"Hypercomplex Numbers in geometry and Physics" 2 (18), Vol 9, 2012
j018

Content of Issue

SOME PROBLEMS OF MATHEMATICAL PHYSICS IN POLYNUMBERS FIELD THEORY
2012jnq | Pavlov D.G., Kokarev S.S.  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia; RSEC Logos, Yaroslavl, Russia, geom2004@mail.ru, logos-center@mail.ru

Some mathematical aspects of the invariant scalar operator Оn, that appears in polynumbers field theory, are considered together with some characteristic properties of its kernel. Concrete expressions for Berwald-Moor metrics and operator On in isotropic cylindrical, non-isotropic cylindrical and general spherical coordinate systems are derived in case n=3. Part of the results is expressed in terms of special functions, which are hyperbolic analogies of trigonometric ones, spherical harmonics and Legendre polynomials. General kind of radial part of n is calculated for arbitrary Оn. The problem of finding of hyperbolic field, generated by homogeneously charged hyperbolic sphere, is solved. We show, that there is no separable solutions to hyperbolic wave equation with cylindrical symmetry.

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LADDER REPRESENTATION OF NONDEGENERATE POLYNUMBERS
2012joq | Garasko G.I.  // Electrotechnical Institute of Russia, Moscow, Russia, gri9z@mail.ru, gri9z.wordpress.com

The paper gives a generalization of exponential representation of nondegenerate polynumbers, which are named as ladder representation. It given on the base of hypercomplex numbers H4. The representation arise an iterative process, which can be finite or infinite. Also a new approach to understanding of notions of length and angle in polynumber spaces are proposed.

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DIFFERENTIAL GEOMETRY OF FINSLER-SPACETIME TANGENT BUNDLE
2012jpq | Brandt Howard  // U.S. Army Research Laboratory, Adelphi, USA, howard.e.brandt.civ@mail.mil

I draw on my earlier work to review various aspects of the differential geometry of a Finsler-spacetime tangent bundle, all based on the possible existence of a physical upper bound on proper acceleration. In particular, the bundle connection and associated differential geometric fields are calculated for a Finsler-spacetime tangent bundle particularized for the case of a statioinary measuring device.

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ON THE INVARIANCE GROUPS OF THE BERWALD-MOOR METRIC OF ORDER TWO AND THREE
2012jrq | Neagu M., Raeisi-Dehkordi H.  // University Transilvania of Brasov, Brasov, Romania; Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran, mircea.neagu@unitbv.ro, hengameh_62@aut.ac.ir

In this paper we describe the groups of local transformations of coordinates which preserve unchanged on tangent bundles the two and three dimensional Berwald-Moґor metrics. Some algebraic properties of these groups are studied. Finally, we suggest the possible structure of these transformations in the general n-dimensional case.

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THE LORENTZ GROUP IS A BASE FOR DESCRIBING INTERACTION OF BOSONS AND FERMIONS WITH PSEUDO-RIEMANNIAN STRUCTURE OF A BACKGROUND SPACE-TIME. WHAT SHOULD BE INSTEAD FOR FINSLERIAN SPACE-TIME MODELS?
2012jsq | Red'kov V.M., Kisel V.V., Ovsiyuk E.M.  // Institute of Physics, National Academy of Sciences of Belarus, Minsk, Belarus; Belarus State Pedagogical University, Minsk, Belarus; Mozyr State Pedagogical University, Mozyr, Belarus, v.redkov@dragon.bas-net.by, e.ovsiyuk@mail.ru

A brief overview of the basics of the theory of wave equations of elementary particles in the presence of external gravitational field, described as a pseudo-Riemannian structure of space-time, is given. Covariant generalization of the wave equations, set in Minkowski space, are presented for bosons and fermions equally, is presented as the result of a single tetrad recipe by Tetrode-Weyl-Fock-Ivanenko, based on representations of the Lorentz group. The Lorentz group plays a unifying role in describing the fields of all particles (with different spins, massive and massless) in the flat and in curved space-time. The difference lies in the fact that in flat space Lorentz group acts as a global symmetry of the wave equations; and in a pseudo-Riemannian space, it plays a role of local symmetry group (dependent on coordinates). Particular attention is given to the Dirac and Maxwell fields. Because any new theory of physical space-time can be expected to cover already developed and proven models, the question naturally arises: for what should be replaced the method of describing the interactions of elementary particles with a pseudo-Rimannian geometric background, if the space-time endowed with a Finsler structure.The answer to this question, if possible, should be fairly universal and independent of the magnitude of the spin of a particle or its mass. The general answer to this question would provide us with the better understanding what we can expect in physics from the use of Finsler geometry, in the most radical aspect as a basic new geometry of physical space-time.One can also put a more particular question: what effective physical media can be described by using a generalized Maxwell electrodynamics in space-time with Finsler geometry.

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DIFFERENTIAL OPERATORS OF BICOMPLEX FUNCTION AND ISOTROPIC RELATIVISTIC AND ELECTRODYNAMICS EQUATIONS
2012jtq | Goryunov A.V.  // Turan-Astana University, Astana, Kazakhstan, avgor@hotbox.ru

Concepts of bicomplex function and its differential operators in bicomplex space are considered. Interrelations of differential operators in bicomplex space and differential operators in 4-space are obtained. Thus the possibility of calculation of derivatives of bicomplex function on 4-space variables is achieved. As result, the main differential isotropic equations of theory of relativity and electrodynamics are obtained as direct consequence of related bicomplex algebraic formulas of preceded paper.

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THE DIFFERENTIAL ALGEBRA OF BIQUATERNIONS IN EQUATIONS OF MATHEMATICAL AND THEORETICAL PHYSICS
2012jzq | Alexeyeva L.A.  // Institute of mathematics and mathematical modelling, Almaty, Kazakhstan, alexeeva@math.kz

The functional space of biquaternions is considered on Minkovskiy space. Here the scalar-vector biquaternions representation is used which was offered by W. Hamilton for quaternions. With introduction of differential operator - a mutual complex gradient (bigradients), which generalize the notion of a gradient on biquaternions space, biquaternionic wave (biwave) equations are considered, their invariance for group of the Lorence-Puancare transformations is proved and their generalized solutions have been obtained. Biquaternionic form of generalized Maxwell-Dirac equation is constructed and its decisions are researched on base of the differential biquaternions algebra. Its generalized decisions are built with use of scalar potential. The new equation for these potential are constructed which unites known equations of quantum mechanics (Klein-Gordon and Schrodinger Eq.). The nonstationary, steady-state and harmonic on time scalar fields and generated by them the spinors and spinors fields in biquaternionic form are constructed.

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FINSLERIAN APPROACH TO THE ELECTROMAGNETIC INTERACTION IN THE PRESENCE OF ISOTOPIC FIELD-CHARGES AND A KINETIC FIELD
2012jxq | Darvas Gyorgy  // Symmetrion, Budapest, Hungary, darvasg@iif.hu

This paper deals with the application of the isotopic field-charge spin theory to the electromagnetic interaction. First there is derived a modified Dirac equation in the presence of a velocity dependent gauge field and isotopic field charges (namely Coulomb and Lorentz type electric charges, as well as gravitational and inertial masses). This equation is compared with the classical Dirac equation and there are discussed the consequences [6, 34, 35, 37]. There is shown that since the presence of isotopic field-charges would distort the Lorentz invariance of the equation, there is a transformation, which restores the invariance, in accordance with the conservation of the isotopic field-charge spin [8]. It is based on the determination of the Fмн field tensor adapted to the above conditions. The presence of the kinetic gauge field makes impossible to assume a flat electromagnetic interaction field. The connection field, which determines the curvature, is derived from the covariant derivative of the kinetic (velocity dependent) gauge field. In this case, there appears a velocity dependent metric, what involves a (velocity arrowed) direction-dependent, that means, Finsler geometry [11, 14]. The option of such a «theory of the electrons» (with the words of Dirac) was shown in the extension of his theory in [23]. This paper is an attempt to a further extension.

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FINSLER GEOMETRY IN TERMS OF WORLD FUNCTION
2012jvq | Rylov Yu.A.  // Institute for Problems in Mechanics RAS, Moscow, Russia, rylov@ipmnet.ru

It is shown that the space-time geometry should be formulated in terms of the world function, because only description in terms of world function admits one to recognize similar geometrical objects in regions of the space-time geometry with different geometries. The Berwald-Moor geometry formulated in terms of the world function appears to be multivariant geometry, which hardly can be used as a space-time geometry, because in this geometry the world lines wobbling of free particles differs from the real wobbling.

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GEOMETRIC PROPERTIES OF EINSTEINS LAW OF ADDITION OF VELOCITIES AND QUATERNIONIC LAW OF ADDITION
2012jwq | Ahmad Mushfiq  // Rajshahi University, Rajshahi, Bangladesh, mushfiqahmad@gmail.com

If velocities u and v add up to give w. The three velocities form a triangle. The same velocities, but in the opposite direction, -v and -u should add up to give -w. Isotropy of space requires that the reversal of direction should reverse the order of addition - -v should come before -u. Lorentz Einstein addition does not fulfill this requirement and Wigner rotation in invoked to correct it. Reciprocal symmetric transformation, we are proposing, maintains the isotropy of space and Wigner rotation is not needed.

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