"Hypercomplex Numbers in geometry and Physics" 2 (16), Vol 8, 2011
j016

Content of Issue

ANALYTIC, DIFFERENTIAL-GEOMETRIC AND ALGEBRAIC PROPERTIES OF SMOOTH FUNCTION OVER POLYNUMBERS
2011jnw | Pavlov D.G., Kokarev S.S.  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia; RSEC Logos, Yaroslavl, Russia, geom2004@mail.ru

The paper is a brief review of results on the theory of differentiable functions of polynumbers variable Pn → Pn and of its applications. We define derivative of a function of polynumbers variable, basing on special classification of degenerated (i. e. irreversible) polynumbers and on the theorem stating general form of R-linear mapping Pn → Pn. Then we define holomorphic function of polynumbers variable as subclass of differentiable functions by the set of differential conditions (polynumbers analog of Cauchy-Riemannian conditions), which in isotropic basis have the form: kdf = 0, (k = 1, . . . , n−1) where kd= Ckd, C - conjugation in algebra Pn. Some generalized classes of holomorphic functions Gnka1 ,ka2 ,...,kar are defined by monomic differential equations, which can be classified by the set of vectors of non-negative integer n-dimensional lattice Zn+ . The question of holomorphic continuation of some smooth function from submanifolds of Pn to Pn is discussed. We derive polynumbers version of Cauchy theorem and Cauchy integral formulae together with possible multidimensional generalization the first one. Using symmetric Berwald-Moor form we develop symmetric analog of differential forms calculus (Symmetric product, Hodge star and external differential). We analyze transformation properties of derivatives of scalar polynumbers functions and of those geometrical objects, that can be constructed from these derivatives. In particular, we construct real scalar invariants, appropriate for Lagrangian formalism in polynumbers field theory. Basing on supports algebra we formulate tangent construction, playing important role in physical interpreting of polynumbers field theory. The formula for Levi-Civita connections coefficients concordant with Berwald-Moore form and formula for volume form based on n-root metric of Finsler type in even dimensions are derived. Also we consider some deformational aspects of smooth function of polynumbers variable and prove the statement, that any R-algebra can be embedded into space of bilinear forms over Pn. The paper can be treated as preliminary sketch of general theory of functions of polynumbers variable (TFPV).

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FINSLER GEOMETRY IN THEORY OF GRAVITATION
2011jow | Vladimirov Yu.S.  // Moscow State University, Moscow, Russia, yusvlad@rambler.ru

Special relativistic theory in its traditional form formulates theory as incomplete. Particle movement dynamic equation is fixed in accordance with relativistic theory principles while particle condition is fixed in non-relativistic form. Ignoring the non-relativistic idea of particle condition we manage to construct a single formal description for determinate and nondeterminate particles which leads to the necessity of multivariant spacetime geometry. Quantum principles are based on multivariant geometry and lose role of the first physical principles. The frame concept of elementary particles gives relativistic description of particle condition which turns out to be applicable for the case of discrete and multivariant spacetime geometry. The frame concept finishes transition from nonrelativistic physics to relativistic one and realizes complete geometrization of physics.

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THE BERWALD-MOOR METRIC IN NILPOTENT DIRAC SPINOR SPACE
2011jpw | Rowlands Peter  // University of Liverpool, Liverpool, UK, p.rowlands@liverpool.ac.uk

The nilpotent version of the Dirac equation can be constructed on the basis of the algebra of a double vector space or complexified double quaternions. This algebra is isomorphic to the standard gamma matrix algebra, with 64 units which can be produced by just 5 generators. The H4 algebra used in the Berwald-Moor metric is a distinct subalgebra of this 64-part algebra. The creation of the 5 generators requires the rotation symmetry of one of the two component vector spaces to be preserved while the other is broken. It is convenient to identify the respective spaces as an observable real space and an unobservable vacuum space, with corresponding physical properties. In combination the 5 generators produce a nilpotent structure which can be identified as a fermionic wavefunction or solution of the Dirac equation. The spinors required to generate the 4 components of the wavefunction can be derived from first principles and have exactly the same form as the four components of the Berwald-Moor metric. They also incorporate the units of the H4 algebra in an identical way. The spinors produce a zero product which can be interpreted in terms of a fermionic singularity arising from the distortion introduced into the vacuum (or spinor) space by the application of a nilpotent condition.

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PHYSICAL FINSLER COORDINATES IN SPACETIME
2011jrw | Brandt Howard E.  // U.S. Army Research Laboratory, Adelphi, USA, howard.e.brandt.civ@mail.mil

In Finsler geometry a Finsler coordinate is a coordinate in the tangent space manifold of a given base manifold. As such it has been given various definitions in the relativity and field theory literature and often even remains undefined physically. Physically meaningful coordinates of a point in the tangent bundle of spacetime are the spacetime and fourvelocity coordinates of the measuring device. It is here emphasized that the four-velocity of the measuring device need not be the same as the four-velocity of the measured object, either classically or quantum mechanically. The four-velocity of a measured particle excitation of a Finslerian quantum field in the tangent space manifold of spacetime is not a suitable physical Finsler coordinate. The role of the Finsler coordinate is elaborated in a detailed example involving a Finslerian quantum field and associated microcausality.

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GEOMETRIZATION OF PHYSICS: DISCRETE SPACE-TIME GEOMETRY AND RELATIVITY THEORY
2011jsw | Rylov Yu.A.  // Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia, rylov@ipmnet.ru

Conventional form of the special relativity theory formulates the theory in an unaccomplished form. The dynamic equations of the particle motion are written in accordance with the relativity principles, whereas the particle state is described in the nonrelativistic form. Ignoring the nonrelativistic concept of particle state, one succeeds to construct an uniform formalism for description of deterministic and indeterministic particles, which leads to a necessity of multivariant space-time geometry. The quantum principles are founded by existence of the multivariant space-time geometry and lose the role of prime physical principles. Skeleton conception of the elementary particles realizes relativistic description of the particle state, which appears to be adequate in the case of discrete and multivatiant space-time geometry. The skeleton conception accomplishes transition from nonrelativistic physics to the relativistic one and realizes complete geometrization of physics.

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HYPERCOMPLEX NUMBERS AND THE ALGEBRAIC SYSTEM OF GENETIC ALPHABETS. ELEMENTS OF ALGEBRAIC BIOLOGY
2011jpw | Petoukhov S.V.  // Mechanical Engineering Institute, RAS, Moscow, Russia, spetoukhov@gmail.com

This article presents some results of investigation of the multi-level system of moleculargenetic alphabets by means of matrix methods from theory of noise-immunity coding. These studies have revealed links of the system of alphabets with some systems of hypercomplex numbers (Hamilton quaternions and Cockle split-quaternions and their extensions), Kronecker families of matrices, orthogonal systems of Rademacher functions and Walsh functions, Hadamard matrices etc. Structural parallels are shown between molecular-genetic alphabets and a system of inheritance of traits in holistic organisms, which obeys the Mendel laws and which is reflected in genetic Punnett squares. The system of molecular-genetic alphabets, common to all living organisms, possesses algebraic properties which lead to a new - algebraic - way of cognition of living matter. This cognition is related with development of algebraic biology associated with hypercomplex numbers. Living matter, providing the transmission of hereditary information in the chain of generations, is presented as information entity deeply algebraic in its nature.

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LINE INTEGRATION AND SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS OVER CAYLEY-DICKSON ALGEBRAS
2011jqw | Ludkovsky S.V.  // Moscow State Technical University MIREA, Moscow, Russia, sludkowski@mail.ru

This article presents some results of investigation of the multi-level system of moleculargenetic alphabets by means of matrix methods from theory of noise-immunity coding. These studies have revealed links of the system of alphabets with some systems of hypercomplex numbers (Hamilton quaternions and Cockle split-quaternions and their extensions), Kronecker families of matrices, orthogonal systems of Rademacher functions and Walsh functions, Hadamard matrices etc. Structural parallels are shown between molecular-genetic alphabets and a system of inheritance of traits in holistic organisms, which obeys the Mendel laws and which is reflected in genetic Punnett squares. The system of molecular-genetic alphabets, common to all living organisms, possesses algebraic properties which lead to a new - algebraic - way of cognition of living matter. This cognition is related with development of algebraic biology associated with hypercomplex numbers. Living matter, providing the transmission of hereditary information in the chain of generations, is presented as information entity deeply algebraic in its nature.

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