Quaternion physical quantities

Quaternions consist of a scalar plus a vector and result from multiplication or division of vectors by vectors. Division of vectors is equivalent to multiplication divided by a scalar. Quaternions as used here consist of the scalar product with positive sign plus the vector product with sign determined by the right-hand rule. Units are specified by the multiplication process. Trigonometric functions are quaternions with units that can satisfy Hamilton's requirements. The square of a trigonometric quaternion is a real number provided that the product of the scalar number and the vector is not commutative. Maxwell's electromagnetic equations for empty space can be represented by a single quaternion equation.     

Revisiting Quaternion Dual Electrodynamics

Dual electrodynamics and corresponding Maxwell’s equations (in the presence of monopole only) are revisited from dual symmetry and accordingly the quaternionic reformulation of field equations and equation of motion is developed in simple, compact and consistent manner.     

Quaternion Dirac Equation and Supersymmetry

Quaternion Dirac equation has been analyzed and its supersymmetrization has been discussed consistently. It has been shown that the quaternion Dirac equation automatically describes the spin structure with its spin up and spin down components of two component quaternion Dirac spinors associated with positive and negative energies. It has also been shown that the supersymmetrization of quaternion Dirac equation works well for different cases associated with zero mass, nonzero mass, scalar potential and generalized electromagnetic potentials. Accordingly we have discussed the splitting of supersymmetrized Dirac equation in terms of electric and magnetic fields.     

四元数在量子力学中的应用

把双四元数推广到了二级双四元数，并设计了一种态函数的四元数表示法，从而用四元数表述了相对论量子力学，使四元数物理学形成了系统，用四元数表示的算符和状态，对于导出算符间的对易关系和状态的洛伦兹变换性质是方便的。     

彩色图像四元数矩不变量的研究

为了研究彩色图像的矩不变量特性,采用四元数进行彩色图像处理,以充分利用彩色图像的整体信息,实现彩色图像RGB并行处理。本文把传统灰度图像的矩不变量理论推广应用到四元数层面上来,定义了彩色图像的四元数矩并构造了该矩函数的仿射不变量。实验结果表明:所提出的彩色图像的四元数矩不变量的稳定性要优于L.V.Gool等人提出的彩色矩仿射不变量,其σ/u值提高了2个数量级。所提出的四元数仿射矩不变量可以作为模式识别中彩色目标的特征描述子来实现彩色图像目标的识别与跟踪。     

Foundations of the Quaternion Quantum Mechanics

We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.     

彩色图像四元数矩不变量的研究

为了研究彩色图像的矩不变量特性,采用四元数进行彩色图像处理,以充分利用彩色图像的整体信息,实现彩色图像RGB并行处理。本文把传统灰度图像的矩不变量理论推广应用到四元数层面上来,定义了彩色图像的四元数矩并构造了该矩函数的仿射不变量。实验结果表明：所提出的彩色图像的四元数矩不变量的稳定性要优于L.V.Gool等人提出的彩色矩仿射不变量,其σ/u值提高了2个数量级。所提出的四元数仿射矩不变量可以作为模式识别中彩色目标的特征描述子来实现彩色图像目标的识别与跟踪。     

Quaternion Octonion Reformulation of Quantum Chromodynamics

We have made an attempt to develop the quaternionic formulation of Yang–Mill’s field equations and octonion reformulation of quantum chromo dynamics (QCD). Starting with the Lagrangian density, we have discussed the field equations of SU(2) and SU(3) gauge fields for both cases of global and local gauge symmetries. It has been shown that the three quaternion units explain the structure of Yang–Mill’s field while the seven octonion units provide the consistent structure of SU(3) _C gauge symmetry of quantum chromo dynamics.     

Quaternion wave equations in curved space-time

The quaternion formulation of relativistic quantum theory is extended to include curvilinear coordinates and curved space-time. This provides a promising framework for further exploration of a unified quantum/gravity theory.     

Global Structure of Quaternion Polynomial Differential Equations

In this paper we mainly study the global structure of the quaternion Bernoulli equations \({\dot q=aq+bq^n}\) for \({q\in {\mathbb{H}}}\), the quaternion field and also some other form of cubic quaternion differential equations. By using the Liouvillian theorem of integrability and the topological characterization of 2–dimensional torus: orientable compact connected surface of genus one, we prove that the quaternion Bernoulli equations may have invariant tori, which possesses a full Lebesgue measure subset of \({{\mathbb{H}}}\). Moreover, if n = 2 all the invariant tori are full of periodic orbits; if n = 3 there are infinitely many invariant tori fulfilling periodic orbits and also infinitely many invariant ones fulfilling dense orbits.     

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 λ 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 ×SU(2) _L ×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 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.     

An Information-Theoretic Perspective on Proper Quaternion Variational Autoencoders

Variational autoencoders are deep generative models that have recently received a great deal of attention due to their ability to model the latent distribution of any kind of input such as images and audio signals, among others. A novel variational autoncoder in the quaternion domain H, namely the QVAE, has been recently proposed, leveraging the augmented second order statics of H-proper signals. In this paper, we analyze the QVAE under an information-theoretic perspective, studying the ability of the H-proper model to approximate improper distributions as well as the built-in H-proper ones and the loss of entropy due to the improperness of the input signal. We conduct experiments on a substantial set of quaternion signals, for each of which the QVAE shows the ability of modelling the input distribution, while learning the improperness and increasing the entropy of the latent space. The proposed analysis will prove that proper QVAEs can be employed with a good approximation even when the quaternion input data are improper.     

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.     

Higher Spin Quaternion Waves in the Klein-Gordon Theory

Electromagnetic interactions are discussed in the context of the Klein-Gordon fermion equation. The Mott scattering amplitude is derived in leading order perturbation theory and the result of the Dirac theory is reproduced except for an overall factor of sixteen. The discrepancy is not resolved as the study points into another direction. The vertex structures involved in the scattering calculations indicate the relevance of a modified Klein-Gordon equation, which takes into account the number of polarization states of the considered quantum field. In this equation the d’Alembertian is acting on quaternion-like plane waves, which can be generalized to representations of arbitrary spin. The method provides the same relation between mass and spin that has been found previously by Majorana, Gelfand, and Yaglom in infinite spin theories.     

A Real Quaternion Spherical Ensemble of Random Matrices

One can identify a tripartite classification of random matrix ensembles into geometrical universality classes corresponding to the plane, the sphere and the anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the anti-sphere with truncations of unitary matrices. This paper focusses on an ensemble corresponding to the sphere: matrices of the form Y=A ^?1 B, where A and B are independent N×N matrices with iid standard Gaussian real quaternion entries. By applying techniques similar to those used for the analogous complex and real spherical ensembles, the eigenvalue joint probability density function and correlation functions are calculated. This completes the exploration of spherical matrices using the traditional Dyson indices β=1,2,4. We find that the eigenvalue density (after stereographic projection onto the sphere) has a depletion of eigenvalues along a ring corresponding to the real axis, with reflective symmetry about this ring. However, in the limit of large matrix dimension, this eigenvalue density approaches that of the corresponding complex ensemble, a density which is uniform on the sphere. This result is in keeping with the spherical law (analogous to the circular law for iid matrices), which states that for matrices having the spherical structure Y=A ^?1 B, where A and B are independent, iid matrices the (stereographically projected) eigenvalue density tends to uniformity on the sphere.     

Quaternion epipolar decomposition for camera pose identification and animation

In the literature of computer vision, computer graphics and robotics, the use of quaternions is exclusively related to 3D rotation representation and interpolation. In this research we found how epipoles in multi-camera systems can be used to represent camera poses in the quaternion domain. The rotational quaternion is decomposed in two epipole rotational quaternions and one z axis rotational quaternion. Quadratic form of the essential matrix is also related to quaternion factors. Thus, five pose parameters are distributed into three independent rotational quaternions resulting in measurement error separation at camera pose identification and greater flexibility at virtual camera animation. The experimental results refer to the design of free viewpoint television.