Paravector

The name paravector is used for the combination of a scalar and a vector in any Clifford algebra, known as geometric algebra among physicists. This name was given by J. G. Maks in a doctoral dissertation at Technische Universiteit Delft, Netherlands, in 1989. The complete algebra of paravectors along with corresponding higher grade generalizations, all in the context of the Euclidean space of three dimensions, is an alternative approach to the spacetime algebra (STA) introduced by David Hestenes. This alternative algebra is called algebra of physical space (APS). For Euclidean spaces, the fundamental axiom indicates that the product of a vector with itself is the scalar value of the length squared (positive) Writing and introducing this into the expression of the fundamental axiom we get the following expression after appealing to the fundamental axiom again which allows to identify the scalar product of two vectors as As an important consequence we conclude that two orthogonal vectors (with zero scalar product) anticommute The following list represents an instance of a complete basis for the space, which forms an eight-dimensional space, where the multiple indices indicate the product of the respective basis vectors, for example The grade of a basis element is defined in terms of the vector multiplicity, such that According to the fundamental axiom, two different basis vectors anticommute, or in other words, This means that the volume element squares to Moreover, the volume element commutes with any other element of the algebra, so that it can be identified with the complex number , whenever there is no danger of confusion. In fact, the volume element along with the real scalar forms an algebra isomorphic to the standard complex algebra. The volume element can be used to rewrite an equivalent form of the basis as The corresponding paravector basis that combines a real scalar and vectors is which forms a four-dimensional linear space.