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Advantages of Geometric Algebra as a New Coordinate-free Approach to Complex Media

Matos, S.A. ; Paiva, C. R. ; Barbosa, A.

Advantages of Geometric Algebra as a New Coordinate-free Approach to Complex Media, Proc Encuentro Ibérico de Electromagnetismo Computacional - EIEC , Monfragüe- Cáceres, Spain, Vol. -, pp. - - -, May, 2010.

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A coordinate-free approach is the best way to address, in the most general form, the electromagnetic characteristics of a broad class of unbounded media – as in the case of anisotropic and bianisotropic media. The most used coordinate-free formalism is the tensor (or dyadic) approach. The authors have been developing a new trend on the analysis of plane wave propagation in anisotropic and bianisotropic media [1-3]. A natural question arises: why use geometric algebra instead of tensor analysis? In fact, there are no free lunches: there is certainly an extra effort required for those who do not master geometric algebra. Nevertheless, some advantages can be pointed out: i) a greater analytical simplicity, which can reduce severely the mathematical calculus and provide more compact and general ways of describing the problem; ii) a geometric perspective is gained, which can bring a new physical insight into the problem; iii) dimension independent: the structure of this algebra remains the same for spacetime. We should stress, that the Gibbs calculus cannot be generalized to four dimensions. Nevertheless, in the authors’ opinion the major advantage is to be able to separate the concept of anisotropy from the tensor formalism. Instead of using the usual tensor methods, we use the more geometric perspective of linear operators, which geometric algebra provides, following a synthetic approach that deals with geometric objects as abstract entities – as opposed to an algebraic approach that deals with the components of geometric objects as composite entities.
In this communication we will present a general discussion of the main advantages of Clifford algebra formalism over the tensor calculus, showing several examples of how geometric algebra can provide a deeper geometrical and physical insight.