Multiforms, Extensors, Canonical and Metric Clifford Algebras

Abstract

In this chapter, we introduce the concepts of multiforms, extensors, the canonical and metric Clifford algebras. Given a real vector space V with dim V = n, multiforms are elements of a space denoted ∧V, each multiform being a sum of non homogeneous k-forms (0 ≤ k ≤ n). We define the exterior product of those objects. Next we introduce a Euclidean product (called canonical product) in V and extend it to the space ∧V, defining also the right and left contracted products of multiforms. Those concepts permit us to introduce the canonical Clifford product in ∧V. We, thus, get a very powerful algebraic structure, the canonical Clifford algebra of ∧V, denoted $\mathcal{C}\ell(V,\cdot )$ that is utilized as a basic calculational tool in the rest of the text. Next, we introduce the concept of extensors, linear mappings from Cartesian products $\wedge V \times....\times \wedge V \rightarrow \wedge V$ and study with details the properties of so-called (1,1)-extensors and some important extensors related to it (in particular its extensions), which will appear in different occasions in the following chapters. In particular, they will play a crucial role in our formulation of the differential geometry of manifolds (Chapter 4) and our theory of the gravitational field (Chapter 5). Equipped with the concept of extensor, we introduce in ∧V a metrical extensor g of arbitrary signature (an object directly associated to a metric tensor g of the same signature) and extend it to ∧V. Given a g, we can define a corresponding metric Clifford algebra. We introduce a pseudo-Euclidean metric extensor ηin V and find a (1,1)-extensor h which relates η to g (the deformation of η by h). Moreover, we derive an important formula which permits obtaining h (modulus a Lorentz transformation) in terms of a given g. Several remarkable results are then presented, such as the golden rule relating the scalar, contracted and Clifford products associated to different metric extensors. With the golden rule, it becomes possible also to relate easily thorough a nice formula, the Hodge star operators associated with two different metric extensors. The chapter ends with a section presenting a set of useful identities that are used several times in the remainder of the book.