A derivation of Maxwell equations in quaternion space

Provides an introduction to the concept and derivation of the Quaternions. Then derives several basic algebraic relationships involving quaternions, culminating with how quaternions can be used to describe 3d rotations. Then discusses the computational advantages that quaternions have over other 3d rotation methods, and provides an example of a classical mechanics rotation problem being solved with quaternions. Then discusses Frobenius's Theorem and the potential physical implication that it has. Overall aim is to provide a good educational resource for the intuition of quaternions and their applications.

2021

Quaternions were appeared through Lagrangian formulation of mechanics in Symplectic vector space. Its general form was obtained from the Clifford algebra, and Frobenius’ theorem, which says that “ the only finite-dimensional real division algebra are the real field R, the complex field C and the algebra H of quaternions” was derived. They appear also through Hamilton formulation of mechanics, as elements of rotation groups in the symplectic vector spaces. Quaternions were used in the solution of 4-dimensional Dirac equation in QED, and also in solutions of Yang-Mills equation in QCD as elements of noncommutative geometry. We present how quaternions are formulated in Clifford Algebra, how it is used in explaining rotation group in symplectic vector space and parallel transformation in holonomic dynamics. When a dynamical system *E-mail address: [email protected].

2021

In this paper, we use four-dimensional quaternionic algebra to describing space-time geometry in curvature form. The transformation relations of a quaternionic variable are established with the help of basis transformations of quaternion algebra. We deduced the quaternionic covariant derivative that explains how the quaternion components vary with scalar and vector fields. The quaternionic metric tensor and geodesic equation are also discussed to describing the quaternionic line element in curved space-time. Moreover, we discussed an expression for the Riemannian Christoffel curvature tensor in terms of the quaternionic metric tensor. We have deduced the quaternionic Einstein’s field-like equation which shows an equivalence between quaternionic matter and geometry.

2021

In this work, we use the concept of quaternion time and demonstrate that it can be applied for description of four-dimensional space-time intervals. Real quaternions form a normed division algebra and we suggest that this is the main advantage of quaternions over other mathematical representations of space-time. First, we use the quaternion norm for the description of the measurement process. We demonstrate that the quaternion time interval together with the finite speed of light signal propagation allow for a simple intuitive understanding of the time interval measurement by a moving observer. We derive a quaternion form of Lorentz time dilation and show that the norm of the quaternion time corresponds to the traditional expression of the Lorentz transformation. We determine that the space-time interval in the observer reference frame is given by a conjugate quaternion expression, which is essential for proper definition of the quaternion derivative in the observer reference frame....

In this work, we use real quaternions and the basic concept of the final speed of light in an attempt to enhance the standard description of special relativity. First, we demonstrate that it is possible to introduce a quaternion time domain where a coordinate point is described by a quaternion time. We show that the time measurement is a function of the observer location, even for stationary frames of reference. We introduce a moving observer, which leads to the traditional Lorentz relation for the time interval. We show that the present approach can be used in stationary, moving, or rotating frames of reference, unlike the traditional special relativity, which applies only to the inertial moving frames. Then, we use the quaternion formulation of space-time and mass-energy equivalence to extend the quaternion relativity to space, mass, and energy. We demonstrate that the transition between the particle and observer reference frames is equivalent to space inversion and can be described mathematically by quaternion conjugation. On the other hand, physical measurements are described by the quaternion norm and consequently are independent of the quaternion conjugation, which seems to be the quaternion formulation of the relativity principle.

2020

We show that the quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing 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 demonstra...

Symmetry, 2022

There are a total of 64 possible multiplication rules that can be defined starting with the generalized imaginary units first introduced by Hamilton. Of these sixty-four choices, only eight lead to non-commutative division algebras: two are associated to the left- and right-chirality quaternions, and the other six are generalizations of the split-quaternion concept first introduced by Cockle. We show that the 4×4 matrix representations of both the left- and right-chirality versions of the generalized split-quaternions are algebraically isomorphic and can be related to each other by 4×4 permutation matrices of the C2×C2 group. As examples of applications of the generalized quaternion concept, we first show that the left- and right-chirality quaternions can be used to describe Lorentz transformations with a constant velocity in an arbitrary spatial direction. Then, it is shown how each of the generalized split-quaternion algebras can be used to solve the problem of quantum-mechanical ...

Assuming the fundamental importance of division algebras, we introduced the division algebra of quaternions as a framework for description of four-dimensional space-time intervals. We derived the quaternion form of the Lorentz time dilation and presented an intuitive physical interpretation of the space-time transformations in the source and observer reference frames. We showed that the resulting physical interpretation is inseparable from experimental measurements. Also, we suggested that the positive direction of the quaternion stationary time is the arrow-of-time and that the length of the quaternion space-time interval is positive in the direction from the beginning to end. Then, we used quaternion algebra to develop quaternion calculus by choosing the correct quaternion form of the differential operators. We applied the new differentiation to the generalized potential function and suggested that the results can be interpreted as the two unified force fields. We repeated the differentiation, assuming that the second derivative of the potential function can be interpreted as the unified matter density. This resulted in the four unified matter density equations and a unified quaternion form of Maxwell equations. Then, we applied the unified fields and unified Maxwell equations to electromagnetic and gravitational interactions. Notably, the expressions for electromagnetic and gravitational interactions are similar as they they were derived from the same quaternion potential function, pointing to a possible procedure for unification of electromagnetism and gravitation. The novel components appearing in expressions for the force fields and Maxwell equations originated from scalar fields and velocity dependent potentials, and require further theoretical and experimental investigation. In particular, we showed that the gradient of the scalar field generates the main current density component for slow varying fields. Consequently, the proposed quaternion framework may be suitable as the foundation for a classical unified theory of space-time and matter, while enhancing the traditional theories of special and general relativity.

The correct specification of the concept of physical fields requires a platform in which these physical fields can be defined. This platform represents a base model that emerges from a Hilbert lattice, a vector space, and a number system. The number system must be an associative division ring. Dynamic fields require the selection of the quaternionic number system. Quaternionic fields are constructed eigenspaces of normal operators in a quaternionic Hilbert space. The base model supports symmetry-related fields and a field that always and everywhere exists. It acts as a repository for dynamic geometric data.

arXiv: General Physics, 2018

In this work, we use real quaternions and the basic concept of the final speed of light in an attempt to enhance the standard description of special relativity. First, we demonstrate that it is possible to introduce a quaternion time domain where a coordinate point is described by a quaternion time. We show that the time measurement is a function of the observer location, even for stationary frames of reference. We introduce a moving observer, which leads to the traditional Lorentz relation for the time interval. We show that the present approach can be used in stationary, moving, or rotating frames of reference, unlike the traditional special relativity, which applies only to the inertial moving frames. Then, we use the quaternion formulation of space-time and mass-energy equivalence to extend the quaternion relativity to space, mass, and energy. We demonstrate that the transition between the particle and observer reference frames is equivalent to space inversion and can be describ...