CHAPTER FIVE Maxwell Equations in Geometric Algebra

2004

American Institute of Physics Conference Series, 2010

We review the canonical analysis of Maxwell's theory in five dimensions (5D). In the first part we summarize the canonical analysis of the free theory in Minkowski space-time. As is well known the theory has two first class constraints and therefore two gauge conditions are needed in order to fix the gauge completely. In particular we discuss the properties of the axial gauge. In the second part we discuss the theory in a different 5D background. We consider a space-time in which the fifth dimension is a circle (Kaluza-Klein dimensional reduction). We show that the axial gauge is not consistent with the periodic boundary conditions in the fifth coordinate. As a consequence it is necessary to introduce a different gauge, the "almost axial gauge".

2025

Many papers have been published over the years that either conjecture or even (claim to) prove the universality of the form of Maxwell's equations. We present yet another derivation of Maxwell's equations and discuss the conclusions suggested by Maxwell universality, namely the logical inevitability of the Lorentz transformations and the mathematical inconsistency of Newtonian physics. I. INTRODUCTION Some years ago, Burns provided yet another proof that the form of Maxwell's equations is "universal" [1] in the sense that Maxwell's equations can be mathematically derived from the continuity equation and the dimension of space-time. This means that whenever a locally conserved ("substantial") quantity like charge or mass moves in a space with three spatial (and one temporal) dimension(s), it will be the source of vector fields of the Maxwellian form. The question whether Maxwell's equation have one of many possible forms of linear coupled partial differential equations, or whether their form can be derived, has been subject in numerous publications in the past. The first paper on this subject, known to the author, was written by H. Hertz and dates back to 1884 (!) [2]. But Hertz did not claim to have proven the unique and universal form of Maxwell's equations (see also Refs. [3, 4]).

Indian Journal of Pure and Applied Mathematics

The well-known algebraic classi]ieation of the electromagnetic tensor ]ield is used to provide the s~)ace-time mani]old o] general relativity with the latest technique of d/iNerential geometry.

Eprint Arxiv Physics 9703028, 1997

On the basis of the ordinary mathematical methods we discuss new classes of solutions of the Maxwell's equations discovered in the papers by D. Ahluwalia, M. Evans and H. M'unera et al.

During the recent years so called geometric techniques have become popular in computational electromagnetism. In this paper, exploiting differential geometry and manifolds, we first give a meaning to what is meant by geometric approaches. Therafter we examine some implications of such geometry in numerical analysis of electromagnetic field and wave problems.

Journal of the Optical Society of America A, 2011

In this paper, we use geometric algebra to describe the polarization ellipse and Stokes parameters. We show that a solution to Maxwell's equation is a product of a complex basis vector in Jackson and a linear combination of plane wave functions. We convert both the amplitudes and the wave function arguments from complex scalars to complex vectors. This conversion allows us to separate the electric field vector and the imaginary magnetic field vector, because exponentials of imaginary scalars convert vectors to imaginary vectors and vice versa, while exponentials of imaginary vectors only rotate the vector or imaginary vector they are multiplied to. We convert this expression for polarized light into two other representations: the Cartesian representation and the rotated ellipse representation. We compute the conversion relations among the representation parameters and their corresponding Stokes parameters. And finally, we propose a set of geometric relations between the electric and magnetic fields that satisfy an equation similar to the Poincaré sphere equation.

1979

It was Heaviside [1] who first called attention to the invariance of Maxwell equations under the transformations E → ±H, H → ±E. Then Larmor [2] and Rainich [3] generalized this fact and demonstrated that Maxwell equations well invariant under the one-parametrical group of transformations of a kind E → E cos θ + H sin θ,

The object of this contribution is twofold. On one hand, it rises some general questions concerning the definition of the electromagnetic field and its intrinsic properties, and it proposes concepts and ways to answer them. On the other hand, and as an illustration of this analysis, a set of quadratic equations for the electromagnetic field is presented, richer in pure radiation solutions than the usual Maxwell equations, and showing a striking property relating geometrical optics to all the other Maxwell solutions.

2018

Conventional quantum mechanical qubits can be lifted to states as 3 + valued operators that act on observables [1]. That operators may be implemented via the two types of Maxwell equations' solution polarizations [2]. Solution of Maxwell equation in geometric algebra formalism gives g-qubits which are exact lifts of conventional qubits. Therefore, it unambiguously reveals actual meaning of complex parameters of qubits of the commonly accepted Hilbert space quantum mechanics and, particularly, directly demonstrates the option of instant nonlocality of states.

Arxiv preprint arXiv:0807.1382, 2008

In this paper, we define energy-momentum density as a product of the complex vector electromagnetic field and its complex conjugate. We derive an equation for the spacetime derivative of the energy-momentum density. We show that the scalar and vector parts of this equation are the differential conservation laws for energy and momentum, and the imaginary vector part is a relation for the curl of the Poynting vector. We can show that the spacetime derivative of this energy-momentum equation is a wave equation. Our formalism is Dirac-Pauli-Hestenes algebra in the framework of Clifford (Geometric) algebra Cl4,0.

2010

The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is desirable to represent it in the simplest and most intuitive formalism possible. We show firstly, that Clifford's geometric algebra, provides a significantly simpler representation than the conventional bra-ket notation, and secondly, that the basis defined by the states of maximum and minimum weight in the Grover search space, allows a simple visualization of the Grover search analogous to the precession of a spin-1 2 particle. Using this formalism we efficiently solve the exact search problem, as well as easily representing more general search situations. We do not claim the development of an improved algorithm, but show in a tutorial paper that geometric algebra provides extremely compact and elegant expressions with improved clarity for the Grover search algorithm. Being a key algorithm in quantum computing and one of the most studied, it forms an ideal basis for a tutorial on how to elucidate quantum operations in terms of geometric algebra-this is then of interest in extending the applicability of geometric algebra to more complicated problems in fields of quantum computing, quantum decision theory, and quantum information.

Journal of Modern Physics, 2020

It is often claimed that Maxwell's electromagnetic equations were originally written in terms of quaternions. Once returned to that form and treated with left and right hand operators as in the mathematics of P. M. Jack, a new seventh scalar electromagnetic field component emerges with possible relations to clean energy extraction and gravitation. It is the purpose here to examine this approach afresh and see how it might link up with other fairly recent, but little known, work in the field. Again, as with the usual form of Maxwell's equations, a new scalar wave equation is derived but, on this occasion, due to the presence of the scalar component of the quaternion, that equation exhibits a wave speed greater than the speed of light. Historical and present uses within military and humanitarian contexts are considered briefly.

2022

The electromagnetic potential A i is a quadrivector in Minkowski space-time x k and its gradient a i k is a tensor of rank two whose elements are the sixteen partial derivatives ∂A i /∂x k. We study in this article the properties of a family of tensors resulting from [a i k ]. We first introduce the covariant tensor [a ki ]. Four initial tensors are obtained by separating [a i k ] on the one hand, and [a ki ] on the other hand into their symmetric and antisymmetric parts. These are (s i k , [f i k ], [S ki ], [F ki ]). As the lowering-raising index operations and symmetrization-antisymmetrization operations do not commute, these four tensors are different. We associate a Lagrangian density L to the determinant of [a i k ] which is invariant in an operation of symmetry of the Poincaré group. In the first part of the article, we show that there is a particular coordinate system where the scalar potential obeys the Hemholtz equation. The solutions allow to describe the "electromagnetic particles", characterized by three quantum numbers n, and m. We give the tensors corresponding to the first five solutions. They describe energy and electric charge distributions. The condition of existence of these particles is related to a property of the electron described in the Wheeler-Feynman's absorber theory. In a second part, we first check that [F ki ] is the usual electromagnetic tensor whose components are the electric and magnetic fields. We prove that Maxwell's equations are obtained by applying the principle of least action to the 4-potential endowed with L. The source terms (ρ and − → j) are expressed in terms of the components of [S ki ]. The results obtained are covariant. The formulation of these tensors being independant of scale, they unify the human and the electron scales, giving a new way to understand elementary particles.