Answer by Norm Cimon for Geometrical meaning of Grassmann Algebra

2011-07-18T23:36:02Z

The best synopsis, one developed by a real teacher, is from David Hestenes who not coincidentally developed a relatively complete curriculum built around it, and an elegant symbolism. The outer product is simply the multi-dimensional extension of the notion of a directed line segment to include directed planes and directed volume elements. William Clifford incorporated this into the surprisingly straightforward notion of a geometric algebra, something that's been a dream of mathematicians for a very long time. In one of the truly unfortunate events in mathematical history, Clifford died of tuberculosis at the age of 34, derailing his ideas for almost one hundred years, and giving us the bizarre pantheon of often incompatible systems we currently use.

What does geometric algebra have for elements? They are called multi-vectors and they have scalar, vector, bi-vector (a directed plane), and - in general - multi-vector components. What Clifford defined was the product of multi-vectors as the sum of the inner (dot) product and Grassman's outer product. Geometric algebra is built up from this basic operation, and the rules that guide it, the key one being that the outer-product is anti-symmetric: reverse the order of the elements in the outer-product and the sign changes. That one simple rule turns out to be the key to unlocking one of the greatest advances in the history of mathematics and physics, an algebra for manipulating objects in space.

Just as a complex numbers "tag" real and imaginary parts, objects in geometric algebra are "tagged" by the basis elements which are extended to include, not just one dimensional basis elements, but multi-dimensional basis elements as well - those directed planes and volume elements mentioned above. It's an extremely elegant and very seductive extension of linear algebra to obtain an algebra that unifies the hodgepodge of systems currently used including differential geometry, matrix algebra, vector algebra and tensors. I'm not making this up, it really does this.

Geometric algebra also manipulates geometric objects in space without having to resort to coordinates. It provides what computer scientists might call a "wrapper" for complex numbers, vectors, rotors, spinors, and the physical concepts derived from those objects. In the initial paper I read, the third page was devoted to showing, almost off-handedly, that geometric algebra has an almost trivial isomorphism that is in every way equivalent to the complex numbers. But there's more. Difficult concepts in physics emerge naturally, almost casually, through the manipulation of geometric space using the algebra. It's greatest benefit may be this: It allows "specialists" to actually talk to each other. Who woulda thunk it!

Here are some links: Hestenes, D. - A Unified Language for Mathematics and Physics Gull, S; Lasenby, A; Doran, C - Imaginary Numbers Are Not Real - The Geometric Algebra of Spacetime Hestenes, D - Reforming the Mathematical Language of Physics Lasenby, J; Lasenby A; Doran, C - A Unified Language for Physics and Engineering in the 21st Century

It can't be emphasized too strongly that geometric algebra is not just another technique. It is, instead, an all encompassing framework.

The best mathematical introduction I've found, to date, is from Alan Macdonald of Luther College in Iowa. His freely available paper, A Survey of Geometric Algebra and Geometric Calculus, is exceptional, but you have to be prepared to spend time with it. It is rigorous and comprehensive, and every page is a new adventure as you learn the operations of geometric algebra.

Macdonald also has a recently (2009) published book, the first undergraduate text to cover both linear algebra and geometric algebra.

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Answer by Norm Cimon for Geometrical meaning of Grassmann Algebra