Clifford Algebras

Abstract

One of the fundamental reasons for the success of complex analysis is that it is possible, using complex numbers, to express geometrical ideas. Basically this is due to the fact that points in the complex plane are represented by complex numbers. This way one can express vector addition, rotations (by multiplication with a complex number with modulus 1), reflection in the real axis (by the mapping z → $\bar{z}$ ) and others in a straightforward way. In this chapter Clifford algebras (sometimes called geometrical algebras) are introduced, which allow similar constructs for dimensions greater than two. Unfortunately some properties of the complex number field, such as commutativity or the existence of an inverse for every nonzero number, are lost. On the other hand, Clifford algebras allow the manipulation of several notions which are specific to higher dimensional geometry, such as k-blades. Basically, a Clifford algebra is constructed from a finite-dimensional space with a scalar product (not necessarily positive definite), introducing an algebra multiplication which both reflects the properties of this scalar product and of the outer product.