In the Dark

Yesterday we had a very nice pedagogical seminar in the Department of Theoretical Physics by one of our PhD students, Gert Vercleyen, who talked about something that isn’t really to do with his main research topic. A departmental seminar is a good environment for research students to gain experience giving presentations. Anyway, the abstract for this talk was:

Anyone doing a degree in physics, engineering, or mathematics will, at a very early stage, need to learn how to deal with vectors. Typically the theory of vectors comes with several products, like the dot product which is useful for determining lengths and angles, the cross product which allows one to find orthogonal vectors, and in 2D the complex product which allows one to easily describe rotations and dilations. Each of these products has its benefits and problems. The dot product is not invertible, the complex product only works in 2D, and the cross product has too many issues to put in this abstract. The goal of the talk is to present an alternative product of vectors, the geometric product, that works in any dimension, allows one to get geometric data, and can be used to apply geometric transformations. I will describe how the usual products can be obtained from the geometric product and work out various examples. 

I was familiar with the basic ideas of this approach (related to Clifford algebra) which encompasses many ideas used frequently in theoretical physics – including quaternions for example (I have to mention them as I’m in Ireland) – in a single elegant formalism. I have never actually used it for anything however. Maybe that will change, though, as many interesting ideas suggested themselves during the talk.

If you’d like to learn a bit more at an introductory level about Geometric Algebra you could do a lot worse than read this paper which, unbelievably, is almost 30 years old. I mentioned at the end of the talk that the first author of this paper, Steve Gull, taught the first course in Mathematics for Natural Sciences I took when I was in the first year at Cambridge way back in 1982. Although he crammed a huge amount into that course, including the “standard” way of talking about vectors, rotations thereof using matrices, and a bit of cartesian tensors, he didn’t talk about Geometric Algebra.

I do think however that there is a case for starting in Year 1 with geometric algebra instead of the way we do it nowadays, not least because as well as being an elegant formalism it lends itself very easily to computational implementation; indeed, I note that there is a Python implementation of Clifford Algebra (which I have not yet played with). Also I think it’s harder to “unlearn” traditional methods and adapt to new ones as you get older.