Skip to navigation

What is Clifford Algebra?

Clifford Algebra is a formulation of algebra which unifies and extends complex numbers and the familiar vector algebra.  The basis of the algebra is the Clifford product of two vectors a and b which is written ab. The product has two parts, a scalar part and a bivector part.  The scalar part is symmetric and corresponds with the usual dot product.

The bivector part is antisymmetric and can be thought of as a directed area, defining a plane.

Thus the product can be written

A series of Clifford Algebras can be formed with different dimensions.   Each vector can be represented as an ordered list of n scalar values a_i.

These are the scalar factors in the sum of products of the scalars with unit vectors e_i in each of the n dimensions, where each product of a basis vector with a scalar commutes i.e.

 so that

The e_i are usually, but not necessarily, orthogonal.  If they are then the following relationships apply

and if

then

The full Clifford Algebra is made up of all possible combinations of Clifford products of 0, 1, 2, 3, ...n of the basis vectors. The terms with zero basis vectors are the scalars.  The total number of elements is 2ⁿ.  The individual terms have the order corresponding to the number of basis vectors in the term.

The most general object in any Clifford algebra is the sum of multiples of all these terms.

1.1 Clifford Algebra in Two Dimensions

In this very simple Clifford Algebra there are two dimensions which can be represented by two orthogonal unit vectors e₁ and e₂ which are such that

e₁e₂ is termed the unit bivector.  A most interesting result is that

so that there are four members of the basis for the algebra

and the last one squares to minus one and thus forms a way of building the complex algebra, if the definition of i is taken as

 For example, if a vector a is defined as

then multiplying e₁ by a gives

and multiplying a by e₁ gives which is the complex conjugate of the first result.  The whole of the rest of complex algebra can be built up from here.

1.2 Where to find out more.

The best starting point I know for introductory material is the web site of the Geometric Algebra Research Group at Cambridge University.  My thanks to them for help and encouragement.  I welcome feedback and discussion of Clifford Algebra.

John Fletcher October 2002, updated March 2003.

Site Search