About a month ago I interrupted my progress thru Penrose’s The Road to Reality to make some notes on Clifford algebras. I have answered most of the questions that initially concerned me but naturally came up with yet more questions. The notes are largely chronological as I explored. Ultimately I want to connect this with Quantum Logic and Probability Theory.

The most fruitful web sources that I have found are the Wikipedia article and John Baez’s Clifford stuff. Both of these presentations are mixtures of concrete and abstract notions. You can proceed with the concrete development quite satisfactorily but be lost in details. The abstract (base free) notions require more sophistication, but fewer details. The same pattern is found in vector spaces, but magnified a few times over. My notes here are mostly concrete with a few references to base free notions.

I have made use of programs to nail down concepts and at the same time have the corroboration that computers can provide. The programs are frankly crutches but have aided me greatly. They are “super-concrete”. The current programs prove nothing but provide strong checks for the sorts of errors that I most often make in proofs. The programs are especially helpful in falsifying or corroborating conjectures. Indeed the programs are the only novel thing in these pages on Clifford algebras. They are somewhat parallel to similar programs on division algebras which follow ideas dating back to Hamilton.

Penrose motivates Clifford algebras as a tool to study rotations in n dimensions, as quaternions can be used to show that 3D rotations of 4π are different from rotations of 2π. This seems like a small point but it may provide a necessary advantage of Clifford algebras over orthogonal matrices in modeling rotations. I have an intuitive non-algebraic proof of this but only for three dimensions. “Spinors” may be reified as Clifford numbers.

This all takes place in an n dimensional Euclidean space, or equivalently a vector space with quadratic form, called “V” in these pages. I use “orientation” to mean some reposition of all the points of a rigid body which leaves the origin fixed. A reflection is an orientation here. A real orthogonal matrix specifies a unique orientation and conversely. “Rotation” here is an equivalence class of paths thru orientation space. A rotation ends up at some particular orientation. If n>2 then exactly two equivalence classes end up at any particular orientation. A quaternion and its negative represent the same orientation but different rotations.

The Questions

My initial questions were I found in the Wikipedia article that the clifford numbers that define rotations are collectively the “Clifford group”. I now have a program, Cg?, that tests a Clifford number for being in the Clifford group and another, Om, that computes an orthogonal matrix from a Clifford number in the group. I define the “unit group” to be that subset of the Clifford group whose elements have length = 1. (Length(c) = √Q(c) = (sqrt (sp c c))). The unit group is a “double cover” of O(n) which is the set of rigid rotations, including reflections. I believe that computing a Clifford number from an orthogonal matrix is tantamount to finding the complex eigenvalues of the real orthogonal matrix, a well studied problem. I have no program to do this.