Английская Википедия:Clifford algebra

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In mathematics, a Clifford algebraШаблон:Efn is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As [[algebra over a field|Шаблон:Math-algebras]], they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems.Шаблон:SfnШаблон: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.Шаблон:Efn

Introduction and basic properties

A Clifford algebra is a unital associative algebra that contains and is generated by a vector space Шаблон:Math over a field Шаблон:Math, where Шаблон:Math is equipped with a quadratic form Шаблон:Math. The Clifford algebra Шаблон:Math is the "freest" unital associative algebra generated by Шаблон:Math subject to the conditionШаблон: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 Шаблон:Math is its multiplicative identity. 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 Шаблон:Math is a finite-dimensional real vector space and Шаблон:Math is nondegenerate, Шаблон:Math may be identified by the label Шаблон:Math, indicating that Шаблон:Math has an orthogonal basis with Шаблон:Math elements with Шаблон:Math, Шаблон:Math with Шаблон:Math, and where Шаблон: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 Шаблон:Math may be written as the tensor algebra Шаблон:Math, that is, the direct sum of the tensor product of Шаблон:Math copies of Шаблон:Math over all Шаблон: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 Шаблон:Math for all elements Шаблон:Math. The product induced by the tensor product in the quotient algebra is written using juxtaposition (e.g. Шаблон:Math). Its associativity follows from the associativity of the tensor product.

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

If the characteristic of the ground field Шаблон:Math is not Шаблон: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 Шаблон:Math, via the polarization identity.

Quadratic forms and Clifford algebras in characteristic Шаблон:Math form an exceptional case. In particular, if Шаблон:Math it is not true that a quadratic form uniquely determines a symmetric bilinear form satisfying Шаблон:Math, nor that every quadratic form admits an orthogonal basis. Many of the statements in this article include the condition that the characteristic is not Шаблон: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 Шаблон:Math then the Clifford algebra Шаблон:Math is just the exterior algebra Шаблон:Math. For nonzero Шаблон:Math there exists a canonical linear isomorphism between Шаблон:Math and Шаблон:Math whenever the ground field Шаблон:Math does not have characteristic two. 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 Шаблон: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 Шаблон:Math be a vector space over a field Шаблон:Math, and let Шаблон:Math be a quadratic form on Шаблон:Math. In most cases of interest the field Шаблон:Math is either the field of real numbers Шаблон:Math, or the field of complex numbers Шаблон:Math, or a finite field.

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

The quadratic form Шаблон:Math may be replaced by a (not necessarily symmetric) bilinear form Шаблон:Math that has the property Шаблон:Math, in which case an equivalent requirement on Шаблон: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 Шаблон: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 Шаблон:Math, namely the tensor algebra Шаблон:Math, and then enforce the fundamental identity by taking a suitable quotient. In our case we want to take the two-sided ideal Шаблон:Math in Шаблон: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 Шаблон: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 productШаблон:Sfn to distinguish it from the exterior product and the scalar product.

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

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

Basis and dimension

Since Шаблон:Math comes equipped with a quadratic form Шаблон:Math, in characteristic not equal to Шаблон:Math there exist bases for Шаблон: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 Шаблон: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 Шаблон:Math over Шаблон:Math is Шаблон:Math and Шаблон:Math is an orthogonal basis of Шаблон:Math, then Шаблон:Math is free over Шаблон: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 (Шаблон:Math) is defined as the multiplicative identity element. For each value of Шаблон:Math there are [[Binomial coefficient|Шаблон: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 Шаблон:Math and Шаблон:Math is isomorphic to Шаблон:Math or Шаблон:Math, where Шаблон:Math is a full matrix ring with entries from Шаблон:Math, Шаблон:Math, or Шаблон:Math. For a complete classification of these algebras see Classification of Clifford algebras.

Real numbers

Шаблон:Main

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 Шаблон:Math is the dimension of the vector space. The pair of integers Шаблон:Math is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted Шаблон:Math The Clifford algebra on Шаблон:Math is denoted Шаблон:Math The symbol Шаблон:Math means either Шаблон:Math or Шаблон:Math depending on whether the author prefers positive-definite or negative-definite spaces.

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

A few low-dimensional cases are:

Complex numbers

One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space of dimension n 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 Шаблон: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 Шаблон:Math with the standard quadratic form by Шаблон:Math.

For the first few cases one finds that

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

Examples: constructing quaternions and dual quaternions

Quaternions

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

Let the vector space Шаблон:Math be real three-dimensional space Шаблон:Math, and the quadratic form be the usual quadratic form. Then, for Шаблон:Math in Шаблон: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 Шаблон:Math and Шаблон:Math given by <math display="block"> v w + w v = 2 (v \cdot w) .</math>

Denote a set of orthogonal unit vectors of Шаблон:Math as Шаблон: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 Шаблон: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 Шаблон:Math defines the even subalgebra Шаблон: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 Шаблон: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 Шаблон: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 Clifford algebra of real four-dimensional space with a degenerate quadratic form.Шаблон:SfnШаблон:Sfn

Let the vector space Шаблон:Math be real four-dimensional space Шаблон:Math and let the quadratic form Шаблон:Math be a degenerate form derived from the Euclidean metric on Шаблон:Math For Шаблон:Math in Шаблон: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 Шаблон:Math onto the Шаблон:Math hyperplane.

The Clifford product of vectors Шаблон:Math and Шаблон: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 Шаблон:Math as Шаблон: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 Шаблон:Math has 16 components. The linear combination of the even degree elements defines the even subalgebra Шаблон: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 Шаблон:Math and the dual unit Шаблон: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 Шаблон: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 Шаблон:Math and Шаблон:Math alternate signs an even number of times, and show the dual unit Шаблон:Math commutes with the quaternion basis elements Шаблон:Math

Examples: in small dimension

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

Dimension 1

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

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

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

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

Dimension 2

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

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

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

Properties

Relation to the exterior algebra

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

The easiest way to establish the isomorphism is to choose an orthogonal basis Шаблон:Math for Шаблон:Math and extend it to a basis for Шаблон:Math as described above. The map Шаблон: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 only works if the basis Шаблон: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 Шаблон:Math is Шаблон:Math, one can also establish the isomorphism by antisymmetrizing. Define functions Шаблон: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 Шаблон:Math elements, Шаблон:Math. Since Шаблон:Math is alternating it induces a unique linear map Шаблон:Math. The direct sum of these maps gives a linear map between Шаблон:Math and Шаблон: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 Шаблон:Math. Recall that the tensor algebra Шаблон:Math has a natural filtration: Шаблон:Math, where Шаблон:Math contains sums of tensors with order Шаблон:Math. Projecting this down to the Clifford algebra gives a filtration on Шаблон: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 Шаблон: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 Шаблон:Math in Шаблон:Math for all Шаблон: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 Шаблон:Math.Шаблон:Efn

Clifford algebras are Шаблон:Math-graded algebras (also known as superalgebras). Indeed, the linear map on V defined by Шаблон:Math (reflection through the origin) preserves the quadratic form Q 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 Шаблон:Math is an involution (i.e. it squares to the identity) one can decompose Шаблон:Math into positive and negative eigenspaces of Шаблон: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 Шаблон: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 Шаблон:Math the structure of a Шаблон:Math-graded algebra. The subspace Шаблон:Math forms a subalgebra of Шаблон:Math, called the even subalgebra. The subspace Шаблон:Math is called the odd part of Шаблон:Math (it is not a subalgebra). Шаблон:Math-grading plays an important role in the analysis and application of Clifford algebras. The automorphism Шаблон:Math is called the main involution or grade involution. Elements that are pure in this Шаблон:Math-grading are simply said to be even or odd.

Remark. The Clifford algebra is not a Шаблон:Math-graded algebra, but is Шаблон:Math-filtered, where Шаблон:Math is the subspace spanned by all products of at most Шаблон:Math elements of Шаблон: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 Шаблон:Math-grading.

The even subalgebra Шаблон:Math of a Clifford algebra is itself isomorphic to a Clifford algebra.Шаблон:EfnШаблон:Efn If Шаблон:Math is the orthogonal direct sum of a vector Шаблон:Math of nonzero norm Шаблон:Math and a subspace Шаблон:Math, then Шаблон:Math is isomorphic to Шаблон:Math, where Шаблон:Math is the form Шаблон:Math restricted to Шаблон:Math and multiplied by Шаблон: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 Шаблон:Math, which extends the sequence Шаблон:Block indent

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

Antiautomorphisms

In addition to the automorphism Шаблон:Math, there are two antiautomorphisms that play an important role in the analysis of Clifford algebras. Recall that the tensor algebra Шаблон: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 Шаблон:Math is invariant under this reversal, this operation descends to an antiautomorphism of Шаблон:Math called the transpose or reversal operation, denoted by Шаблон:Math. The transpose is an antiautomorphism: Шаблон:Math. The transpose operation makes no use of the Шаблон:Math-grading so we define a second antiautomorphism by composing Шаблон: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.Шаблон:Efn

Note that all of these operations are involutions. One can show that they act as Шаблон:Math on elements which are pure in the Шаблон:Math-grading. In fact, all three operations depend only on the degree modulo Шаблон:Math. That is, if Шаблон:Math is pure with degree Шаблон: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:

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

Clifford scalar product

When the characteristic is not Шаблон:Math, the quadratic form Шаблон:Math on Шаблон:Math can be extended to a quadratic form on all of Шаблон:Math (which we also denoted by Шаблон: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 Шаблон:Math denotes the scalar part of Шаблон:Math (the degree-Шаблон:Math part in the Шаблон: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 Шаблон:Math are elements of Шаблон:Math – this identity is not true for arbitrary elements of Шаблон:Math.

The associated symmetric bilinear form on Шаблон: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 Шаблон:Math. The bilinear form on all of Шаблон:Math is nondegenerate if and only if it is nondegenerate on Шаблон:Math.

The operator of left (respectively right) Clifford multiplication by the transpose Шаблон:Math of an element Шаблон:Math is the adjoint of left (respectively right) Clifford multiplication by Шаблон: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 Шаблон:Math, the vector space Шаблон:Math is finite-dimensional and that the associated symmetric bilinear form of Шаблон:Math is nondegenerate.

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

The structure of Clifford algebras can be worked out explicitly using the following result. Suppose that Шаблон:Math has even dimension and a non-singular bilinear form with discriminant Шаблон:Math, and suppose that Шаблон:Math is another vector space with a quadratic form. The Clifford algebra of Шаблон:Math is isomorphic to the tensor product of the Clifford algebras of Шаблон:Math and Шаблон:Math, which is the space Шаблон:Math with its quadratic form multiplied by Шаблон: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 only on the signature Шаблон:Math. This is an algebraic form of Bott periodicity.

Lipschitz group

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

In this section we assume that Шаблон:Math is finite-dimensional and the quadratic form Шаблон: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 Шаблон:Math maps Шаблон:Math, where Шаблон:Math is the main involution defined above.

The Lipschitz group Шаблон:Math is defined to be the set of invertible elements Шаблон:Math that stabilize the set of vectors under this action,Шаблон:Sfn meaning that for all Шаблон:Math in Шаблон: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 Шаблон:Math that preserves the quadratic form Шаблон:Math, and so gives a homomorphism from the Lipschitz group to the orthogonal group. The Lipschitz group contains all elements Шаблон:Math of Шаблон:Math for which Шаблон:Math is invertible in Шаблон:Math, and these act on Шаблон:Math by the corresponding reflections that take Шаблон:Math to Шаблон:Math. (In characteristic Шаблон:Math these are called orthogonal transvections rather than reflections.)

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

Шаблон:Details

In arbitrary characteristic, the spinor norm Шаблон: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 Шаблон:Math of non-zero elements of Шаблон:Math. It coincides with the quadratic form Шаблон:Math of Шаблон:Math when Шаблон: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 Шаблон:Math, Шаблон:Math, or Шаблон:Math on Шаблон:Math. The difference is not very important in characteristic other than 2.

The nonzero elements of Шаблон:Math have spinor norm in the group (Шаблон:Math of squares of nonzero elements of the field Шаблон:Math. So when Шаблон:Math is finite-dimensional and non-singular we get an induced map from the orthogonal group of Шаблон:Math to the group Шаблон:Math, also called the spinor norm. The spinor norm of the reflection about Шаблон:Math, for any vector Шаблон:Math, has image Шаблон:Math in Шаблон: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 \mbox{Pin}_V(K)  &\to \mbox{O}_V(K)  \to K^\times/\left(K^\times\right)^2, \\
 1 \to \{\pm 1\} \to \mbox{Spin}_V(K) &\to \mbox{SO}_V(K) \to K^\times/\left(K^\times\right)^2.

\end{align}</math>

Note that in characteristic Шаблон:Math the group Шаблон: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 Шаблон:Math for the algebraic group of square roots of 1 (over a field of characteristic not Шаблон: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 \mbox{Pin}_V \rightarrow \mbox{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(\mbox{Pin}_V; K) \to H^0(\mbox{O}_V; K) \to H^1(\mu_2; K).</math>

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

Spin and pin groups

Шаблон:Details

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

The pin group Шаблон:Math is the subgroup of the Lipschitz group Шаблон:Math of elements of spinor norm Шаблон:Math, and similarly the spin group Шаблон:Math is the subgroup of elements of Dickson invariant Шаблон:Math in Шаблон:Math. When the characteristic is not Шаблон:Math, these are the elements of determinant Шаблон:Math. The spin group usually has index Шаблон: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 Шаблон:Math. If Шаблон:Math does not have characteristic Шаблон:Math this is just the group of elements of the orthogonal group of determinant Шаблон:Math. If Шаблон:Math does have characteristic Шаблон:Math, then all elements of the orthogonal group have determinant Шаблон:Math, and the special orthogonal group is the set of elements of Dickson invariant Шаблон:Math.

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

In the common case when Шаблон: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 Шаблон:Math has dimension at least Шаблон:Math. Further the kernel of this homomorphism consists of Шаблон:Math and Шаблон:Math. So in this case the spin group, Шаблон:Math, is a double cover of Шаблон:Math. Please note, however, that the simple connectedness of the spin group is not true in general: if Шаблон:Math is Шаблон:Math for Шаблон:Math and Шаблон:Math both at least Шаблон:Math then the spin group is not simply connected. In this case the algebraic group Шаблон:Math is simply connected as an algebraic group, even though its group of real valued points Шаблон: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.Шаблон:Which

Spinors

Clifford algebras Шаблон:Math, with Шаблон:Math even, are matrix algebras which have a complex representation of dimension Шаблон:Math. By restricting to the group Шаблон: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 Шаблон:Math then it splits as the sum of two half spin representations (or Weyl representations) of dimension Шаблон:Math.

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

Шаблон:Details To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The pin group, Шаблон:Math is the set of invertible elements in Шаблон: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 Шаблон:Math. The spin group consists of those elements of Шаблон: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 Шаблон:Math.

Let Шаблон:Math be the automorphism which is given by the mapping Шаблон:Math acting on pure vectors. Then in particular, Шаблон:Math is the subgroup of Шаблон:Math whose elements are fixed by Шаблон: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 Шаблон:Math.) Then the spin group lies within Шаблон:Math.

The irreducible representations of Шаблон: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 Шаблон: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 Шаблон:Math as a pin representation in either signature Шаблон:Math or Шаблон: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 Шаблон:Math manifolds.

Physics

Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra with a basis generated by the matrices Шаблон: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 Шаблон:Math is the matrix of a quadratic form of signature Шаблон:Math (or Шаблон:Math corresponding to the two equivalent choices of metric signature). These are exactly the defining relations for the Clifford algebra Шаблон:Math, whose complexification is Шаблон:Math which, by the classification of Clifford algebras, is isomorphic to the algebra of Шаблон:Math complex matrices Шаблон:Math. However, it is best to retain the notation Шаблон: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 Шаблон: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 Шаблон: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 Шаблон:Math convention, hence fits in Шаблон:Math.Шаблон: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,Шаблон: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.Шаблон:SfnШаблон:Sfn

Computer vision

Clifford algebras have been applied in the problem of action recognition and classification in computer vision. Rodriguez et alШаблон: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.Шаблон:Efn
  • Clifford algebras may be generalized to a form of degree higher than quadratic over a vector space.Шаблон:Sfn

See also

Шаблон:Portal Шаблон:Div col

Шаблон:Div col end

Notes

Шаблон:Notelist

Citations

Шаблон:Reflist

References

Шаблон:Refbegin

Шаблон:Refend

Further reading

Шаблон:Refbegin

Шаблон:Refend

External links

Шаблон:Number systems Шаблон:Industrial and applied mathematics Шаблон:Authority control

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