"Hypercomplex Numbers in geometry and Physics" 2 (20), Vol 10, 2013
j020

Content of Issue

Geometry and physics of holomorphic functions in polynumbers field theory
2013jgz | 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

A physical-geometric interpreting of holomorphic functions over polynumbers variable for a number of holomorphicity classes is investigated with using tangent construction, developed in [5]. It is shown, that any concrete choice of holomorphic function (polynumber potential) defines some field-theoretical model with background space-time of GR together with tensor fields of a various ranges. The question on local causal structure of pseudoRiemannian space-time, obtained by tangent construction in Berwald-Moor space, is investigated in general form. It is shown, that the only two causal types of space-times with signature(+,−, −,−) and(+,+,−,−) can be generated by tangent construction. The systems of differential equations, defining polynumber potential for Schwarzschild metric and cosmological FRW-metrics are derived.

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Material point nonrelativistic movement in spherical potential field in view of space-time expansion
2013jkz | Garas’ko G.I.  // Electrotechnical Institute of Russia, Moscow, Russia, gri9z@yandex.ru, gri9z.wordpress.com

Movement of material point in Newton potential with singularity in origin of coordinates is considered. Differential equation describing dependence of square of orbital velosity on distance from origin of coordinates is obtained and its approximate solution is presented. In the case of large distance from origin of coordinates, square of orbital velosity go to non-zero value, which depends on space-time expansion increment.

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Finsler geometry
2013jlz | Garas’ko G.I.  // Electrotechnical Institute of Russia, Moscow, Russia, gri9z@yandex.ru, gri9z.wordpress.com

If before Finsler geometry pretends only on geometrization of classical mechanics than after formulation of Finsler geometry self-sufficiency principle we can speaking that the geometry pretends on whole physics geometrization. From that principle follow field theory equations and electromagnetic field and gravitational field naturally unified in four-dimensional pseudo-Riemannian space and in curved Berwald-Moore space. Energy momentum tensor concerned with conservation laws follows from E. Neter theorem. In weak fields approach from Finsler geometry self-sufficiency principle follows linear field theory equations for several independent fields. In opposite case field equations becomes nonlinear and fields becomes non-independent that leads to superposition principle nonfulfillment. In any Finsler space exists a field or some fields in that space may be supplemented with field, which make sense of action as function of coordinates and analogous to real part of complex potential on Euclidean plane. We propose name such potential as conformal potential. Nondegenerate polynumbers are finslerian spaces, which are very interesting itself and possibly may use in physics. For any finslerian space is possible to build equation analogous to Schrodinger equation or Klein-Gordon equation. This means that the geometry allows further quantum-mechanical development.

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Dual spaces, particle singularities and quartic geometry
2013jmz | Peter Rowlands  // University of Liverpool, Liverpool, UK, p.rowlands@liverpool.ac.uk

Relativistic quantum mechanics and the properties of Dirac fermions can be generated in a particularly powerful way using two vector spaces which are commutative to each other and which contain identical information. The apparently broken symmetry between the two spaces observed through the quadratic geometry of ordinary space becomes a perfect and unbroken symmetry in the quartic geometry which defines the single physical quantity through which the two spaces can be combined.

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Physical Finsler Coordinates for Classical Motion
2013jnz | Howard E. Brandt  // U.S. Army Research Laboratory, Adelphi, USA, howard.e.brandt.civ@mail.mil

It was argued in earlier work that the four-velocity of a measured quantum particle excitation of a Finslerian quantum field in the tangent space manifold of spacetime is not a suitable Finsler coordinate, whereas the four velocity of the measuring device relative to the vacuum is a suitable Finsler coordinate. Furthermore, in the present work, it is argued that the physical Finsler coordinate for describing the classical motion of a macroscopic object is the four-velocity of the classical object, which in effect acts as a measuring device measuring the characteristics of the metric field. Specifically, geodesic motion of a macroscopic object in a Finslerian spacetime is considered, where the appropriate physical Finsler coordinate is the four-velocity of the object undergoing geodesic motion. It is also claimed that for a macroscopic object, such as a macroscopic measuring device, consisting of more than Avogadro’s number of atoms, any supposed quantum state is negligibly small, so that for all practical purposes the object is best described by classical mechanics. It is argued that this and the above follow from a reasonable upper bound on physically possible proper acceleration.

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Polyadic operations on the sets of matrix-valued functions
2013joz | Gal’mak A.M.  // Mogilev State University of Food Technology, Mogilev, Belarus, halm54@mail.ru

Main object of the present paper are polyadic operations on the setМJ (P). Elements of the set are functions defined on the nonempty setJwith values, which belong to set of all matrixМ(P)with elements from some ringP. Such polyadic operations for the first time introduce E. Post. He consider the case J={1, . . . , m−1}, where C – field of complex number.

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On possible effects of the spinor structures in quantum physics
2013jpz | Elena Ovsiyuk, Olga Veko, Alexandru Oan˘a, Mircea Neagu, Vladimir Balan, Victor Red’kov  // Mozyr State Pedagogical University, Mozyr, Belarus; University Transilvania of Bra¸sov, Bra¸sov, Romania; University Politehnica of Bucharest, Bucharest, Romania; B.I. Stepanov Institute of Physics, NAS of Belarus, v.redkov@dragon.bas-net.by

The paper discusses the following topics: the spinor structure of space models; the relation between the Dirac–Schwinger quantization rule and the superposition principle in quantum mechanics; the manifestation of spinor space structure in classifying the solutions of the Dirac equation and for the matrix elements which are related to physical quantities; spinors in polarization optics; the Jones formalism for completely and partly polarized light; General Relativity and Riemannian space-time models with spinor structure and tetrad (vierbein) formalism.

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Spinors, matrix structures, and projective geometry in polarization optics
2013jrz | Elena Ovsiyuk, Olga Veko, Mircea Neagu, Vladimir Balan, Victor Red’kov  // Mozyr State Pedagogical University, Mozyr, Belarus; University Transilvania of Brashov, Brashov, Romania; University Politehnica of Bucharest, Bucharest, Romania; B.I. Stepanov Institute of Physics, NAS of Belarus, v.redkov@dragon.bas-net.by

The paper discusses the role played by Mueller and Jones formalisms in polarization optics, by addressing the following aspects: restriction to theSU(2) symmetry, non-relativistic Stokes 3-vectors; Cartan 2-spinors in polarization optics; Jones 4-spinors for partially polarized light; the linear groupSL(4,) and the classification of 1-parametric Mueller matrices; semi-group structure and classification of degenerate Mueller matrices.

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Finsler-Berwald space with very special relativity
2013jsz | Narasimhamurthy S.K., G.N. Latha Kumari  // Kuvempu University, Shimoga, Karnataka, India, nmurthysk@hotmail.com, nslathams@gmail.com

The symmetry of space time is described by using the so called isometric group. The generators of isometric group are directly connected with the Killing vectors [18]. In this paper, we present an explicit connection between the symmetries in the VSR and isometric group of Finsler space. The Killing vectors in Finsler space are constructed in a systematic way. Further, the solutions of Killing equations are present explicitly in the isometric symmetry of Finsler spaces. The Killing vectors of Finsler-Berwald space are given and we proved that the 4-dimensional Finsler-Berwald space with constant curvature has 15 independent Killing vectors.

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The Einstein equations for case of nonholonomic distributions with Berwald-Moor metric
2013jtz | Galaev S.V.  // Saratov State University, Saratov, Russia, sgalaev@mail.ru

The notion of an extended connection closing sub-Finslerian space codimension 1 is introduced. On the zero-curvature distribution of sub-Finslerian space with the Finsler metric an almost contact K¨ahlerian space is obtained.

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Nonholonomic geodesic in space with for Berwald-Moor metric
2013jzz | Bukusheva A.V.  // Saratov State University, Saratov, Russia, bukusheva@list.ru

On a smooth five-dimensional manifold is considered distribution of codimension 1 with the Finsler metric type Berwald-Moor. We define the intrinsic connection associated with a given metric structure.

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Redshift for a weak gravitational field in Finsler space of events Berwald-Moor
2013jxz | Zaripov R.G.  // Institute of Mechanics and Engineering, Kazan Science Center, Russian Academy of Sciences, Kazan, Russia, zaripov@mail.knc.ru

The weak gravitational field in curved Finsler space of events Berwald-Moor is considered. From the equations of a geodesic line classical equations of motion of a particle in a limiting case for non-Newtonian three-dimensional space in a gravitational field are given. Linear equations for a metric tensor and their solutions are reduced. The problem on redshift is considered.

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Investigation of role of irreducible torsion components under plane wave propagation in Riemannian-Cartan
2013jvz | Shcherban V.N.  // Moscow State Pedagogical University, Moscow, Russia, vovan-ru1@yandex.ru

For quadratic lagrangian of general type obtained variational equations of gravitational field in Riemannian-Catran space. Structure of irreducible torsion components under plane wave propagation in Riemannian-Catran space is investigated.

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