Introduction to Clifford Algebras

Clifford algebras of lower-dimensional Euclidean spaces and Minkowski spacetime are discussed and identified with their matrix images. The geometric significance of quaternions and bivectors is explored in 3D and 4D. Pauli’s introduction of spin is reviewed in terms of Clifford algebras. The exterior algebra and contractions are introduced and related to the Clifford algebra. The exterior and shuffle products of the Grassmann-Cayley algebra are applied to the join and meet; it is shown that the shuffle product is like an exterior product stepping downwards from the volume element. Spin groups and conformal groups are discussed in lower dimensions. Then Clifford algebras are defined over arbitrary fields for arbitrary quadratic forms as well as for nonsymmetric bilinear forms.