Narrow Results

Using the quaternion framework (Q-math), we show that the specific mathematical equations born in “quaternion medium” in physical units become known physical laws. In particular, it is shown how one can discover immanently hidden “geometric physical laws”: Cartesian frames, equations of electrodynamics, Q-vector formulation of the relativity theory. One can also find the linked logical chain between laws of quantum, classical, and relativistic mechanics.

In this paper, we present a study comparing the performance of unscented and extended Kalman filtering for improving model for trajectory interpolation. Specifically, we examine orientation motion with quaternions, which are critical for appreciating direction. Our experimental results and analysis indicate that unscented Kalman filtering performs equivalently with extended Kalman filtering.

The Cayley-Dickson procedure has been used to construct the root systems of SO(8), SO(9) and the exceptional Lie algebra F₄ in terms of quaternions. The Aut(F₄) has a simple presentation of the form (O, O) ⊕ (O, O)^* where O represents the binary octahedral group. Two sets of quaternionic root system of F₄, symbolically written [F₄, F₄] = F₄ + e₇F₄, describe the octonionic root system of E₈ where the root system of the exceptional Lie algebra E₇ are represented by the imaginary octonions. The automorphism group of the octonionic root system of E₇ preserving the octonion algebra is the Chevalley group G₂(2) where the maximal subgroups of G₂(2) are the automorphism groups of the octonionic root systems of the maximal Lie algebras E₆, SU(2) × SO(12) and SU(8) of E₇. Another pairing of the quaternionic roots of the form [F₄, F₄] = F₄ + σF₄, constitutes the root system of E₈ in terms of icosians where half of the roots describe the roots of the noncrystallographic Coxeter group H₄. The relevance of H₄ to E₈ and the maximal subgroups of H₄ have been discussed. Polyhedral structures obtained from the quaternionic root systems of H₄ are described as the orbits of H₃.

The papers introduces a new complex of differential forms which provides a fine resolution for the sheaf of regular functions in two quaternionic variables and the sheaf of monogenic functions in two vector variables. The paper announces some applications of this complex to the construction and interpretation of sheaves of quaternionic and Clifford hyperfunctions as equivalence classes of such differential forms.

We consider a generalized Vekua equation in biquaternionic formalism where the Cauchy-Riemann operator is replaced by the differential operator D of Dirac. For particular classes we construct differential operators of higher order which give a relation between the monogenic functions as solutions of Dw = 0 and the generalized pseudoanalytic functions as solutions of the generalized Vekua equation. This is done by considering a corresponding differential equation of second order. Using generating functions in the sense of L. Bers we can give further representations of such functions and we can obtain related pseudoanalytic functions of the second kind as solutions of another differential equation of first order.

This is generally a literal translation, terms such as versor and parataxis in use at the time are not updated to contemporary style, rather defined and briefly compared in an appendix. The decision to translate Lemaître’s 1948 essay arose not merely because of personal interest in Lemaître’s ‘persona’, Physicist - Priest, but from an ever increasing interest in Quaternions and more recent discovery of correspondence with Octonions in additional dimensional (XD) brane topological phase transitions (currently most active research arena in all physics), and to make this particular work available to readers interested in any posited historical value during the time quaternions were still considered a prize of some merit, which as well-known, were marginalized soon thereafter (beginning mid1880’s) by the occluding dominance of the rise of vector algebra; indeed, Lemaître himself states in his introduction: “Since elliptic space plays an increasingly important part …I have thought that an exposition … could present some utility even if the specialists … must bear the judgment that it contains nothing really new”, but also because the author feels quaternions (likely in conjunction with octonions), extended into elliptical and hyperbolic XD spaces, especially in terms of the ease they provide in simplifying the Dirac equation, will be essential facilitators in ushering in post-standard model physics of unified field mechanics. The translator comes to realizes that 3^rd regime Natural Science (Classical → Quantum → Unified Field Mechanics) will be described by a reformulated M-Theoretic topological field theory, details of which will be best described by Quaternion-Octonion correspondence.

Emergence of the experimental evidence of E₈ in the analysis of one dimensional Ising-model invokes further studies of the Coxeter-Weyl groups generated by reflections regarding their applications to polytopes. The Coxeter group W(H₄) which describes the Platonic Polytopes 600-cell and 120-cell in 4D singles out in the mass relations of the bound states of the Ising model for it is a maximal subgroup of the Coxeter-Weyl group W(E₈). There exists a one-to-one correspondence between the finite subgroups of quaternions and the Coxeter-Weyl groups of rank 4 which facilitates the study of the rank-4 Coxeter-Weyl groups. In this paper we study the systematic classifications of the 3D-polyhedra and 4D-polytopes through their symmetries described by the rank-3 and rank-4 Coxeter-Weyl groups represented by finite groups of quaternions. We also develop a technique on the constructions of the duals of the polyhedra and the polytopes and give a number of examples. Applications of the rank-2 Coxeter groups have been briefly mentioned.

Physics at the fundamental level can be effectively reduced to an explanation of the structures and interactions of fermions. Fermions appear to be singularities rather than extended objects, but there is no obvious way of creating such structures within the 3-dimensional space of observation. However, the algebra associated with the Dirac equation appears to suggest that the fermion requires a double, rather than a single, vector space, and this would seem to be confirmed by the double rotation required by spin ½ objects, and the associated effects of zitterberwegung and Berry phase shift. Further investigation of the second ‘space’ reveals that it is, in effect, an ‘antispace’, which contains the same information as real space but in a less accessible form. The two spaces effectively cancel to produce a norm 0 (nilpotent) object which has exactly the mathematical structure required to be a fermionic singularity.

We make a preliminary exploratory study of higher dimensional (HD) orthogonal forms of the quaternion algebra in order to explore putative novel Nilpotent/Idempotent/Dirac symmetry properties. Stage-1 transforms the dual quaternion algebra in a manner that extends the standard anticommutative 3-form, i, j, k into a 5D/6D triplet. Each is a copy of the others and each is self-commutative and believed to represent spin or different orientations of a 3-cube. The triplet represents a copy of the original that contains no new information other than rotational perspective and maps back to the original quaternion vertex or to a second point in a line element. In Stage-2 we attempt to break the inherent quaternionic property of algebraic closure by stereographic projection of the Argand plane onto rotating Riemann 4-spheres. Finally, we explore the properties of various topological symmetries in order to study anticommutative - commutative cycles in the periodic rotational motions of the quaternion algebra in additional HD dualities.

Dimensionality has been a much discussed subject since Minkowski formalized special relativity by extending 3D space to 4D space-time. However, there has never been any consensus on the number of dimensions that nature requires and there has been no explanation of why dimensions are needed at all. It is proposed here that dimensions originate in the theory of numbers, that extending the number of dimensions beyond the 3 required by Euclidean space necessarily requires a fundamental change in the meaning of the concept, and that, although various algebraic techniques allow such extension of dimensionality, the structures required always ensure that the number of dimensions and their fundamental characteristics remain ambiguous, leaving the final question unanswerable.