The Clifford Product

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Section 6.4 The Clifford Product

What is the relationship between the Clifford algebra considered in Section 6.2 and the exterior algebra considered in Section 6.3? These algebras have the same size, but different products. However, it is easily seen that the two products agree so long as the elements being multiplied are orthogonal! Thus, the Clifford product can be viewed as a generalization of the exterior product that incorporates information about the inner product.

More precisely, we can introduce a new bilinear, associative product on \(V\text{,}\) written as \(\vee\) (“vee”), and given by

\begin{equation} v\vee w = v\wedge w + g(v,w)\tag{6.4.1} \end{equation}

and use associativity to determine the form of this product on elements of \(\Cl(V)\) with rank other than \(1\text{.}\) It is then reasonably straightforward to verify that this new product is in fact precisely the Clifford product.

Again, this construction extends easily to the case of vector fields, allowing for the realization of Clifford algebras on the (dual) vector fields on any manifold.