Quaternion Basis Research Articles

Scalar Breit interaction for molecular calculations.

Variational treatment of the Dirac-Coulomb-Gaunt or Dirac-Coulomb-Breit two-electron interaction at the Dirac-Hartree-Fock level is the starting point of high-accuracy four-component calculations of atomic and molecular systems. In this work, we introduce, for the first time, the scalar Hamiltonians derived from the Dirac-Coulomb-Gaunt and Dirac-Coulomb-Breit operators based on spin separation in the Pauli quaternion basis. While the widely used spin-free Dirac-Coulomb Hamiltonian includes only the direct Coulomb and exchange terms that resemble nonrelativistic two-electron interactions, the scalar Gaunt operator adds a scalar spin-spin term. The spin separation of the gauge operator gives rise to an additional scalar orbit-orbit interaction in the scalar Breit Hamiltonian. Benchmark calculations of Aun (n = 2-8) show that the scalar Dirac-Coulomb-Breit Hamiltonian can capture 99.99% of the total energy with only 10% of the computational cost when real-valued arithmetic is used, compared to the full Dirac-Coulomb-Breit Hamiltonian. The scalar relativistic formulation developed in this work lays the theoretical foundation for the development of high-accuracy, low-cost correlated variational relativistic many-body theory.

Relativistic Fermion and Boson Fields: Bose-Einstein Condensate as a Time Crystal

In a basis of the space-time coordinate frame four quaternions discovered by Hamilton can be used. For subsequent reproduction of the coordinate frame these four quaternions are expanded to four 4 × 4 matrices with real-valued matrix coefficients −0 and 1. This group set is isomorphic to the SU(2) group. Such a matrix basis introduces extra six degrees of freedom of matter motion in space-time. There are three rotations about three space axes and three boosts along these axes. Next one declares the differential generating operators acting on the energy-momentum density tensor written in the above quaternion basis. The subsequent actions of this operator together with its transposed one on the above tensor lead to the emergence of the gravitomagnetic equations that are like the Maxwell equations. Wave equations extracted from the gravitomagnetic ones describe the propagation of energy density waves and their vortices through space. The Dirac equations and their reduction to two equations with real-valued functions, the quantum Hamilton-Jacobi equations and the continuity equations, are considered. The Klein-Gordon equations arising on the mass shell hints to the alternation of the paired fermion fields and boson ones. As an example, a Feynman diagram of an electron–positron time crystal is illustrated.

COORDINATE TRANSFORMATIONS OF THREE-PHASE VARIABLES USING QUATERNIONS

Among the variety of modern approaches to the mathematical description of the power quality indicators during the processes of transmission, distribution, conversion and calculation of the ac electric power, the representation of three-phase models in the form of a purely imaginary quaternion located in a separate subspace of the four-dimensional hypercomplex space allows, in relation to the generally accepted method of analyzing linear circuits, for example, symmetrical components with the selection of a direct, reverse and zero phase sequence for the fundamental harmonic, to take into a more complete account the features of energy consumption, especially in the presence of distortion in the modified forms of harmonic signals. In addition, the division of the quaternion into scalar (real) and partial (imaginary) makes it possible to significantly simplify the subsequent analytical processing of synthesis of a power converters control signals for active filtering and power supply of autonomous loads of an arbitrary type, including a single-phase configuration, by extracting from its composition individual components responsible for both the amplitude-phase asymmetry and the nonlinearity of the characteristics.
 
 The main algorithmic principles of organizing control structures as part of three-phase systems of various functional purposes, as a rule, are based on the conversion of reference signals and current values ​​of measured currents and voltages into state coordinates obtained by rotating the three-dimensional space plane by a given angle. At the same time, the calculated ratios for the numerical determination of the initial variables transformed by rotation in the quaternion basis are a function of only four kinematic parameters, which, other things being equal, leads to a simplification of the control law in relation to the traditional vector-matrix approach using nine direction cosines with six connection equations. In this regard, this paper is devoted to the applied problems of implementing linear transformations by E. Clarke and R.H. Park in terms of four-dimensional hypercomplex numbers, in compliance with the additional requirement of the invariance of scalar quantities after the transition.

Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System

The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains; however, it can be used as a substitute derivative in specific domains.

Quaternion Algebra on 4D Superfluid Quantum Space-Time: Can Dark Matter Be a Manifestation of the Superfluid Ether?

Quaternions are a natural framework of 4D space-time, where the unit element relates to time, and three others relate to 3D space. We define a quaternion set of differential torsion operators (shifts with rotations) that act to the energy-momentum tensor written on the same quaternion basis. It results in the equations of gravity-torsion (gravitomagnetic) fields that are similar to Maxwell’s equations. These equations are parent equations, generating the following equations: (a) equations of the transverse gravity-torsion waves; (b) the vorticity equation describing vortices orbital speed of which grows monotonically in the vortex core but far from it, it goes to a permanent level; (c) the modified Navier–Stokes equation leading to the Schrödinger equation in the nonrelativistic limit and to the Dirac equation in the relativistic limit. The Ginsburg–Landau theory of superfluidity resulting from the Schrödinger equation shows the emergence of coupled proton-antiproton pairs forming the Bose–Einstein condensate. In the final part of the article, we describe Samokhvalov’s experiment with rotating nonelectric, nonferromagnetic massive disks in a vacuum. It demonstrates an unknown force transferring the rotational moment from the driving disk to a driven one. It can be a manifestation of the dark matter. For studying this phenomenon, we propose a neutron interference experiment that is like the Aharonov–Bohm one.

Quaternionic comparisons of electromagnetism using Lorentz transformations

In this paper, by defining properties of the real and complex quaternion algebras, the electromagnetic field equations including monopoles have been studied in $$\hbox {S}'$$ reference systems, which moves with the velocity u according to the fixed reference system $$\hbox {S}$$. Quaternionic source and wave equalities have been derived in a more compact and elegant manner in $$\hbox {S}'$$ system. Besides, Poynting theorem has been firstly obtained in moving media. Additionally, the comparisons between $$\hbox {S}$$ and $$\hbox {S}'$$ reference systems have been separately done for field, source, potential and wave equations in quaternion basis elements. Furthermore, Poynting theorems have been compared by means of $$\hbox {S}$$ and $$\hbox {S}'$$ systems for the first time.

Eigen-Decomposition of Quaternions

This paper introduces biquaternion eigen-decomposition theory (via Peirce decomposition) with respect to a selected quaternion with a non-zero vector part. The eigen-decomposition allows evaluation of polynomials and power series with real coefficients as functions of quaternions. This extension of analytic functions to functions of quaternions requires only standard complex function evaluation. The theory also applies to quaternion rotations. The theory uses biquaternion calculations indicated by matrix methods via the algebraic isomorphism between Hamilton’s biquaternions and appropriate $$4\times 4$$ complex matrices. The isomorphism preserves algebraic structure. In particular, the left and right biquaternion multiplication by the selected quaternion maps to left and right matrix multiplication, respectively. This unifies the representation of the left and right quaternion multiplication as a linear map into a single matrix form. This matrix, as a linear operator, acts on matrices, so that the eigenvectors have matrix form that maps into the biquaternions. Use of an alternate quaternion basis results in a similarity transform of the representation matrix, preserving eigenvalues across change of basis. The similarity transform allows simple eigenvector calculation. The matrix for the selected quaternion has two identical, complex conjugate pairs of eigenvalues. Each pair corresponds to two complex conjugate pairs of eigenvector biquaternions, an idempotent pair and a nilpotent pair. Idempotent and nilpotent eigenvectors correspond to the commuting and non-commuting parts, respectively, of quaternion multiplication.

To Principles of Quantum Theory Construction

New insight to the principles of the quantum theory construction is given. It is based on the symmetry study of main differential equations of mechanics and electrodynamics. It has been shown, that differential equations, which are invariant under transformations of groups, which are symmetry groups of mathematical numbers (considered within the frames of the number theory) determine the mathematical nature of the quantities, incoming in given equations. The substantial consequence of the given consideration is the proof of the main postulate of quantum mechanics, that is, the proof of the statement, that to any quantum mechanical quantity can be set up into the correspondence the Hermitian matrix. It is shown, that a non-abelian character of the multiplicative group of the quaternion ring leads to the nonapplicability of the quaternion calculus for the construction of new versions of quantum mechanics directly. The given conclusion seems to be actual, since there is a number of modern publications with the development of the quantum mechanics theory using the quaternions with the standard basis {e, i, j, k}. The correct way for the construction of new versions of quantum mechanics on the quaternion base is discussed in the paper presented. It is realized by means of the representation of the quaternions through the basis of the linear space of complex numbers over the field of real numbers, under the multiplicative group of which the equations of the dynamics of mechanical systems are invariant. At the same time the quaternion calculus is applicable in electrodynamics, at that the new versions of quantum electrodynamics can be constructed by an infinite number of the ways corresponding to an infinite number of the matrix representations of the standard quaternion basis {e, i, j, k}. The given conclusion is the consequence of the high symmetry of Maxwell equations.

The Schrödinger equation as a consequence of the quaternion algebra stability

It is shown that the requirement of invariance of a quaternion basis under rotations and conformal transformations of the pre-geometric spin surface leads to an equation generalizing that of quantum mechanics.

Quaternion Octonion Reformulation of Grand Unified Theories

In this paper, Grand Unified theories are discussed in terms of quaternions and octonions by using the relation between quaternion basis elements with Pauli matrices and Octonions with Gell Mann \lambda matrices. Connection between the unitary groups of GUTs and the normed division algebra has been established to re-describe the SU(5)gauge group. We have thus described the SU(5)gauge group and its subgroup SU(3)_{C}\times SU(2)_{L}\times U(1) by using quaternion and octonion basis elements. As such the connection between U(1) gauge group and complex number, SU(2) gauge group and quaternions and SU(3) and octonions is established. It is concluded that the division algebra approach to the the theory of unification of fundamental interactions as the case of GUTs leads to the consequences towards the new understanding of these theories which incorporate the existence of magnetic monopole and dyon.

Fermion Flavors in Quaternion Basis and Infrared QCD

I analyze the lattice simulation data of the Domain Wall Fermion in quaternion basis. As pointed out by Atiyah and Ward, the minimum action solution for SU(2) Yang–Mills fields in Euclidean 4-space correspond, via Penrose twistor transform, to algebraic bundles on the complex projective 3-space. Assuming dominance of correlation between the fermions on the domain walls via exchange of instantons, I extract parameters necessary for defining gauge fields of Atiyah–Ward ansatz. The QCD effective coupling in the infrared and the relation between the number of flavors and the infrared fixed point is investigated. Consequence of this lepton flavor assignment to phenomenology of baryons is also discussed.

Matrix realization of dual quaternionic electromagnetism

AbstractIn this paper, a new representational model based on dual quaternionic matrices is proposed for classical electromagnetism. After demonstrating the isomorphic matrix representations of dual quaternions, Maxwell’s equations and the constitutive relations for electromagnetism are expressed in terms of dual quaternionic matrices. For this purpose, new 8 × 8 matrices connected with quaternion basis elements have been introduced.

QUATERNION SOLUTIONS OF FOUR-DIMENSIONAL LIOUVILLE AND SINE-GORDON EQUATIONS

The quaternion solutions of four-dimensional Liouville and sine-Gordon equations are explicity obtained by using the quaternion basis.

A functional calculus for sesquihermitian operators on quaternionic Hilbert space

AbstractA continuous real-linear operator A = A0 + i1A1 + i2A2 + i3A3 on a quaternionic Hilbert space is called sesquihermitian if the linear operators Av are Hermitian; this condition is independent of the choice of quaternion basis (i1,i2,i3). The joint spectral distribution of the Av provides a functional calculus for sesquihermitian operators and real-valued C∞-functions on . This calculus is independent of the quaternion basis and extends naturally to quaternion-valued functions to give a continuous quaternion-linear mapping from the algebra of these functions to that of sesquihermitian operators. The mapping is not, in general, multiplicative unless the Av commute, in which case it agrees with that for several commuting operators on complex Hilbert space.