Abstract
A new unification of the Maxwell equations is given in the domain of Clifford algebras with in a fashion similar to those obtained with Pauli and Dirac algebras. It is shown that the new electromagnetic field multivector can be obtained from a potential function that is closely related to the scalar and the vector potentials of classical electromagnetics. Additionally it is shown that the gauge transformations of the new multivector and its potential function and the Lagrangian density of the electromagnetic field are in agreement with the transformation rules of the second-rank antisymmetric electromagnetic field tensor, in contrast to the results obtained by applying other versions of Clifford algebras.
Highlights
Clifford algebras provide a unifying structure for Euclidean, Minkowski, and multivector spaces of all dimensions
We discuss its applications to electromagnetism and obtain a new electromagnetic field multivector, which is closely related to the scalar and vector potentials of the classical electromagnetics
We have shown that in the framework of the Clifford algebra defined in Equation (3), the Maxwell equations in vacuum reduce to a single equation in a fashion similar to that in other types of Clifford algebras
Summary
Clifford algebras provide a unifying structure for Euclidean, Minkowski, and multivector spaces of all dimensions. He showed that the Maxwell equations in vacuum are equivalent to the equation of holomorphy in Minkowski space-time. Imaeda [9] showed that Maxwell equation in vacuum are equivalent to the condition of holomorphy for functions of a real biquaternion variable. We apply a different Clifford algebra to the Maxwell equations of electromagnetism, and we show how this formulation relates to the classical theory in a straightforward manner resulting in two main formulas; the first is a simplistic rendering of Maxwell’s equations in a short formula ( ) ∂F − ∂0 F + F* = −ρ + J (1). We discuss its applications to electromagnetism and obtain a new electromagnetic field multivector, which is closely related to the scalar and vector potentials of the classical electromagnetics. We give the matrix representation of the electromagnetic field multivector and its Lorentz transformation
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