Octonionic geometry

Gogberashvili, Merab

doi:10.1007/s00006-005-0003-2

Cited by 31 publications

(34 citation statements)

References 3 publications

(4 reference statements)

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“…Although this assumption is qualified by the fact that time contributes an opposite sign to the metric distance and so distinct from a regular fourdimensional Cartesian vector. Note that the octonions, being the generalization of quaternions, have also been considered as an expanded arena for spacetime [9][10][11][12].…”

Section: Algebraic Formulations Of Timementioning

confidence: 99%

Time As a Geometric Property of Space

et al. 2016

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The proper description of time remains a key unsolved problem in science. Newton conceived of time as absolute and universal which "flows equably without relation to anything external." In the nineteenth century, the four-dimensional algebraic structure of the quaternions developed by Hamilton, inspired him to suggest that he could provide a unified representation of space and time. With the publishing of Einstein's theory of special relativity these ideas then lead to the generally accepted Minkowski spacetime formulation of 1908. Minkowski, though, rejected the formalism of quaternions suggested by Hamilton and adopted an approach using four-vectors. The Minkowski framework is indeed found to provide a versatile formalism for describing the relationship between space and time in accordance with Einstein's relativistic principles, but nevertheless fails to provide more fundamental insights into the nature of time itself. In order to answer this question we begin by exploring the geometric properties of three-dimensional space that we model using Clifford geometric algebra, which is found to contain sufficient complexity to provide a natural description of spacetime. This description using Clifford algebra is found to provide a natural alternative to the Minkowski formulation as well as providing new insights into the nature of time. Our main result is that time is the scalar component of a Clifford space and can be viewed as an intrinsic geometric property of three-dimensional space without the need for the specific addition of a fourth dimension.

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Section: Algebraic Formulations Of Timementioning

confidence: 99%

Time As a Geometric Property of Space

et al. 2016

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“…Let us define the interval in (2+1)-Minkowski space by the element of split quaternions [12][13][14][15],…”

Section: Boostsmentioning

confidence: 99%

“…The geometry of space-time cannot be described apart from the way it is observed, it is just a reflection of the symmetries of the physical signals we receive and of the algebra used in the measurement process [12][13][14][15]. Since all observable quantities we extract from single measurements are real in geometrical applications it is possible to restrict ourselves to the field of real numbers.…”

Section: Introductionmentioning

confidence: 99%

Split quaternions and particles in (2+1)-space

Gogberashvili

2014

Eur. Phys. J. C

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It is well known that quaternions represent rotations in 3D Euclidean and Minkowski spaces. However, the product by a quaternion gives rotation in two independent planes at once and to obtain single-plane rotations one has to apply half-angle quaternions twice from the left and on the right (with inverse). This 'double-cover' property is a potential problem in the geometrical application of split quaternions, since the (2+2)-signature of their norms should not be changed for each product. If split quaternions form a proper algebraic structure for microphysics, the representation of boosts in (2+1)-space leads to the interpretation of the scalar part of quaternions as the wavelengths of particles. The invariance of space-time intervals and some quantum behaviors, like noncommutativity and the fundamental spinor representation, probably also are algebraic properties. In our approach the Dirac equation represents the CauchyRiemann analyticity condition and two fundamental physical parameters (the speed of light and Planck's constant) emerge from the requirement of positive definiteness of the quaternionic norms.

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“…An octonionic geometry in Ref. [4] is described. In this paper a new geometrical interpretation of the products of octonionic basis units is presented and eight real parameters of octonions are interpreted as the spacetime coordinates, momentum and energy.…”

Section: Non-associativity In Physicsmentioning

confidence: 99%

A Non-Associative Quantum Mechanics

Dzhunushaliev

2006

Found Phys Lett

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A non-associative quantum mechanics is proposed in which the product of three and more operators can be non-associative one. The multiplication rules of the octonions define the multiplication rules of the corresponding operators with quantum corrections. The self-consistency of the operator algebra is proved for the product of three operators. Some properties of the non-associative quantum mechanics are considered. It is proposed that some generalization of the non-associative algebra of quantum operators can be helpful for understanding of the algebra of field operators with a strong interaction.

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Octonionic geometry

Cited by 31 publications

References 3 publications

Time As a Geometric Property of Space

Time As a Geometric Property of Space

Split quaternions and particles in (2+1)-space

A Non-Associative Quantum Mechanics

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