finsler geometry, hypercomplex numbers and physics
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Murom, FERT-2016
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Roger Penrose - 2013
Moscow, FERT-2012
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FinslerSchool "Wood Lake"
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About this site

There is a close relation between notion of Number and such fundamental physical categories as space, time, matter, field and almost no one has doubt about this. Usually, this relation is associated with such particular numbers as real and complex numbers, sometime on add quaternions and octavas. The organizers of this site do not deny the fundamental role of these numbers. However, they pay attention to the fact that there are others generalizations of number, for example, hypercomplex numbers, waiting for interpretation of their link with physics and geometry. The problem of physical and geometrical interpretation of hypercomplex numbers is an actual task all the more because spaces associated with hypercomplex numbers belong to the class of Finsler spaces, the more general class than Riemannian manifolds. The progress in physics often was conditioned by new geometrical views and we hope that the explanation of geometrical background must lead to new qualitative consequences in physics and for this once.

Einstein was forced to get out of the classic Euclid geometry while creating the relativistic theory. He replaced it with the Riemann one. It is natural to assume that the future development of physics also will demand some new geometry. Those may become the Finsler geometry, which is more general than the Minkovskiy geometry. It is fundamentally important, that the points of Finsler spaces in some cases may be expressed in terms of hypercomplex numbers, algebras with special exclusive properties.

On the background of the infinite variety and baffling complexity of Finsler spaces surprises the existence of simple and wonderful exceptions. In particular, these Finsler Spaces demonstrate their direct link with hypercomplex numbers possessing usual associative and commutative properties. Unfortunately, today there are no theorems which would classify all selected thus Finsler Spaces. However, being based only on applied aspects of natural links of geometry with modern physics it is possible to note the algebras of the quaternions above the field of the complex numbers (biquaternions) and above the ring of dual (diquaternions), and also algebras of the complex numbers above the complex numbers ( biucomplex numbers) and dual numbers above the dual numbers (quadrinumbers). All this spaces possess the multiplicative norms of the fourth order, and at the same time they demonstrate close connection with the fundamental for physicists group of Lorenz.

Taking into account that the variety of natural for Finsler spaces invariants, as a rule, essentially exceeds a quantity of corresponding parameters of quadratic manifolds, it is very tempting to attempt to use such geometry for the modeling of different physical phenomena, up to comparison of one of them with the real space-time, instead of habitual Riemannian geometry. The presence of such invariants leads to the fact that in many Finsler spaces appear not only interesting linear, but also some special nonlinear transformations. The certain analogue of such transformations are the conformal transformation of usual Euclidians spaces, wich preserve not distances, but angles. Sins in the spaces with the higher degree of fundamental form, than quadratic, besides lengths and angles they appear the fundamentally new metric properties - that and the number of equiform transformation (nonlinear, as a rule), it grows not only in the quantitative, but also in the qualitative plan. The development of Finsler geometry, going side by side with the expansion of the concept of the number brings the perceptible benefit to both these divisions of mathematics. Thus, for the first appears the possibility to solve one of the key problems of the geometry - natural and simple means to generalize the concept of angle, after introducing instead of the scalar product, connected with the symmetrical bilinear form, symmetrical polyscalar forms from three and more vectors. Such approach shows that the idea of Finsler metric tenzor, introduced by Cartan, proves to be not completely effective and it requires replacement by another tenzor with rank more than two.

The XI International Conference "Finsler Extensions of Relativity Theory" (FERT-2015) was held in Murom (Russia)
The X International Conference "Finsler Extensions of Relativity Theory" (FERT-2014) was held in Brasov (Romania)
The IX International Conference "Finsler Extensions of Relativity Theory" (FERT-2013) was held 26--30 August 2013 in Debrecen (Humgary)
Roger Penrose in Moscow
The VIII International Conference "Finsler Extensions of Relativity Theory" (FERT-2012) was held 25 June -- 1 July 2012 in Moscow and Fryazino, Russia
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