Algebrodinamics: Primodial Light, Particles-Caustics and Flow of Time
2004jas | Kassandrov V. V.

In the field theories with twistor structure particles can be identified with (spacially bounded) caustics of null geodesic congruences defined by the twistor field. As a realization, we consider the ``algebrodynamical'' approach based on the field equations which originate from noncommutative analysis (over the algebra of biquaternions) and lead to the complex eikonal field and to the set of gauge fields associated with solutions of the eikonal equation. Particle-like formations represented by singularities of these fields possess ``elementary'' electric charge and other realistic ``quantum numbers'' and manifest self-consistent time evolution including transmutations. Related concepts of generating ``World Function'' and of multivalued physical fields are discussed. The picture of Lorentz invariant light-formed aether and of matter born from light arises then quite naturally. The notion of the Time Flow identified with the flow of primodial light (``pre-Light'') is introduced in the context.

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On some questions of four dimensional topology: a survey of modern research
2004jar | Mikhailov R. V.

Our physical intuition distinguishes four dimensions in a natural correspondence with material reality. Four dimensionality plays special role in almost all modern physical theories. High dimensional quantum fields theory and string theory are considered often together with their compactifications, i. e. the main space, describing the reality is a product of a four-dimensional manifold with some compact high-dimensional space. In this way we come to the well-known Kaluza-Klein model and ten-dimension superstring theory.
It is an interesting fact that the dimension four is a more complicated dimension from pure mathematical point of view. It seems that there is a contradiction with our intuition in understanding of the dimension concept, really, new dimensions give us new complexity. But it is not true in general. Additional dimensions often give a new freedom. It is natural that we must have some golden mean in this approach, in which we don't have a necessary freedom, but low-dimensional methods weakly work. In topology this mean is dimension four.
The goal of this note is to give a small survey of some problems in four-dimensional topology.

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Quaternions: Algebra, Geometry and Physical Theories
2004jaq | Yefremov A. P.

A review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups of Q-units transformations leaving Q-multiplication rule form-invariant are determined. A series of mathematical and physical applications is offered, among them use of Q-triads as a moveable frame, analysis of Q-spaces families, Q-formulation of Newtonian mechanics in arbitrary rotating frames, and realization of a Q-Relativity model comprising all effects of Special Relativity and admitting description of kinematics of non-inertial motion. A list of "Quaternionic Coincidences" is presented revealing surprising interconnection between basic relations of some physical theories and Q-numbers mathematics.

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Three-numbers, which cube of norm is nondegenerate three-form
2004jap | Garas`ko G. I.

Arbitrary three-form can be put in a canonical form. The requirement of existence of two-parametric Abelian Lie group to play the role of group of symmetry for three-form admits selecting the three-forms that correspond to three-numbers and finding all the three-numbers which cube of norm is a non-degenerate three-form with respect to a special coordinate system. There are exactly two (up to isomorphism) such sets of hypercomplex numbers, namely the sets: C₃, H₃. They can be regarded as generalizations of complex and binary (hyperbolic) bi-numbers to the case of three-numbers.

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On Possibility of Theoretical Disproof of Ether
2004eli | A. A. Eliovich  // PFUR

We discuss the problem: is it possible to disprove the ether conception on basis of classical electrodynamics equations, without appealing to experiment? We criticize recent T. A. Perevozskij's attempt to do it. We show that such mental experiments (with static fields and without changing of references frames) can't disprove even rough etherdynamical theories (although they can uncover unusual effects in such theories) and give no information for classical ether theories. After this we adduce the general arguments which show why it isn't possible to disprove ether on pure theoretical basis. We discuss the methodological context of the ether question and the notion of ether as interesting methodological instrument of natural science.

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Concerning the generalized Lorentz symmetry and the generalization of the Dirac equation
2004bg | G. Yu. Bogoslovsky, H. F. Goenner

The work is devoted to the generalization of the Dirac equation for a flat locally anisotropic, i.e., Finslerian space–time. At first we reproduce the corresponding metric and a group of the generalized Lorentz transformations, which has the meaning of the relativistic symmetry group of such event space. Next, proceeding from the requirement of the generalized Lorentz invariance we find a generalized Dirac equation in its explicit form. An exact solution of the nonlinear generalized Dirac equation is also presented.

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Concerning quaternions: finite displacements of solid bodies and points.
2002han | Khanukaev Yu. I.

We examine the quaternions technique as an alternative to vector and matrix description of spatial finite displacements of solid bodie. We also give the quaternionic description of Lorentz transformation.

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The Octonions
2002bae | Baez John C.  // Department of Mathematics University of California, baez@math.ucr.edu

From ArXiv:math.RA/0105155 v4 23 Apr 2002
The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.

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Finslerian Spaces Possessing Local Relativistic Symmetry
1999bgg | G. Yu. Bogoslovsky, H. F. Goenner

It is shown that the problem of a possible violation of the Lorentz transformations at Lorentz factors \gamma 5 x 10^10 , indicated by the situation which has dev eloped in the physics of ultra-high energy cosmic rays (the absence of the gzk cutoOE), has a nontrivial solution. Its essence consists in the discovery of the so-called generalized Lorentz transformations which seem to correctly link the inertial reference frames at any values of c. Like the usual Lorentz transformations, the generalized ones are linear, possess group properties and lead to the Einstein law of addition of 3-velocities. However, their geometric meaning turns out to be differen t: they serv e as relativistic symmetry transformations of a flat anisotropic Finslerian event space rather than of Minkowski space. Consideration is given to two typ es of Finsler spaces which generalize locally isotropic Riemannian space-time of relativit y theory, e.g. Finsler spaces with a partially and entirely broken local 3D isotropy. The investigation advances argumen ts for the corresp onding generalization of the theory of fundamen tal interactions and for a sp eci® c searc h for physical effects due to local anisotropy of space-time.

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On the possibility of phase transitions in the geometric structure of space-time
1998bbg | G. Yu. Bogoslovsky, H. F. Goenner

It is shown that space-time may be not only in a state which is described by Riemann geometry but also in states which are described by Finsler geometry. Transitions between various metric states of space-time have the meaning of phase transitions in its geometric structure. These transitions together with the evolution of each of the possible metric states make up the general picture of space-time manifold dynamics.

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1992ego

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Quaternionic analysis
1979sud | Sudbery Antony

The richness of the theory of functions over the complex field makes it natural to look for a similar theory for the only other non-trivial real associative division algebra, namely the quaternions. Such a theory exists and is quite far-reaching, yet it seems to be little known. It was not developed until nearly a century after Hamilton's discovery of quaternions. Hamilton himself and his principal followers and expositors, Tait and Joly, only developed the theory of functions of a quaternion variable as far as it could be taken by the general methods of the theory of functions of several real variables (the basic ideas of which appeared in their modern form for the first time in Hamilton's work on quaternions). They did not delimit a special class of regular functions among quaternion-valued functions of a quaternion variable, analogous to the regular functions of a complex variable.
This may have been because neither of the two fundamental definitions of a regular function of a complex variable has interesting consequences when adapted to quaternions; one is too restrictive, the other not restrictive enough. The functions of a quaternion variable which have quaternionic derivatives, in the obvious sense, are just the constant and linear functions (and not all of them); the functions which can be represented by quaternionic power series are just those which can be represented by power series in four real variables.
In 1935 R Fueter proposed a deашnition of "regular" for quaternionic functions by means of an analogue of the Cauchy-Riemann equations. He showed that this definition led to close analogues of Cauchy's theorem Cauchy's integral formula, and the Laurent expansion. In the next twelve years Fueter and his collaborators developed the theory of quaternionic analysis.
The theory developed by Fueter and his school is incomplete in some ways, and many of their theorems are neither so general nor so rigorously proved as present-day standards of exposition in complex analysis would require. The purpose of this paper is to give a self-contained account of the main line of quaternionic analysis which remedies these deficiencies, as well as adding a certain number of new results. By using the exterior differential calculus we are able to give new and simple proofs of most of the main theorems and to clarify the relationship between quaternionic analysis and complex analysis.

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Forms permitting composition
1970sch | Schafer R. D.

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Forms degree n permiting composition
1963sch | Schafer R. D.

Important Article about hypercomplex numbers.

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