finsler geometry, hypercomplex numbers and physics
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Finsler Prize
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Non-Commercial Foundation for Finsler Geometry Research and
Moscow Bauman State Technical University
establish the Prize for the solution of the following problem:

“Construct a unified geometrical theory
of the gravitational and electromagnetic fields
on the base of the 4-dimensional Finsler space
with the Berwald-Moor metric,
or prove the impossibility of such theory”.

        The goal of the Constitutors of the Prize is to stimulate the research exploiting the hyper complex algebras as the universal codes of Geometry and Physics. First of all, the Constitutors are interested in the research connected with the poly-numbers – the commutative associative algebras – as the natural generalizations of real and complex numbers preserving their main arithmetic properties. Recently, it turned out that the poly-numbers are closely connected with various Finsler geometries [5]. These geometries are the generalizations of Riemannian geometries that are the base of General Relativity Theory and other modern geometric field theories. One of the poly-numbers’ classes leads to the Finsler geometry with Berwald-Moor metric (ds4 = dξ1234). Its basic invariants have the powers higher than two, and this makes the difference with the Riemannian and other common geometries. The change of the quadratic metric to the higher order one implies the qualitatively new geometrical ideas, and the Constitutors express the belief that this opens the unexpected and fruitful perspectives in fundamental Physics.
        Although Finsler geometry did not demonstrate its advantages over Riemannian constructions in Physics for almost a hundred years of its existence, it is this geometry that promises a lot now. This is due to the generalization of one of the main objects in many geometries – the scalar product which becomes not the bilinear symmetrical form but the poly-linear form [4]. The natural consequence of such generalization is the need to pass from the common two-index Finsler metric tensor depending on point and direction to its poly-index generalization depending only on point.
        It is known that the attempts to unify several fundamental interactions on the base of geometry with the Riemann metric (Einstein, Weyl [1]) failed, while the more successful ones (Kaluza-Klein, superstrings, branes etc.) use more than four dimensions. If the problem of the unification of the gravitation and electromagnetism appears to be solvable in four dimensions on the base of a Finsler metric, this fact will undoubtedly stimulate both the Finsler geometry research in Physics and the theory of hypercomplex numbers in Mathematics.
        According to the competition conditions, the fundamental metric should be that of Berwald-Moor. This metrics provides the limit transitions not only to the Galilean space but also to the Minkowsky space, thus, making probable the corresponding unified field theory contain both classical and relativistic Physics. Besides, the geometry with this metric can be classified as poly-metric one, and the role of poly-metric geometries among all Finsler geometries was underlined by P.K.Rashevski [3]. This role is partly revealed in the case of complex plane that is an example of a bimetric space.

Competition conditions

1. It is principal that the competitive paper provides the following:
- the unification of the above mentioned fundamental interactions must be purely geometrical (that is must go along with Einstein, Weyl, Kaluza theories) and ensue from the same principles;
- the basic space-time geometry arises from the change of Minkowski metric to the Finsler type Berwald-Moor metric which has the form of (ds4 = dξ1234) in a special basis;
- the space-time must be 4-dimensional;
- the author who proves the impossibility of such theory can also apply for the Prize.

2. Only papers that have been previously published in the reviewed Physical or Mathematical Journal are accepted for the competition. To ensure the priority the authors are recommended to use the ArXive system.

3. The papers for the competition are accepted from August 1, 2007 to August 1, 2017. The papers sent after the last date or after the official announcement of the Prize award (according to the postmark) will not be regarded as competitive.

4. The paper must be written in Russian and English or in English and sent by a registered letter to: “Finsler Prize”, Department of Physics, Bauman MSTU, 2-ya Baumanskaya str., Bld.5, Moscow, Russian Federation.

5. The decision on the Prize award is taken by the Prize Committee as papers become available. The Committee consists of not less than five persons and automatically includes the members of the International Scientific Committee of the next to go annual Conference “Finsler Extensions of the Relativity Theory”.

6. The papers sufficing the criteria mentioned in items 1-4 must be considered and estimated by all the members of the Prize Committee not later than in 6 months after the reception date.

7. Every member of the Prize Committee considers the papers and votes individually; the joint decision is not envisaged.

8. The vote results are sent to the authors but are closed. No review is sent to the authors.

9. The decision on the Prize award is considered to be adopted if there are more than 3/4 positive votes.

10. If there are two or more papers solving the problem equivalently, the priority is given to the paper that has the earlier publication date. As such the date of the acceptance of the paper in a Journal or the date of its publication in the ArXiv system is considered.

11. The Prize amount is a ruble equivalent to the 50000 (fifty thousands) USD according to the Russian Federation Central Bank course on the pay date.

12. The general sponsor and the guarantor of the Prize is the open joint-stock company “ĚÎÇÝŇ” which is a member of the “Antares”-group of Russian enterprises.

13. The Prize money are paid in rubles not later than in 2 months after the official announcement of the Prize Committee which is to appear at the web-site www.polynumbers.ru

14. The taxes and other obligatory payments are paid by the winner according to the laws of Russian Federation.

References

1. Weyl H. Space, time, matter. M, Yanus, 1996
2. Rashevski P.K. Riemann geometry and tensor analysis. M, Nauka, 1967
3. Rashevski P.K. Poly-metric geometry. Proc. Sem. on vector and tensor analysis and applications in Geometry, Mechanics and Physics, vol. V. Ed. Kagan V.F., M-L, OGIZ, 1941
4. Pavlov D.G. Generalization of the scalar product axioms. Hypercomplex numbers in Geometry and Physics, 1 (1), Vol. 1, 2004
5. Garas’ko G.I., Pavlov D.G. Geometry of the non-degenerate poly-numbers. Hypercomplex numbers in Geometry and Physics, 1 (7), Vol. 4, 2007
6. Garas’ko G.I. Field theory and Finsler spaces. Hypercomplex numbers in Geometry and Physics, 2 (6), Vol. 3, 2006
7. www.polynumbers.ru


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