Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing.
The idea of being the "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a universal property, as done below.
Clifford multiplication together with the distinguished subspace is strictly richer than the exterior product since it makes use of the extra information provided by Q.
The universal characterization of the Clifford algebra shows that the construction of Cl(V, Q) is functorial in nature.
The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms.
The symbol Cln(R) means either Cln,0(R) or Cl0,n(R), depending on whether the author prefers positive-definite or negative-definite spaces.
Thus, for each dimension n, up to isomorphism there is only one Clifford algebra of a complex vector space with a nondegenerate quadratic form.
Denote a set of orthogonal unit vectors of R3 as {e1, e2, e3}, then the Clifford product yields the relations
In this section, dual quaternions are constructed as the even subalgebra of a Clifford algebra of real four-dimensional space with a degenerate quadratic form.
Denote a set of mutually orthogonal unit vectors of R4 as {e1, e2, e3, e4}, then the Clifford product yields the relations
The exchanges of e1 and e4 alternate signs an even number of times, and show the dual unit ε commutes with the quaternion basis elements i, j, k. Let K be any field of characteristic not 2.
For dim V = 1, if Q has diagonalization diag(a), that is there is a non-zero vector x such that Q(x) = a, then Cl(V, Q) is algebra-isomorphic to a K-algebra generated by an element x that satisfies x2 = a, the quadratic algebra K[X] / (X2 − a).
Given a vector space V, one can construct the exterior algebra ⋀V, whose definition is independent of any quadratic form on V. It turns out that if K does not have characteristic 2 then there is a natural isomorphism between ⋀V and Cl(V, Q) considered as vector spaces (and there exists an isomorphism in characteristic two, which may not be natural).
One can show that this map is independent of the choice of orthogonal basis and so gives a natural isomorphism.
In addition to the automorphism α, there are two antiautomorphisms that play an important role in the analysis of Clifford algebras.
Recall that the tensor algebra T(V) comes with an antiautomorphism that reverses the order in all products of vectors:
In this section we assume that characteristic is not 2, the vector space V is finite-dimensional and that the associated symmetric bilinear form of Q is nondegenerate.
The Lipschitz group contains all elements r of V for which Q(r) is invertible in K, and these act on V by the corresponding reflections that take v to v − (⟨r, v⟩ + ⟨v, r⟩)r/Q(r).
If V is a finite-dimensional real vector space with a non-degenerate quadratic form then the Lipschitz group maps onto the orthogonal group of V with respect to the form (by the Cartan–Dieudonné theorem) and the kernel consists of the nonzero elements of the field K. This leads to exact sequences
Over other fields or with indefinite forms, the map is not in general onto, and the failure is captured by the spinor norm.
The spinor norm of the reflection about r⊥, for any vector r, has image Q(r) in K×/(K×)2, and this property uniquely defines it on the orthogonal group.
The pin group, Pinp,q is the set of invertible elements in Clp,q that can be written as a product of unit vectors:
The spin group consists of those elements of Pinp,q that are products of an even number of unit vectors.
Conversely, since the pin group is generated by unit vectors, all of its irreducible representation are induced in this manner.
To classify the pin representations, one need only appeal to the classification of Clifford algebras.
In the case of a (pseudo-)Riemannian manifold, the tangent spaces come equipped with a natural quadratic form induced by the metric.
This is in the (3, 1) convention, hence fits in Cl3,1(R)C.[12] The Dirac matrices were first written down by Paul Dirac when he was trying to write a relativistic first-order wave equation for the electron, and give an explicit isomorphism from the Clifford algebra to the algebra of complex matrices.
The use of Clifford algebras to describe quantum theory has been advanced among others by Mario Schönberg,[i] by David Hestenes in terms of geometric calculus, by David Bohm and Basil Hiley and co-workers in form of a hierarchy of Clifford algebras, and by Elio Conte et al.[13][14] Clifford algebras have been applied in the problem of action recognition and classification in computer vision.
Rodriguez et al[15] propose a Clifford embedding to generalize traditional MACH filters to video (3D spatiotemporal volume), and vector-valued data such as optical flow.
The authors demonstrate the effectiveness of the Clifford embedding by recognizing actions typically performed in classic feature films and sports broadcast television.