Clifford algebra

In mathematics, a Clifford algebraTemplate:Efn is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As [[algebra over a field|Template:Math-algebras]], they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems.Template:SfnTemplate:Sfn The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford (1845–1879).

The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (pseudo-)Riemannian Clifford algebras, as distinct from symplectic Clifford algebras.Template:Efn

Introduction and basic properties

A Clifford algebra is a unital associative algebra that contains and is generated by a vector space Template:Math over a field Template:Math, where Template:Math is equipped with a quadratic form Template:Math. The Clifford algebra Template:Math is the "freest" unital associative algebra generated by Template:Math subject to the conditionTemplate:Efn <math display="block">v^2 = Q(v)1\ \text{ for all } v\in V,</math> where the product on the left is that of the algebra, and the Template:Math on the right is the algebra's multiplicative identity (not to be confused with the multiplicative identity of Template:Math). 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.

When Template:Math is a finite-dimensional real vector space and Template:Math is nondegenerate, Template:Math may be identified by the label Template:Math, indicating that Template:Math has an orthogonal basis with Template:Math elements with Template:Math, Template:Math with Template:Math, and where Template:Math indicates that this is a Clifford algebra over the reals; i.e. coefficients of elements of the algebra are real numbers. This basis may be found by orthogonal diagonalization.

The free algebra generated by Template:Math may be written as the tensor algebra Template:Math, that is, the direct sum of the tensor product of Template:Math copies of Template:Math over all Template:Math. Therefore one obtains a Clifford algebra as the quotient of this tensor algebra by the two-sided ideal generated by elements of the form Template:Math for all elements Template:Math. The product induced by the tensor product in the quotient algebra is written using juxtaposition (e.g. Template:Math). Its associativity follows from the associativity of the tensor product.

The Clifford algebra has a distinguished subspace Template:Math, being the image of the embedding map. Such a subspace cannot in general be uniquely determined given only a Template:Math-algebra that is isomorphic to the Clifford algebra.

If Template:Math is invertible in the ground field Template:Math, then one can rewrite the fundamental identity above in the form <math display="block">uv + vu = 2\langle u, v\rangle1\ \text{ for all } u,v \in V,</math> where <math display="block">\langle u, v \rangle = \frac{1}{2} \left( Q(u + v) - Q(u) - Q(v) \right)</math> is the symmetric bilinear form associated with Template:Math, via the polarization identity.

Quadratic forms and Clifford algebras in characteristic Template:Math form an exceptional case in this respect. In particular, if Template:Math it is not true that a quadratic form necessarily or uniquely determines a symmetric bilinear form that satisfies Template:Math,Template:Sfn Many of the statements in this article include the condition that the characteristic is not Template:Math, and are false if this condition is removed.

As a quantization of the exterior algebra

Clifford algebras are closely related to exterior algebras. Indeed, if Template:Math then the Clifford algebra Template:Math is just the exterior algebra Template:Math. Whenever Template:Math is invertible in the ground field Template:Math, there exists a canonical linear isomorphism between Template:Math and Template:Math. That is, they are naturally isomorphic as vector spaces, but with different multiplications (in the case of characteristic two, they are still isomorphic as vector spaces, just not naturally). Clifford multiplication together with the distinguished subspace is strictly richer than the exterior product since it makes use of the extra information provided by Template:Math.

The Clifford algebra is a filtered algebra; the associated graded algebra is the exterior algebra.

More precisely, Clifford algebras may be thought of as quantizations (cf. quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra.

Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.

Universal property and construction

Let Template:Math be a vector space over a field Template:Math, and let Template:Math be a quadratic form on Template:Math. In most cases of interest the field Template:Math is either the field of real numbers Template:Math, or the field of complex numbers Template:Math, or a finite field.

A Clifford algebra Template:Math is a pair Template:Math,Template:EfnTemplate:Sfn where Template:Math is a unital associative algebra over Template:Math and Template:Math is a linear map Template:Math that satisfies Template:Math for all Template:Math in Template:Math, defined by the following universal property: given any unital associative algebra Template:Math over Template:Math and any linear map Template:Math such that <math display="block">j(v)^2 = Q(v)1_A \text{ for all } v \in V</math> (where Template:Math denotes the multiplicative identity of Template:Math), there is a unique algebra homomorphism Template:Math such that the following diagram commutes (i.e. such that Template:Math):

File:CliffordAlgebra-01.png

The quadratic form Template:Math may be replaced by a (not necessarily symmetricTemplate:Sfn) bilinear form Template:Math that has the property Template:Math, in which case an equivalent requirement on Template:Math is <math display="block"> j(v)j(v) = \langle v, v \rangle 1_A \quad \text{ for all } v \in V .</math>

When the characteristic of the field is not Template:Math, this may be replaced by what is then an equivalent requirement, <math display="block"> j(v)j(w) + j(w)j(v) = ( \langle v, w \rangle + \langle w, v \rangle )1_A \quad \text{ for all } v, w \in V , </math> where the bilinear form may additionally be restricted to being symmetric without loss of generality.

A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that contains Template:Math, namely the tensor algebra Template:Math, and then enforce the fundamental identity by taking a suitable quotient. In our case we want to take the two-sided ideal Template:Math in Template:Math generated by all elements of the form <math display="block">v\otimes v - Q(v)1</math> for all <math>v\in V</math> and define Template:Math as the quotient algebra <math display="block">\operatorname{Cl}(V, Q) = T(V) / I_Q .</math>

The ring product inherited by this quotient is sometimes referred to as the Clifford productTemplate:Sfn to distinguish it from the exterior product and the scalar product.

It is then straightforward to show that Template:Math contains Template:Math and satisfies the above universal property, so that Template:Math is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra Template:Math. It also follows from this construction that Template:Math is injective. One usually drops the Template:Math and considers Template:Math as a linear subspace of Template:Math.

The universal characterization of the Clifford algebra shows that the construction of Template:Math is Template:Em in nature. Namely, Template:Math can be considered as a functor from the category of vector spaces with quadratic forms (whose morphisms are linear maps that preserve the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (that preserve the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras.

Basis and dimension

Since Template:Math comes equipped with a quadratic form Template:Math, in characteristic not equal to Template:Math there exist bases for Template:Math that are orthogonal. An orthogonal basis is one such that for a symmetric bilinear form <math display="block">\langle e_i, e_j \rangle = 0 </math> for <math> i\neq j</math>, and <math display="block">\langle e_i, e_i \rangle = Q(e_i).</math>

The fundamental Clifford identity implies that for an orthogonal basis <math display="block">e_i e_j = -e_j e_i</math> for <math>i \neq j</math>, and <math display="block">e_i^2 = Q(e_i).</math>

This makes manipulation of orthogonal basis vectors quite simple. Given a product <math>e_{i_1}e_{i_2}\cdots e_{i_k}</math> of distinct orthogonal basis vectors of Template:Math, one can put them into a standard order while including an overall sign determined by the number of pairwise swaps needed to do so (i.e. the signature of the ordering permutation).

If the dimension of Template:Math over Template:Math is Template:Math and Template:Math is an orthogonal basis of Template:Math, then Template:Math is [[Free module|free over Template:Math]] with a basis <math display="block">\{e_{i_1}e_{i_2}\cdots e_{i_k} \mid 1\le i_1 < i_2 < \cdots < i_k \le n\text{ and } 0\le k\le n\}.</math>

The empty product (Template:Math) is defined as being the multiplicative identity element. For each value of Template:Math there are [[Binomial coefficient|Template:Math]] basis elements, so the total dimension of the Clifford algebra is <math display="block">\dim \operatorname{Cl}(V, Q) = \sum_{k=0}^n \binom{n}{k} = 2^n.</math>

Examples: real and complex Clifford algebras

The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms.

Each of the algebras Template:Math and Template:Math is isomorphic to Template:Math or Template:Math, where Template:Math is a full matrix ring with entries from Template:Math, Template:Math, or Template:Math. For a complete classification of these algebras see Classification of Clifford algebras.

Real numbers

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

Clifford algebras are also sometimes referred to as geometric algebras, most often over the real numbers.

Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form: <math display="block">Q(v) = v_1^2 + \dots + v_p^2 - v_{p+1}^2 - \dots - v_{p+q}^2 ,</math> where Template:Math is the dimension of the vector space. The pair of integers Template:Math is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted Template:Math The Clifford algebra on Template:Math is denoted Template:Math The symbol Template:Math means either Template:Math or Template:Math, depending on whether the author prefers positive-definite or negative-definite spaces.

A standard basis Template:Math for Template:Math consists of Template:Math mutually orthogonal vectors, Template:Math of which square to Template:Math and Template:Math of which square to Template:Math. Of such a basis, the algebra Template:Math will therefore have Template:Math vectors that square to Template:Math and Template:Math vectors that square to Template:Math.

A few low-dimensional cases are:

• Template:Math is naturally isomorphic to Template:Math since there are no nonzero vectors.
• Template:Math is a two-dimensional algebra generated by Template:Math that squares to Template:Math, and is algebra-isomorphic to Template:Math, the field of complex numbers.
• Template:Math is a two-dimensional algebra generated by Template:Math that squares to Template:Math, and is algebra-isomorphic to the split-complex numbers.
• Template:Math is a four-dimensional algebra spanned by Template:Math. The latter three elements all square to Template:Math and anticommute, and so the algebra is isomorphic to the quaternions Template:Math.
• Template:Math is isomorphic to the algebra of split-quaternions.
• Template:Math is an 8-dimensional algebra isomorphic to the direct sum Template:Math, the split-biquaternions.
• Template:Math, also called the Pauli algebra,Template:SfnTemplate:Sfn is isomorphic to the algebra of biquaternions.

Complex numbers

One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space of dimension Template:Math is equivalent to the standard diagonal form <math display="block">Q(z) = z_1^2 + z_2^2 + \dots + z_n^2.</math> Thus, for each dimension Template:Math, up to isomorphism there is only one Clifford algebra of a complex vector space with a nondegenerate quadratic form. We will denote the Clifford algebra on Template:Math with the standard quadratic form by Template:Math.

For the first few cases one finds that

• Template:Math, the complex numbers
• Template:Math, the bicomplex numbers
• Template:Math, the biquaternions

where Template:Math denotes the algebra of Template:Math matrices over Template:Math.

Examples: constructing quaternions and dual quaternions

Quaternions

In this section, Hamilton's quaternions are constructed as the even subalgebra of the Clifford algebra Template:Math.

Let the vector space Template:Math be real three-dimensional space Template:Math, and the quadratic form be the usual quadratic form. Then, for Template:Math in Template:Math we have the bilinear form (or scalar product) <math display="block">v \cdot w = v_1 w_1 + v_2 w_2 + v_3 w_3.</math> Now introduce the Clifford product of vectors Template:Math and Template:Math given by <math display="block"> v w + w v = 2 (v \cdot w) .</math>

Denote a set of orthogonal unit vectors of Template:Math as Template:Math, then the Clifford product yields the relations <math display="block"> e_2 e_3 = -e_3 e_2, \,\,\, e_1 e_3 = -e_3 e_1,\,\,\, e_1 e_2 = -e_2 e_1,</math> and <math display="block"> e_1 ^2 = e_2^2 = e_3^2 = 1. </math> The general element of the Clifford algebra Template:Math is given by <math display="block"> A = a_0 + a_1 e_1 + a_2 e_2 + a_3 e_3 + a_4 e_2 e_3 + a_5 e_1 e_3 + a_6 e_1 e_2 + a_7 e_1 e_2 e_3.</math>

The linear combination of the even degree elements of Template:Math defines the even subalgebra Template:Math with the general element <math display="block"> q = q_0 + q_1 e_2 e_3 + q_2 e_1 e_3 + q_3 e_1 e_2. </math> The basis elements can be identified with the quaternion basis elements Template:Math as <math display="block"> i= e_2 e_3, j = e_1 e_3, k = e_1 e_2,</math> which shows that the even subalgebra Template:Math is Hamilton's real quaternion algebra.

To see this, compute <math display="block"> i^2 = (e_2 e_3)^2 = e_2 e_3 e_2 e_3 = - e_2 e_2 e_3 e_3 = -1,</math> and <math display="block"> ij = e_2 e_3 e_1 e_3 = -e_2 e_3 e_3 e_1 = -e_2 e_1 = e_1 e_2 = k.</math> Finally, <math display="block"> ijk = e_2 e_3 e_1 e_3 e_1 e_2 = -1.</math>

Dual quaternions

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.Template:SfnTemplate:Sfn

Let the vector space Template:Math be real four-dimensional space Template:Math and let the quadratic form Template:Math be a degenerate form derived from the Euclidean metric on Template:Math For Template:Math in Template:Math introduce the degenerate bilinear form <math display="block">d(v, w) = v_1 w_1 + v_2 w_2 + v_3 w_3 .</math> This degenerate scalar product projects distance measurements in Template:Math onto the Template:Math hyperplane.

The Clifford product of vectors Template:Math and Template:Math is given by <math display="block">v w + w v = -2 \,d(v, w).</math> Note the negative sign is introduced to simplify the correspondence with quaternions.

Denote a set of mutually orthogonal unit vectors of Template:Math as Template:Math, then the Clifford product yields the relations <math display="block">e_m e_n = -e_n e_m, \,\,\, m \ne n,</math> and <math display="block">e_1 ^2 = e_2^2 = e_3^2 = -1, \,\, e_4^2 = 0.</math>

The general element of the Clifford algebra Template:Math has 16 components. The linear combination of the even degree elements defines the even subalgebra Template:Math with the general element <math display="block"> H = h_0 + h_1 e_2 e_3 + h_2 e_3 e_1 + h_3 e_1 e_2 + h_4 e_4 e_1 + h_5 e_4 e_2 + h_6 e_4 e_3 + h_7 e_1 e_2 e_3 e_4.</math>

The basis elements can be identified with the quaternion basis elements Template:Math and the dual unit Template:Math as <math display="block"> i = e_2 e_3, j = e_3 e_1, k = e_1 e_2, \,\, \varepsilon = e_1 e_2 e_3 e_4.</math> This provides the correspondence of Template:Math with dual quaternion algebra.

To see this, compute <math display="block"> \varepsilon ^2 = (e_1 e_2 e_3 e_4)^2 = e_1 e_2 e_3 e_4 e_1 e_2 e_3 e_4 = -e_1 e_2 e_3 (e_4 e_4 ) e_1 e_2 e_3 = 0 ,</math> and <math display="block"> \varepsilon i = (e_1 e_2 e_3 e_4) e_2 e_3 = e_1 e_2 e_3 e_4 e_2 e_3 = e_2 e_3 (e_1 e_2 e_3 e_4) = i\varepsilon.</math> The exchanges of Template:Math and Template:Math alternate signs an even number of times, and show the dual unit Template:Math commutes with the quaternion basis elements Template:Math.

Examples: in small dimension

Let Template:Math be any field of characteristic not Template:Math.

Dimension 1

For Template:Math, if Template:Math has diagonalization Template:Math, that is there is a non-zero vector Template:Math such that Template:Math, then Template:Math is algebra-isomorphic to a Template:Math-algebra generated by an element Template:Math that satisfies Template:Math, the quadratic algebra Template:Math.

In particular, if Template:Math (that is, Template:Math is the zero quadratic form) then Template:Math is algebra-isomorphic to the dual numbers algebra over Template:Math.

If Template:Math is a non-zero square in Template:Math, then Template:Math.

Otherwise, Template:Math is isomorphic to the quadratic field extension Template:Math of Template:Math.

Dimension 2

For Template:Math, if Template:Math has diagonalization Template:Math with non-zero Template:Math and Template:Math (which always exists if Template:Math is non-degenerate), then Template:Math is isomorphic to a Template:Math-algebra generated by elements Template:Math and Template:Math that satisfies Template:Math, Template:Math and Template:Math.

Thus Template:Math is isomorphic to the (generalized) quaternion algebra Template:Math. We retrieve Hamilton's quaternions when Template:Math, since Template:Math.

As a special case, if some Template:Math in Template:Math satisfies Template:Math, then Template:Math.

Properties

Relation to the exterior algebra

Given a vector space Template:Math, one can construct the exterior algebra Template:Math, whose definition is independent of any quadratic form on Template:Math. It turns out that if Template:Math does not have characteristic Template:Math then there is a natural isomorphism between Template:Math and Template:Math considered as vector spaces (and there exists an isomorphism in characteristic two, which may not be natural). This is an algebra isomorphism if and only if Template:Math. One can thus consider the Clifford algebra Template:Math as an enrichment (or more precisely, a quantization, cf. the Introduction) of the exterior algebra on Template:Math with a multiplication that depends on Template:Math (one can still define the exterior product independently of Template:Math).

The easiest way to establish the isomorphism is to choose an orthogonal basis Template:Math for Template:Math and extend it to a basis for Template:Math as described above. The map Template:Math is determined by <math display="block">e_{i_1}e_{i_2} \cdots e_{i_k} \mapsto e_{i_1}\wedge e_{i_2}\wedge \cdots \wedge e_{i_k}.</math> Note that this works only if the basis Template:Math is orthogonal. One can show that this map is independent of the choice of orthogonal basis and so gives a natural isomorphism.

If the characteristic of Template:Math is Template:Math, one can also establish the isomorphism by antisymmetrizing. Define functions Template:Math by <math display="block">f_k(v_1, \ldots, v_k) = \frac{1}{k!}\sum_{\sigma\in \mathrm{S}_k} \sgn(\sigma)\, v_{\sigma(1)}\cdots v_{\sigma(k)}</math> where the sum is taken over the symmetric group on Template:Math elements, Template:Math. Since Template:Math is alternating, it induces a unique linear map Template:Math. The direct sum of these maps gives a linear map between Template:Math and Template:Math. This map can be shown to be a linear isomorphism, and it is natural.

A more sophisticated way to view the relationship is to construct a filtration on Template:Math. Recall that the tensor algebra Template:Math has a natural filtration: Template:Math, where Template:Math contains sums of tensors with order Template:Math. Projecting this down to the Clifford algebra gives a filtration on Template:Math. The associated graded algebra <math display="block">\operatorname{Gr}_F \operatorname{Cl}(V,Q) = \bigoplus_k F^k/F^{k-1}</math> is naturally isomorphic to the exterior algebra Template:Math. Since the associated graded algebra of a filtered algebra is always isomorphic to the filtered algebra as filtered vector spaces (by choosing complements of Template:Math in Template:Math for all Template:Math), this provides an isomorphism (although not a natural one) in any characteristic, even two.

Grading

In the following, assume that the characteristic is not Template:Math.Template:Efn

Clifford algebras are Template:Math-graded algebras (also known as superalgebras). Indeed, the linear map on Template:Math defined by Template:Math (reflection through the origin) preserves the quadratic form Template:Math and so by the universal property of Clifford algebras extends to an algebra automorphism <math display="block">\alpha: \operatorname{Cl}(V, Q) \to \operatorname{Cl}(V, Q).</math>

Since Template:Math is an involution (i.e. it squares to the identity) one can decompose Template:Math into positive and negative eigenspaces of Template:Math <math display="block">\operatorname{Cl}(V, Q) = \operatorname{Cl}^{[0]}(V, Q) \oplus \operatorname{Cl}^{[1]}(V, Q)</math> where <math display="block">\operatorname{Cl}^{[i]}(V, Q) = \left\{ x \in \operatorname{Cl}(V, Q) \mid \alpha(x) = (-1)^i x \right\}.</math>

Since Template:Math is an automorphism it follows that: <math display="block">\operatorname{Cl}^{[i]}(V, Q)\operatorname{Cl}^{[j]}(V, Q) = \operatorname{Cl}^{[i+j]}(V, Q)</math> where the bracketed superscripts are read modulo 2. This gives Template:Math the structure of a Template:Math-graded algebra. The subspace Template:Math forms a subalgebra of Template:Math, called the even subalgebra. The subspace Template:Math is called the odd part of Template:Math (it is not a subalgebra). Template:Math-grading plays an important role in the analysis and application of Clifford algebras. The automorphism Template:Math is called the main involution or grade involution. Elements that are pure in this Template:Math-grading are simply said to be even or odd.

Remark. The Clifford algebra is not a Template:Math-graded algebra, but is Template:Math-filtered, where Template:Math is the subspace spanned by all products of at most Template:Math elements of Template:Math. <math display="block">\operatorname{Cl}^{\leqslant i}(V, Q) \cdot \operatorname{Cl}^{\leqslant j}(V, Q) \subset \operatorname{Cl}^{\leqslant i+j}(V, Q).</math>

The degree of a Clifford number usually refers to the degree in the Template:Math-grading.

The even subalgebra Template:Math of a Clifford algebra is itself isomorphic to a Clifford algebra.Template:EfnTemplate:Efn If Template:Math is the orthogonal direct sum of a vector Template:Math of nonzero norm Template:Math and a subspace Template:Math, then Template:Math is isomorphic to Template:Math, where Template:Math is the form Template:Math restricted to Template:Math. In particular over the reals this implies that: <math display="block">\operatorname{Cl}_{p,q}^{[0]}(\mathbf{R}) \cong \begin{cases}

 \operatorname{Cl}_{p,q-1}(\mathbf{R}) & q > 0 \\
 \operatorname{Cl}_{q,p-1}(\mathbf{R}) & p > 0

\end{cases}</math>

In the negative-definite case this gives an inclusion Template:Math, which extends the sequence Template:Block indent

Likewise, in the complex case, one can show that the even subalgebra of Template:Math is isomorphic to Template:Math.

Antiautomorphisms

In addition to the automorphism Template:Math, there are two antiautomorphisms that play an important role in the analysis of Clifford algebras. Recall that the tensor algebra Template:Math comes with an antiautomorphism that reverses the order in all products of vectors: <math display="block">v_1\otimes v_2\otimes \cdots \otimes v_k \mapsto v_k\otimes \cdots \otimes v_2\otimes v_1.</math> Since the ideal Template:Math is invariant under this reversal, this operation descends to an antiautomorphism of Template:Math called the transpose or reversal operation, denoted by Template:Math. The transpose is an antiautomorphism: Template:Math. The transpose operation makes no use of the Template:Math-grading so we define a second antiautomorphism by composing Template:Math and the transpose. We call this operation Clifford conjugation denoted <math>\bar x</math> <math display="block">\bar x = \alpha(x^\mathrm{t}) = \alpha(x)^\mathrm{t}.</math> Of the two antiautomorphisms, the transpose is the more fundamental.Template:Efn

Note that all of these operations are involutions. One can show that they act as Template:Math on elements that are pure in the Template:Math-grading. In fact, all three operations depend on only the degree modulo Template:Math. That is, if Template:Math is pure with degree Template:Math then <math display="block">\alpha(x) = \pm x \qquad x^\mathrm{t} = \pm x \qquad \bar x = \pm x</math> where the signs are given by the following table:

Template:Math	Template:Math	Template:Math	Template:Math	Template:Math	…
<math>\alpha(x)\,</math>	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math
<math>x^\mathrm{t}\,</math>	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math
<math>\bar x</math>	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math

Clifford scalar product

When the characteristic is not Template:Math, the quadratic form Template:Math on Template:Math can be extended to a quadratic form on all of Template:Math (which we also denoted by Template:Math). A basis-independent definition of one such extension is <math display="block">Q(x) = \left\langle x^\mathrm{t} x\right\rangle_0</math> where Template:Math denotes the scalar part of Template:Math (the degree-Template:Math part in the Template:Math-grading). One can show that <math display="block">Q(v_1v_2 \cdots v_k) = Q(v_1)Q(v_2) \cdots Q(v_k)</math> where the Template:Math are elements of Template:Math – this identity is not true for arbitrary elements of Template:Math.

The associated symmetric bilinear form on Template:Math is given by <math display="block">\langle x, y\rangle = \left\langle x^\mathrm{t} y\right\rangle_0.</math> One can check that this reduces to the original bilinear form when restricted to Template:Math. The bilinear form on all of Template:Math is nondegenerate if and only if it is nondegenerate on Template:Math.

The operator of left (respectively right) Clifford multiplication by the transpose Template:Math of an element Template:Math is the adjoint of left (respectively right) Clifford multiplication by Template:Math with respect to this inner product. That is, <math display="block">\langle ax, y\rangle = \left\langle x, a^\mathrm{t} y\right\rangle,</math> and <math display="block">\langle xa, y\rangle = \left\langle x, y a^\mathrm{t}\right\rangle.</math>

Structure of Clifford algebras

In this section we assume that characteristic is not Template:Math, the vector space Template:Math is finite-dimensional and that the associated symmetric bilinear form of Template:Math is nondegenerate.

A central simple algebra over Template:Math is a matrix algebra over a (finite-dimensional) division algebra with center Template:Math. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions.

• If Template:Math has even dimension then Template:Math is a central simple algebra over Template:Math.
• If Template:Math has even dimension then the even subalgebra Template:Math is a central simple algebra over a quadratic extension of Template:Math or a sum of two isomorphic central simple algebras over Template:Math.
• If Template:Math has odd dimension then Template:Math is a central simple algebra over a quadratic extension of Template:Math or a sum of two isomorphic central simple algebras over Template:Math.
• If Template:Math has odd dimension then the even subalgebra Template:Math is a central simple algebra over Template:Math.

The structure of Clifford algebras can be worked out explicitly using the following result. Suppose that Template:Math has even dimension and a non-singular bilinear form with discriminant Template:Math, and suppose that Template:Math is another vector space with a quadratic form. The Clifford algebra of Template:Math is isomorphic to the tensor product of the Clifford algebras of Template:Math and Template:Math, which is the space Template:Math with its quadratic form multiplied by Template:Math. Over the reals, this implies in particular that <math display="block"> \operatorname{Cl}_{p+2,q}(\mathbf{R}) = \mathrm{M}_2(\mathbf{R})\otimes \operatorname{Cl}_{q,p}(\mathbf{R}) </math> <math display="block"> \operatorname{Cl}_{p+1,q+1}(\mathbf{R}) = \mathrm{M}_2(\mathbf{R})\otimes \operatorname{Cl}_{p,q}(\mathbf{R}) </math> <math display="block"> \operatorname{Cl}_{p,q+2}(\mathbf{R}) = \mathbf{H}\otimes \operatorname{Cl}_{q,p}(\mathbf{R}). </math> These formulas can be used to find the structure of all real Clifford algebras and all complex Clifford algebras; see the classification of Clifford algebras.

Notably, the Morita equivalence class of a Clifford algebra (its representation theory: the equivalence class of the category of modules over it) depends on only the signature Template:Math. This is an algebraic form of Bott periodicity.

Lipschitz group

The class of Lipschitz groups (Template:AkaTemplate:Sfn Clifford groups or Clifford–Lipschitz groups) was discovered by Rudolf Lipschitz.Template:Sfn

In this section we assume that Template:Math is finite-dimensional and the quadratic form Template:Math is nondegenerate.

An action on the elements of a Clifford algebra by its group of units may be defined in terms of a twisted conjugation: twisted conjugation by Template:Math maps Template:Math, where Template:Math is the main involution defined above.

The Lipschitz group Template:Math is defined to be the set of invertible elements Template:Math that stabilize the set of vectors under this action,Template:Sfn meaning that for all Template:Math in Template:Math we have: <math display="block">\alpha(x) v x^{-1}\in V .</math>

This formula also defines an action of the Lipschitz group on the vector space Template:Math that preserves the quadratic form Template:Math, and so gives a homomorphism from the Lipschitz group to the orthogonal group. The Lipschitz group contains all elements Template:Math of Template:Math for which Template:Math is invertible in Template:Math, and these act on Template:Math by the corresponding reflections that take Template:Math to Template:Math. (In characteristic Template:Math these are called orthogonal transvections rather than reflections.)

If Template:Math is a finite-dimensional real vector space with a non-degenerate quadratic form then the Lipschitz group maps onto the orthogonal group of Template:Math with respect to the form (by the Cartan–Dieudonné theorem) and the kernel consists of the nonzero elements of the field Template:Math. This leads to exact sequences <math display="block"> 1 \rightarrow K^\times \rightarrow \Gamma \rightarrow \operatorname{O}_V(K) \rightarrow 1,</math> <math display="block"> 1 \rightarrow K^\times \rightarrow \Gamma^0 \rightarrow \operatorname{SO}_V(K) \rightarrow 1.</math>

Over other fields or with indefinite forms, the map is not in general onto, and the failure is captured by the spinor norm.

Spinor norm

In arbitrary characteristic, the spinor norm Template:Math is defined on the Lipschitz group by <math display="block">Q(x) = x^\mathrm{t}x.</math> It is a homomorphism from the Lipschitz group to the group Template:Math of non-zero elements of Template:Math. It coincides with the quadratic form Template:Math of Template:Math when Template:Math is identified with a subspace of the Clifford algebra. Several authors define the spinor norm slightly differently, so that it differs from the one here by a factor of Template:Math, Template:Math, or Template:Math on Template:Math. The difference is not very important in characteristic other than 2.

The nonzero elements of Template:Math have spinor norm in the group (Template:Math of squares of nonzero elements of the field Template:Math. So when Template:Math is finite-dimensional and non-singular we get an induced map from the orthogonal group of Template:Math to the group Template:Math, also called the spinor norm. The spinor norm of the reflection about Template:Math, for any vector Template:Math, has image Template:Math in Template:Math, and this property uniquely defines it on the orthogonal group. This gives exact sequences: <math display="block">\begin{align}

 1 \to \{\pm 1\} \to \operatorname{Pin}_V(K)  &\to \operatorname{O}_V(K)  \to K^\times/\left(K^\times\right)^2, \\
 1 \to \{\pm 1\} \to \operatorname{Spin}_V(K) &\to \operatorname{SO}_V(K) \to K^\times/\left(K^\times\right)^2.

\end{align}</math>

Note that in characteristic Template:Math the group Template:Math has just one element.

From the point of view of Galois cohomology of algebraic groups, the spinor norm is a connecting homomorphism on cohomology. Writing Template:Math for the algebraic group of square roots of 1 (over a field of characteristic not Template:Math it is roughly the same as a two-element group with trivial Galois action), the short exact sequence <math display="block"> 1 \to \mu_2 \rightarrow \operatorname{Pin}_V \rightarrow \operatorname{O}_V \rightarrow 1</math> yields a long exact sequence on cohomology, which begins <math display="block"> 1 \to H^0(\mu_2; K) \to H^0(\operatorname{Pin}_V; K) \to H^0(\operatorname{O}_V; K) \to H^1(\mu_2; K).</math>

The 0th Galois cohomology group of an algebraic group with coefficients in Template:Math is just the group of Template:Math-valued points: Template:Math, and Template:Math, which recovers the previous sequence <math display="block"> 1 \to \{\pm 1\} \to \operatorname{Pin}_V(K) \to \operatorname{O}_V(K) \to K^\times/\left(K^\times\right)^2,</math> where the spinor norm is the connecting homomorphism Template:Math.

Spin and pin groups

In this section we assume that Template:Math is finite-dimensional and its bilinear form is non-singular.

The pin group Template:Math is the subgroup of the Lipschitz group Template:Math of elements of spinor norm Template:Math, and similarly the spin group Template:Math is the subgroup of elements of Dickson invariant Template:Math in Template:Math. When the characteristic is not Template:Math, these are the elements of determinant Template:Math. The spin group usually has index Template:Math in the pin group.

Recall from the previous section that there is a homomorphism from the Lipschitz group onto the orthogonal group. We define the special orthogonal group to be the image of Template:Math. If Template:Math does not have characteristic Template:Math this is just the group of elements of the orthogonal group of determinant Template:Math. If Template:Math does have characteristic Template:Math, then all elements of the orthogonal group have determinant Template:Math, and the special orthogonal group is the set of elements of Dickson invariant Template:Math.

There is a homomorphism from the pin group to the orthogonal group. The image consists of the elements of spinor norm Template:Math. The kernel consists of the elements Template:Math and Template:Math, and has order Template:Math unless Template:Math has characteristic Template:Math. Similarly there is a homomorphism from the Spin group to the special orthogonal group of Template:Math.

In the common case when Template:Math is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected when Template:Math has dimension at least Template:Math. Further the kernel of this homomorphism consists of Template:Math and Template:Math. So in this case the spin group, Template:Math, is a double cover of Template:Math. Note, however, that the simple connectedness of the spin group is not true in general: if Template:Math is Template:Math for Template:Math and Template:Math both at least Template:Math then the spin group is not simply connected. In this case the algebraic group Template:Math is simply connected as an algebraic group, even though its group of real valued points Template:Math is not simply connected. This is a rather subtle point, which completely confused the authors of at least one standard book about spin groups.Template:Which

Spinors

Clifford algebras Template:Math, with Template:Math even, are matrix algebras that have a complex representation of dimension Template:Math. By restricting to the group Template:Math we get a complex representation of the Pin group of the same dimension, called the spin representation. If we restrict this to the spin group Template:Math then it splits as the sum of two half spin representations (or Weyl representations) of dimension Template:Math.

If Template:Math is odd then the Clifford algebra Template:Math is a sum of two matrix algebras, each of which has a representation of dimension Template:Math, and these are also both representations of the pin group Template:Math. On restriction to the spin group Template:Math these become isomorphic, so the spin group has a complex spinor representation of dimension Template:Math.

More generally, spinor groups and pin groups over any field have similar representations whose exact structure depends on the structure of the corresponding Clifford algebras: whenever a Clifford algebra has a factor that is a matrix algebra over some division algebra, we get a corresponding representation of the pin and spin groups over that division algebra. For examples over the reals see the article on spinors.

Real spinors

Template:Details To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The pin group, Template:Math is the set of invertible elements in Template:Math that can be written as a product of unit vectors: <math display="block">\mathrm{Pin}_{p,q} = \left\{v_1v_2 \cdots v_r \mid \forall i\, \|v_i\| = \pm 1\right\}.</math> Comparing with the above concrete realizations of the Clifford algebras, the pin group corresponds to the products of arbitrarily many reflections: it is a cover of the full orthogonal group Template:Math. The spin group consists of those elements of Template:Math that are products of an even number of unit vectors. Thus by the Cartan–Dieudonné theorem Spin is a cover of the group of proper rotations Template:Math.

Let Template:Math be the automorphism that is given by the mapping Template:Math acting on pure vectors. Then in particular, Template:Math is the subgroup of Template:Math whose elements are fixed by Template:Math. Let <math display="block">\operatorname{Cl}_{p,q}^{[0]} = \{ x\in \operatorname{Cl}_{p,q} \mid \alpha(x) = x\}.</math> (These are precisely the elements of even degree in Template:Math.) Then the spin group lies within Template:Math.

The irreducible representations of Template:Math restrict to give representations of the pin group. Conversely, since the pin group is generated by unit vectors, all of its irreducible representation are induced in this manner. Thus the two representations coincide. For the same reasons, the irreducible representations of the spin coincide with the irreducible representations of Template:Math.

To classify the pin representations, one need only appeal to the classification of Clifford algebras. To find the spin representations (which are representations of the even subalgebra), one can first make use of either of the isomorphisms (see above) <math display="block">\operatorname{Cl}^{[0]}_{p,q} \approx \operatorname{Cl}_{p,q-1}, \text{ for } q > 0</math> <math display="block">\operatorname{Cl}^{[0]}_{p,q} \approx \operatorname{Cl}_{q,p-1}, \text{ for } p > 0</math> and realize a spin representation in signature Template:Math as a pin representation in either signature Template:Math or Template:Math.

Applications

Differential geometry

One of the principal applications of the exterior algebra is in differential geometry where it is used to define the bundle of differential forms on a smooth manifold. In the case of a (pseudo-)Riemannian manifold, the tangent spaces come equipped with a natural quadratic form induced by the metric. Thus, one can define a Clifford bundle in analogy with the exterior bundle. This has a number of important applications in Riemannian geometry. Perhaps more important is the link to a spin manifold, its associated spinor bundle and Template:Math manifolds.

Physics

Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra that has a basis that is generated by the matrices Template:Math, called Dirac matrices, which have the property that <math display="block">\gamma_i\gamma_j + \gamma_j\gamma_i = 2\eta_{ij} ,</math> where Template:Math is the matrix of a quadratic form of signature Template:Math (or Template:Math corresponding to the two equivalent choices of metric signature). These are exactly the defining relations for the Clifford algebra Template:Math, whose complexification is Template:Math, which, by the classification of Clifford algebras, is isomorphic to the algebra of Template:Math complex matrices Template:Math. However, it is best to retain the notation Template:Math, since any transformation that takes the bilinear form to the canonical form is not a Lorentz transformation of the underlying spacetime.

The Clifford algebra of spacetime used in physics thus has more structure than Template:Math. It has in addition a set of preferred transformations – Lorentz transformations. Whether complexification is necessary to begin with depends in part on conventions used and in part on how much one wants to incorporate straightforwardly, but complexification is most often necessary in quantum mechanics where the spin representation of the Lie algebra Template:Math sitting inside the Clifford algebra conventionally requires a complex Clifford algebra. For reference, the spin Lie algebra is given by <math display="block">\begin{align}

 \sigma^{\mu\nu}
   &= -\frac{i}{4}\left[\gamma^\mu,\, \gamma^\nu\right], \\
 \left[\sigma^{\mu\nu},\, \sigma^{\rho\tau}\right]
   &= i\left(\eta^{\tau\mu}\sigma^{\rho\nu} + \eta^{\nu\tau}\sigma^{\mu\rho} - \eta^{\rho\mu}\sigma^{\tau\nu} - \eta^{\nu\rho} \sigma^{\mu\tau}\right).

\end{align}</math>

This is in the Template:Math convention, hence fits in Template:Math.Template:Sfn

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 result was used to define the Dirac equation and introduce the Dirac operator. The entire Clifford algebra shows up in quantum field theory in the form of Dirac field bilinears.

The use of Clifford algebras to describe quantum theory has been advanced among others by Mario Schönberg,Template:Efn 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.Template:SfnTemplate:Sfn

Computer vision

Clifford algebras have been applied in the problem of action recognition and classification in computer vision. Rodriguez et alTemplate:Sfn propose a Clifford embedding to generalize traditional MACH filters to video (3D spatiotemporal volume), and vector-valued data such as optical flow. Vector-valued data is analyzed using the Clifford Fourier Transform. Based on these vectors action filters are synthesized in the Clifford Fourier domain and recognition of actions is performed using Clifford correlation. The authors demonstrate the effectiveness of the Clifford embedding by recognizing actions typically performed in classic feature films and sports broadcast television.

Generalizations

• While this article focuses on a Clifford algebra of a vector space over a field, the definition extends without change to a module over any unital, associative, commutative ring.Template:Efn
• Clifford algebras may be generalized to a form of degree higher than quadratic over a vector space.Template:Sfn

See also

• Algebra of physical space
• Cayley–Dickson construction
• Classification of Clifford algebras
• Clifford analysis
• Clifford module
• Complex spin structure
• Dirac operator
• Exterior algebra
• Fierz identity
• Gamma matrices
• Generalized Clifford algebra
• Geometric algebra
• Higher-dimensional gamma matrices
• Hypercomplex number
• Octonion
• Paravector
• Quaternion
• Spin group
• Spin structure
• Spinor
• Spinor bundle

References

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• Template:Citation. An advanced textbook on Clifford algebras and their applications to differential geometry.
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• Template:Citation; ibid II (1883) 46; ibid III (1884) 7–9. Summarized in The Collected Mathematics Papers of James Joseph Sylvester (Cambridge University Press, 1909) v III. online and further.
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Further reading

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External links

• Template:Springer
• Planetmath entry on Clifford algebras Template:Webarchive
• A history of Clifford algebras (unverified)
• John Baez on Clifford algebras
• Clifford Algebra: A Visual Introduction
• Clifford Algebra Explorer : A Pedagogical Tool