equivalence of special spinors and vectorscalar sets is discussed within the context of the algebra of the split octonions. University diploma in physics theoretical physics, javakhishvili tbilisi state university ph. Nonstandard analysis nsa is an attempt to found, on a rigorous basis, the notion of in nitesimal that was so in uential in the development and early history of the. Pdf octonionic version of dirac equations semantic. Accordingly, we have made an attempt to investigate the work energy theorem. On the structure of the split octonion algebra springerlink. Octonions are 8dimensional number and thats why instead of system of equations, dirac, as well as maxwell equation, each will be represented with only one equation.
We constructed spin ladder structure and dirac equation with splitoctonions. A nilsquare or nilpotent infinitesimal can then be defined. In this paper, the real, complex octonion algebra and their properties are defined. Furey, walter grant scott research fellow in physics trinity hall. On such manifolds, the space of 2forms splits orthogonally into components. A formulation of the maxwell equations in terms of the split octonions is presented in 5. Standard model particles from split octonions30 zaninetti l. The splitoctonions are used in the description of physical law.
The possible generalization of hamiltons equations for a nonassociative quantum theory is proposed. Pdf the geometrical application of split octonions is considered. In the previous work, we studied split octonions, their mathematical properties, and how they can be. The nonexistence of magnetic monopoles in classical electrodynamics is connected with the use of the associativity limit. Pdf octonion generalization of pauli and dirac matrices. Furthermore, any solution to the free dirac equation for a spinonehalf particle is automatically a solution to the free kleingordon equation. Schnabl higher structures and effective eld theory t. Splitquaternions and the dirac equation pdf free download.
We have thus written the generalized potential, generalized field, and generalized current of dyons in terms of split octonions and accordingly the split octonion forms of generalized dirac maxwells equations are obtained in compact and consistent manner. Newton conceived of time as absolute and universal which flows equably without relation to anything external. Koplinger 37 provided proof to a statement that the dirac equation in physics can be found in conic sedenions. Splitquaternions and the dirac equation springerlink. Isotropic subspaces in splitquaternions and splitoctonions luca vitagliano. The importance of zero divisors in physical applications of split algebras was specially noted in 19. Octonion generalization of pauli and dirac matrices international. In our approach the particle mass and electromagnetic potentials are part of an octonionic gradient function together with the spacetime derivatives. Gogberashvili, octonionic version of dirac equations, international journal. If the constant mass of the universe is assumed then from the equation for schwarzschild radius. Biquaternions and the clifford algebra cl2 video 414. This approach departs from the classical logic used in conventional mathematics by denying the general applicability of the law of excluded middle i. Applied mathematics and computation 84 1 1997, 2747. It is shown that a simple continuity condition in the algebra of split octonions suffices to formulate a system of differential equations that are equivalent to the standard dirac equations.
Ariasmarco the inverse tomography problem on orbisurfaces m. Extra high spinmass ratio of the electron corresponds to overrotating kerr geometry, singularity of which has to be regulated, forming a solitonsource 1. An automorphism of any two octonions, and, gives the equation which. It is shown that a simple continuity condition in the algebra of split octonions suffices to formulate a system of differential equations that. Kevin carmondy, circular and hyperbolic quaternions, octonions, sedionions. Generalized kleingordon field equations with octonion.
I show how the quantization of fundamental brackets leads to a quantum theoretic formalism were the wave functions and operators are spacetime clifford algebra valued, and the role of the schroedinger equation is played by a diraclike equation with the operator of. The quaternions naturally describe rotations in three dimensions, and in fact, all symmetry groups are based on one of these four number systems. There are many other models which, either expected, numerically supported, or proved, have the same statistical properties as the kpz equation when both are viewed on large spacetime scales. This equation represents one of the zero divisors in the algebra of split octonions. Generalized contact bundles and their local structure. On weighted dirac operators and their fundamental solutions for anisotropic media. Pdf rotations in the space of split octonions researchgate. We express the spinor bundle and the dirac operator and write down seibergwitten like equations on such manifolds. A new nonassociative algebra for the quantization of strongly interacting fields is proposed. In our approach particle mass and electromagnetic potentials are part of octonionic.
There are precisely two further generalizations of the real and complex numbers, namely, the quaternions and the octonions. In particular, the kpz equation is a beautiful example for an integrable stochastic system. Multiply the nonconjugated dirac equation by the conjugated wave function from the left and multiply the conjugated equation by the wave function from right and subtract the equations. Geometrical applications of split octonions hindawi. The geometry of the octonions oregon state university. Equation , which we receive from the invariance of the interval 10, is the symbolic form of the fourdimensional diracs equation. Splitquaternions and the dirac equation request pdf. D, in theoretical and mathematical physics, lomonosov moscow state university. Four dimensional euclidean gravity and octonions computer science. Splitquaternionic 2spinor equation in order to construct a splitquaternionic version of the dirac equation, recall that the four 2. This will simplify applying mathematical operations and may lead to new physical results.
Todd wilson department of computer science, csuf title. Accordingly the split octonion forms of generalized dirac maxwells equations are obtained in compact and consistent manner. A process of structure constant factorization is illustrated by the realization of the split octonion. The equation of h 2 p kp 0 for a point p 2 m doesnt mean that p is an umbilic point and weingarten map of m in p can be diagonalizable.
It is shown that a simple continuity condition in the algebra of split octonions suffices to formulate a system of differential equations that are equivalent to the. Division algebras and the standard model some short videos filmed by vincent lavigne seminar by c. Thus in general, the kleingordon equation for the time independent case becomes 4. In mathematics, the octonions are anormed division algebra over the real numbers, usually represented by the capital letter o, using boldface o or blackboard bold. Additionally, the duality transformations of gravitoelectromagnetic situation for two reference systems are also represented. We define selfduality of a 2form by considering the part as the bundle of selfdual 2forms. Diracs operator and maxwells equations in vacuum are derived in the algebra of split octonions. Merab gogberashvili associate professor elementary particles and quantum fields email. Octonionic version of dirac equations international. We interpret this as an equation of continuity for probability with j. Vo mera superstring moduli from string eld theory j. Synthetic differential geometry or smooth infinitesimal analysis have roots in category theory.
The dirac equation a great deal more was hidden in the dirac equation than the author had expected when he wrote it down in 1928. The automorphism group of the octonions is an exceptional lie group. First and second fundamental solutions of the timefractional telegraph equation with laplace or dirac operators. The dirac equation in physics the equation of motion of a free spin 12 particle, like e. Syzygies for the dirac operator in several variables in nonstable range josef silhan. Octonions, or cayley numbers, are used to extract the square root of the classical relativistic hamiltonian, in place of the usual dirac matrices.
One implication of this equivalence is that the usual dirac spinor field can be recast as a vectorscalar field, and this construction is outlined. In the nineteenth century, the fourdimensional algebraic structure of the quaternions developed by hamilton, inspired him to suggest that he could provide a unified representation of space and time. This resulted in switching from one uduality orbit to another, or equivalently, from an. The dirac equation is one of the two factors, and is conventionally taken to be p m 0 31 making the standard substitution, p. The proper description of time remains a key unsolved problem in science. Universe free fulltext pecceiquinn transformations.
Color confinement and spatial dimensions in the complex. This paper demonstrates these properties using an explicit representation of the automorphisms on splitoctonions, the. It should be added, however, that it was dirac who found most of the additional. Generalized splitoctonion electrodynamics springerlink. Splitoctonions are used in the description of physical law. Starting with a brief description of gravitodyons, we reformulate the generalized linear gravitational field equations of gravitodyons in terms of split octonion. How the complex quaternions give each of the lorentz reps. The dirac equation can be thought of in terms of a square root of the kleingordon equation. The broken octonion, however, does have value in leading to the higher. We have thus written the generalized split octonion potential wave equations and the generalized fields equation of dyons in split octonions.
It is shown that simple continuity condition in the algebra of split octonions suffices to formulate system of differential equations that are equivalent to the standard diracs one. For example, the dirac equation in physics the equation of motion of a free spin 12 particle, like e. Merab gogberashvili associate professor elementary. The approximations which lead to classical maxwellheaviside equations from full octonionic equations are given. We express the generalized gravitoheavisidian gh potentials. In this work we find the surfaces with the equation h2 k 0, whose generated by graph of a polynomial under homothetic motion groups in lorentz 3space. We consider 7dimensional pseudoriemannian manifolds with structure group. Frontiers time as a geometric property of space physics. The geometry of the octonions tevian dray, corinne a. In this paper our purpose is to rewrite dirac and maxwell equations in vacuum using split octonions.
Gogberashvili 38 applied the octonions to investigate the electromagnetic field equations. As the subalgebra of conic sedenions, the hyperbolic octonions are used to describe the dirac equation. It is usually assumed that the laws of nature have always been the same as they are now. In the present paper we shall show that the algebra of split octonions, we used in 17 to describe the geometry, su. Dirac himself remarked in one of his talks that his equation was more intelligent than its author. In this paper, we attempt to give toy models of a nonassociative quantum mechanics using. Split octonion reformulation of generalized linear. The question we wish to address in this article is whether one can formulate the dirac equation without availing oneself of complex bispinors.
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