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An introduction to relativistic electrodynamics: Part II: Calculus with complex 4-vectors
Umeå University, Faculty of Science and Technology, Department of Physics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. (UMIT)ORCID iD: 0000-0001-7838-1307
##### Abstract [en]

The conventional way of introducing relativity when teaching electrodynamics is to leave Gibbs' vector calculus for a more general tensor calculus. This sudden change of formalism can be quite problematic for the students and we therefore in this two-part paper consider alternate approaches. The algebra $\mathbb{C}^{2 \times 2}$ of 2-by-2 complex matrices (sometimes presented in the form of Clifford algebra or complex quaternions) may be used for spinor related formulations of special relativity and electrodynamics. In this Part II we use this algebraic structure but with notations that fits in with the formalism of Part I. Each observer $\mathbf{e}_0$ defines a product on the space of complex 4-vectors $\mathbb{M}^{\mathbb{C}} = \mathbb{M} + i\mathbb{M}$ so that $\mathbb{M}^{\mathbb{C}}$ becomes an algebra isomorphic to $\mathbb{C}^{2\times 2}$ with $\mathbf{e}_0$ as algebra unit. The spacetime geometric equations of Part I become complex (spinor related) equations where the antisymmetric 4-dyadics have been replaced by complex 3-vectors, i.e., by elements in $\mathbf{e}_0^{\perp \mathbb{C}} = \mathbf{e}_0^\perp + i\mathbf{e}_0^\perp$. For example, instead of the electromagnetic dyadic field $\mathbf{F}$ we now have the complex field variable $\mathbf{f} = \mathbf{E}/c + i\mathbf{B}$. Some linear algebra together with the formalism of Gibbs' vector calculus (trivially allowing for complex 3-vectors) is sufficient for dealing with the equations in their complex form.

2018. , p. 7
##### Keywords [en]
electrodynamics, special relativity, coordinate-free calculus, spinors
##### National Category
Other Physics Topics
Physics
##### Identifiers
OAI: oai:DiVA.org:umu-152751DiVA, id: diva2:1257612
Available from: 2018-10-22 Created: 2018-10-22 Last updated: 2018-10-25

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Cite
Citation style
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Other style
More styles
Language
• de-DE
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• fi-FI
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• sv-SE
• Other locale
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• rtf