"HyperComplex Numbers in Geometry and Physics" 1 (5), vol. 3, 2006
j005

Content of Issue is in the theme. The journal in one file is below.

On the World function and the relation between geometries
2006jaz | Garas`ko G. I.  // Electrotechnical Institute of Russia, Moscow, gri9z@mail.ru}

It is shown that the World function can be regarded as a link between the qualitatively different geometries with one and the same congruence of the world lines (geodesics). If the space in which the World function is defined is a polynumber space, then the hypothesis of the analyticity of the vector field of the generalized velocities of the world lines lead to the strict limitations on the structure of the World function. Main result: Minkowskian space and polynumber space correspond to the same physical World.

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Construction of the pseudo Riemannian geometry on the base of the Berwald-Moor geometry
2006jay | Garas`ko G. I., Pavlov D. G.  // gri9z@mail.ru; geom2004@mail.ru

The space of the associative commutative hyper complex numbers, H₄, is a 4-dimensional metric Finsler space with the Berwald-Moor metric. It provides the possibility to construct the tensor fields on the base of the analytical functions of the H₄ variable and also in case when this analyticity is broken. Here we suggest a way to construct the metric tensor of a 4-dimensional pseudo Riemannian space (space-time) using as a base the 4-contravariant tensor of the tangent indicatrix equation of the Berwald-Moor space and the World function. The Berwald-Moor space appears to be closely related to the Minkowski space. The break of the analyticity of the World function leads to the non-trivial curving of the 4-dimensional space-time and, particularly, to the Newtonian potential in the non-relativistic limit

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The Octonions
2002bae | Baez John C.  // Department of Mathematics University of California, baez@math.ucr.edu

From ArXiv:math.RA/0105155 v4 23 Apr 2002
The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.

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