Clifford Algebras with Coördinates

We start with an n-dimensional vector space V over the reals R, and a positive definite quadratic form Q on V.

The Clifford algebra is a 2n dimensional Associative algebra C whose multiplication we will define. First there is in C a copy both of R and of V as separate subspaces of C. When we write an expression for an element of V or R, there may be some confusion as to whether we are talking about a member of R, V or C. We shall talk as if R and V were both subsets of C. It turns out that some equations will have two or three readings and a more careful development would prove that these all mean the same thing. This follows from the axioms of associative algebras. This notational ambiguity will initially confuse any careful reader.

For this development we choose a basis B for V which is orthonormal according to Q. (This is always possible.) Q(Σxibi) = Σxi2 summed over the members bi of B. Our multiplication is defined by the requirements that if x and y are distinct elements of B then xx = −1 and xy = −yx. With these rules we can compute arbitrary products in C.

Consider the transitive closure c of B under multiplication but not addition. If one applies association freely and sorts the factors from B, applying xx = −1 and xy = −yx, then no member of B can appear more than once. There is thus in c just one product of elements from B for each of the 2n subsets of B. The empty subset corresponds to the real 1, which is a member of C. These 2n values form a basis for C. As we have seen in the note on associative algebras, knowing the products of the basis vectors of an algebra, lets one compute the product of arbitrary members of the algebra.

This follows the development by Clifford. When physics uses Clifford algebras today, one selects a subset of the elements of B and for each member x of that subset requires that xx=1 instead of xx=−1. This is like the generalization that Minkowski made to Riemmann’s metric tensor by allowing negative elements on the diagonal.

Products of an even number of elements of B form a basis for the even subspace of C. Ditto odd. Each of these spaces is 2n−1 dimensional. Both of these spaces are closed under addition but the product of two odd vectors is even. The sum of even or odd vectors is typically neither. Clifford numbers that model rotations are even.