This book is only an introduction to Clifford algebras. Here are some suggestions for further reading; they are not meant to provide a comprehensive bibliography, but rather to indicate where to go next.

The algebraic environment

Further results about multilinear mappings and tensor products, are given in standard textbooks, such as Cohn [Coh], Jacobson [Jac] and Mac Lane and Birkhoff [MaB]. The results are presented in the more general setting of modules over a commutative ring. This leads to serious problems which do not arise in the vector space case. The idea of considering representations of algebras in terms of modules extends to infinite-dimensional algebras, such as C*-algebras. A good starting point for this is the book by Lance [Lan].

The proof of the existence of the tensor product of two modules over a commutative ring, as described in the remark at the end of Section 3.2, does not lead to a simple description of the structure of the tensor product, and there are also problems with torsion. Tensor products of vector spaces are so much more straightforward that they deserve to be treated separately.

Quadratic spaces

Lam [Lam] is the standard work on quadratic forms, and is a goldmine of mathematics. It considers quadratic forms over fields not of characteristic 2.

Clifford algebras are finite-dimensional algebras. Here we consider the properties of finite-dimensional algebras. We also consider how they can be represented as algebras of endomorphisms of a vector space, or equivalently as algebras of matrices. An alternative way of thinking about this is to consider modules over an algebra; this is important in the theory of Clifford algebras, where such modules appear as spaces of spinors.

Algebras

Again, let K denote either the field R of real numbers or the field C of complex numbers. A finite-dimensional (associative) algebra A over K is a finite-dimensional vector space over K equipped with a law of composition: that is, a mapping (multiplication) (a, b) → ab from A × A into A which satisfies

• (ab)c = a(bc) (associativity),

• a(b + c) = ab + ac,

• (a + b)c = ac + bc,

• λ(ab) = (λa)b = a(λb),

for λ ∈ K and a, b, c ∈ A. (As usual, multiplication is carried out before addition).

An algebra A is unital if there exists 1 ∈ A, the identity element, such that 1a = a1 = a for all a ∈ A. We shall principally be concerned with unital algebras. An algebra A is commutative if ab = ba for all a, b ∈ A.

A mapping φ from an algebra A over K to an algebra B over K is an algebra homomorphism if it is linear, and if φ(ab) = φ(a)φ(b) for a, b ∈ A.

In this chapter, we briefly describe some applications of Clifford algebras to physics. The first of these concerns the spin of an elementary particle, when the spin is 1/2. Pauli introduced the Pauli spin matrices to describe this phenomenon. Physics is concerned with partial differential operators: building on Pauli's ideas, Dirac introduced a first order differential operator, now called the Dirac operator, in order to formulate the relativistic wave equation for an electron; this in turn led to the discovery of the positron. We shall study the Dirac operator more fully in the next chapter. Here we show how it can be used to formulate Maxwell's equation for an electromagnetic field in a particularly simple way, and will also describe the Dirac equation.

Particles with spin 1/2

The Pauli spin matrices were introduced by Pauli to represent the internal angular momentum of particles which have spin 1/2. Let us briefly describe how this can be interpreted in terms of the Clifford algebra A0,3. In quantum mechanics, an observable corresponds to a Hermitian linear operator T on a Hilbert space H, and, when the possible values of the observable are discrete, these possible values are the eigenvalues of T.

The Stern-Gerlach experiment showed that elementary particles have an intrinsic angular momentum, or spin. If x is a unit vector in R3 then the component Jx of the spin in the direction x is an observable.

The material in this chapter should be familiar to the reader, but it is worth reading through it to become familiar with the notation and terminology that is used. We shall not give details; these are given in standard textbooks, such as Mac Lane and Birkhoff [MaB], Jacobson [Jac] or Cohn [Coh].

Groups

A group is a non-empty set G together with a law of composition, a mapping (g, h) → gh from G × G to G, which satisfies:

1. (gh)j = g(hj) for all g, h, j in G (associativity),

2. there exists e in G such that eg = ge = g for all g ∈ G, and

3. for each g ∈ G there exists g-1 ∈ G such that gg-1 = g-1g = e.

It then follows that e, the identity element, is unique, and that for each g ∈ G the inverse g-1 is unique.

A group G is abelian, or commutative, if gh = hg for all g, h ∈ G. If G is abelian, then the law of composition is often written as addition: (g, h) → g + h. In such a case, the identity is denoted by 0, and the inverse of g by -g.

A non-empty subset H of a group G is a subgroup of G if h1h2 ∈ H whenever h1, h2 ∈ H, and h-1 ∈ H whenever h ∈ H.

Clifford algebras find their use in many areas of mathematics: in differential analysis, where operators of Dirac type are used in proofs of the Atiyah-Singer index theorem, in harmonic analysis, where the Riesz transforms provide a higher-dimensional generalization of the Hilbert transform, in geometry, where spin groups illuminate the structure of the classical groups, and in mathematical physics, where Clifford algebras provide a setting for electromagnetic theory, spin 1/2 particles, and the Dirac operator in relativistic quantum mechanics. This book is intended as a straightforward introduction to Clifford algebras, without going on to study any of the above topics in detail (suggestions for further reading are made at the end). This means that it concentrates on the underlying structure of Clifford algebras, and this inevitably means that it approaches the subject algebraically.

The first part is concerned with the background from algebra that is required. The first chapter describes, without giving details, the necessary knowledge of groups and vector spaces that is needed. Any reader who is not familiar with this material should consult standard texts on algebra, such as Mac Lane and Birkhoff [MaB], Jacobson [Jac] or Cohn [Coh]. Otherwise, skim through it, to familiarize yourself with the notation and terminology that is used.

The second chapter deals with algebras, and modules over algebras. It turns out that the algebra H of quaternions has an important part to play in the theory of Clifford algebras, and fundamental properties of this algebra are developed here.