Clifford algebra


In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. 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.
The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as Riemannian Clifford algebras, as distinct from symplectic Clifford algebras.

Introduction and basic properties

A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form. The Clifford algebra is the "freest" algebra generated by V subject to the condition
where the product on the left is that of the algebra, and the 1 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].
Where V is a real vector space and Q is nondegenerate, may be identified by the label Clp,q, indicating that V has an orthogonal basis with p elements with, q with, and where R indicates that this is a Clifford algebra over the reals; i.e. coefficients of elements of the algebra are real numbers.
The free algebra generated by V may be written as the tensor algebra, that is, the sum of the tensor product of n copies of V over all n, and so a Clifford algebra would be the quotient of this tensor algebra by the two-sided ideal generated by elements of the form for all elements. The product induced by the tensor product in the quotient algebra is written using juxtaposition. Its associativity follows from the associativity of the tensor product.
The Clifford algebra has a distinguished subspace V, being the image of the embedding map. Such a subspace cannot in general be uniquely determined given only a K-algebra isomorphic to the Clifford algebra.
If the characteristic of the ground field K is not 2, then one can rewrite this fundamental identity in the form
where
is the symmetric bilinear form associated with Q, via the polarization identity.
Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case. In particular, if it is not true that a quadratic form uniquely determines a symmetric bilinear form satisfying, nor that every quadratic form admits an orthogonal basis. Many of the statements in this article include the condition that the characteristic is not 2, 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 then the Clifford algebra is just the exterior algebra ⋀. For nonzero Q there exists a canonical linear isomorphism between ⋀ and whenever the ground field K does not have characteristic two. That is, they are naturally isomorphic as vector spaces, but with different multiplications. 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 Clifford algebra is a filtered algebra, the associated graded algebra is the exterior algebra.
More precisely, Clifford algebras may be thought of as quantizations 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 V be a vector space over a field K, and let be a quadratic form on V. In most cases of interest the field K is either the field of real numbers R, or the field of complex numbers C, or a finite field.
A Clifford algebra is a pair, where A is a unital associative algebra over K and i is a linear map satisfying for all v in V, defined by the following universal property: given any unital associative algebra A over K and any linear map such that
, there is a unique algebra homomorphism such that the following diagram commutes :

The quadratic form Q may be replaced by a bilinear form that has the property, in which case an equivalent requirement on j is
When the characteristic of the field is not 2, the first requirement may be omitted as it is implied by the second and the bilinear form may 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 V, namely the tensor algebra T, and then enforce the fundamental identity by taking a suitable quotient. In our case we want to take the two-sided ideal IQ in T generated by all elements of the form
and define as the quotient algebra
The ring product inherited by this quotient is sometimes referred to as the Clifford product to distinguish it from the exterior product and the scalar product.
It is then straightforward to show that contains V and satisfies the above universal property, so that Cl is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra. It also follows from this construction that i is injective. One usually drops the i and considers V as a linear subspace of.
The universal characterization of the Clifford algebra shows that the construction of is functorial in nature. Namely, Cl can be considered as a functor from the category of vector spaces with quadratic forms to the category of associative algebras. The universal property guarantees that linear maps between vector spaces extend uniquely to algebra homomorphisms between the associated Clifford algebras.

Basis and dimension

Since V comes equipped with a quadratic form Q, in characteristic not equal to 2 there exist bases for V that are orthogonal. An orthogonal basis is one such that for a symmetric bilinear form
The fundamental Clifford identity implies that for an orthogonal basis
This makes manipulation of orthogonal basis vectors quite simple. Given a product of distinct orthogonal basis vectors of V, one can put them into a standard order while including an overall sign determined by the number of pairwise swaps needed to do so.
If the dimension of V over K is n and is an orthogonal basis of, then is free over K with a basis
The empty product is defined as the multiplicative identity element. For each value of k there are n choose k basis elements, so the total dimension of the Clifford algebra is

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 Cl and Cl is isomorphic to A or, where A is a full matrix ring with entries from R, C, or H. For a complete classification of these algebras see classification of Clifford algebras.

Real numbers

Real Clifford algebras are also sometimes referred to as geometric algebras.
Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:
where is the dimension of the vector space. The pair of integers is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted Rp,q. The Clifford algebra on R is denoted Cl. The symbol Cl means either Cl or Cl depending on whether the author prefers positive-definite or negative-definite spaces.
A standard basis for Rp,q consists of mutually orthogonal vectors, p of which square to +1 and q of which square to −1. Of such a basis, the algebra Cl will therefore have p vectors that square to +1 and q vectors that square to −1.
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
Thus, for each dimension n, 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 Cn with the standard quadratic form by Cl.
For the first few cases one finds that
where denotes the algebra of matrices over C.

Examples: constructing quaternions and dual quaternions

Quaternions

In this section, Hamilton's quaternions are constructed as the even sub algebra of the Clifford algebra Cl.
Let the vector space V be real three-dimensional space R3, and the quadratic form Q be the negative of the usual Euclidean metric. Then, for v, w in R3 we have the bilinear form
Now introduce the Clifford product of vectors v and w given by
This formulation uses the negative sign so the correspondence with quaternions is easily shown.
Denote a set of orthogonal unit vectors of R3 as e1, e2, and e3, then the Clifford product yields the relations
and
The general element of the Clifford algebra Cl is given by
The linear combination of the even degree elements of Cl defines the even subalgebra Cl with the general element
The basis elements can be identified with the quaternion basis elements i, j, k as
which shows that the even subalgebra Cl is Hamilton's real quaternion algebra.
To see this, compute
and
Finally,

Dual quaternions

In this section, dual quaternions are constructed as the even Clifford algebra of real four-dimensional space with a degenerate quadratic form.
Let the vector space V be real four-dimensional space R4, and let the quadratic form Q be a degenerate form derived from the Euclidean metric on R3. For v, w in R4 introduce the degenerate bilinear form
This degenerate scalar product projects distance measurements in R4 onto the R3 hyperplane.
The Clifford product of vectors v and w is given by
Note the negative sign is introduced to simplify the correspondence with quaternions.
Denote a set of mutually orthogonal unit vectors of R4 as e1, e2, e3 and e4, then the Clifford product yields the relations
and
The general element of the Clifford algebra has 16 components. The linear combination of the even degree elements defines the even subalgebra with the general element
The basis elements can be identified with the quaternion basis elements i, j, k and the dual unit ε as
This provides the correspondence of Cl with dual quaternion algebra.
To see this, compute
and
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, and k.

Examples: in small dimension

Let K be any field of characteristic not 2.

Dimension 1

For, if Q has diagonalization diag, that is there is a non-zero vector x such that, then is algebra-isomorphic to a K-algebra generated by an element x satisfying, the quadratic algebra.
In particular, if then is algebra-isomorphic to the dual numbers algebra over K.
If a is a non-zero square in K, then.
Otherwise, is isomorphic to the quadratic field extension K of K.

Dimension 2

For, if Q has diagonalization with non-zero a and b, then is isomorphic to a K-algebra generated by elements x and y satisfying, and.
Thus is isomorphic to the quaternion algebra. We retrieve Hamilton's quaternions when, since.
As a special case, if some x in V satisfies, then.

Properties

Relation to the exterior algebra

Given a vector space V one can construct the exterior algebra ⋀, 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 ⋀ and considered as vector spaces. This is an algebra isomorphism if and only if. One can thus consider the Clifford algebra as an enrichment of the exterior algebra on V with a multiplication that depends on Q.
The easiest way to establish the isomorphism is to choose an orthogonal basis for V and extend it to a basis for as described above. The map is determined by
Note that this only works if the basis 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 K is 0, one can also establish the isomorphism by antisymmetrizing. Define functions by
where the sum is taken over the symmetric group on k elements, Sk. Since fk is alternating it induces a unique linear map. The direct sum of these maps gives a linear map between ⋀ and. 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. Recall that the tensor algebra T has a natural filtration:, where Fk contains sums of tensors with order. Projecting this down to the Clifford algebra gives a filtration on. The associated graded algebra
is naturally isomorphic to the exterior algebra ⋀. Since the associated graded algebra of a filtered algebra is always isomorphic to the filtered algebra as filtered vector spaces, this provides an isomorphism in any characteristic, even two.

Grading

In the following, assume that the characteristic is not 2.
Clifford algebras are Z2-graded algebras. Indeed, the linear map on V defined by preserves the quadratic form Q and so by the universal property of Clifford algebras extends to an algebra automorphism
Since α is an involution one can decompose into positive and negative eigenspaces of α
where
Since α is an automorphism it follows that:
where the bracketed superscripts are read modulo 2. This gives the structure of a Z2-graded algebra. The subspace forms a subalgebra of, called the even subalgebra. The subspace is called the odd part of . This Z2-grading plays an important role in the analysis and application of Clifford algebras. The automorphism α is called the main involution or grade involution. Elements that are pure in this Z2-grading are simply said to be even or odd.
Remark. In characteristic not 2 the underlying vector space of inherits an N-grading and a Z-grading from the canonical isomorphism with the underlying vector space of the exterior algebra ⋀. It is important to note, however, that this is a vector space grading only. That is, Clifford multiplication does not respect the N-grading or Z-grading, only the Z2-grading: for instance if, then, but, not in. Happily, the gradings are related in the natural way:. Further, the Clifford algebra is Z-filtered:
The degree of a Clifford number usually refers to the degree in the N-grading.
The even subalgebra of a Clifford algebra is itself isomorphic to a Clifford algebra. If V is the orthogonal direct sum of a vector a of nonzero norm Q and a subspace U, then is isomorphic to, where −QQ is the form Q restricted to U and multiplied by −Q. In particular over the reals this implies that:
In the negative-definite case this gives an inclusion, which extends the sequence
Likewise, in the complex case, one can show that the even subalgebra of Cl is isomorphic to Cl.

Antiautomorphisms

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 comes with an antiautomorphism that reverses the order in all products of vectors:
Since the ideal IQ is invariant under this reversal, this operation descends to an antiautomorphism of called the transpose or reversal operation, denoted by xt. The transpose is an antiautomorphism:. The transpose operation makes no use of the Z2-grading so we define a second antiautomorphism by composing α and the transpose. We call this operation Clifford conjugation denoted
Of the two antiautomorphisms, the transpose is the more fundamental.
Note that all of these operations are involutions. One can show that they act as ±1 on elements which are pure in the Z-grading. In fact, all three operations depend only on the degree modulo 4. That is, if x is pure with degree k then
where the signs are given by the following table:
k mod 40123…
+−+−k
++−−k/2
+−−+k/2

Clifford scalar product

When the characteristic is not 2, the quadratic form Q on V can be extended to a quadratic form on all of . A basis-independent definition of one such extension is
where ⟨a⟩ denotes the scalar part of a. One can show that
where the vi are elements of V – this identity is not true for arbitrary elements of.
The associated symmetric bilinear form on is given by
One can check that this reduces to the original bilinear form when restricted to V. The bilinear form on all of is nondegenerate if and only if it is nondegenerate on V.
The operator of left Clifford multiplication by the transpose a of an element a is the adjoint of left Clifford multiplication by a with respect to this inner product. That is,
and

Structure of Clifford algebras

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 non-singular. A central simple algebra over K is a matrix algebra over a division algebra with center K. 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 U has even dimension and a non-singular bilinear form with discriminant d, and suppose that V is another vector space with a quadratic form. The Clifford algebra of is isomorphic to the tensor product of the Clifford algebras of U and dim/2dV, which is the space V with its quadratic form multiplied by dim/2d. Over the reals, this implies in particular that
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 depends only on the signature. This is an algebraic form of Bott periodicity.

Lipschitz group

The class of Lipschitz groups was discovered by Rudolf Lipschitz.
In this section we assume that V is finite-dimensional and the quadratic form Q 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 x maps, where α is the main involution defined above.
The Lipschitz group Γ is defined to be the set of invertible elements x that stabilize the set of vectors under this action, meaning that for all v in V we have:
This formula also defines an action of the Lipschitz group on the vector space V that preserves the quadratic form Q, and so gives a homomorphism from the Lipschitz group to the orthogonal group. The Lipschitz group contains all elements r of V for which Q is invertible in K, and these act on V by the corresponding reflections that take v to.
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 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.

Spinor norm

In arbitrary characteristic, the spinor norm Q is defined on the Lipschitz group by
It is a homomorphism from the Lipschitz group to the group K× of non-zero elements of K. It coincides with the quadratic form Q of V when V 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 −1, 2, or −2 on Γ1. The difference is not very important in characteristic other than 2.
The nonzero elements of K have spinor norm in the group 2 of squares of nonzero elements of the field K. So when V is finite-dimensional and non-singular we get an induced map from the orthogonal group of V to the group K×/2, also called the spinor norm. The spinor norm of the reflection about r⊥, for any vector r, has image Q in K×/2, and this property uniquely defines it on the orthogonal group. This gives exact sequences:
Note that in characteristic 2 the group 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 μ2 for the algebraic group of square roots of 1, the short exact sequence
yields a long exact sequence on cohomology, which begins
The 0th Galois cohomology group of an algebraic group with coefficients in K is just the group of K-valued points:, and, which recovers the previous sequence
where the spinor norm is the connecting homomorphism.

Spin and Pin groups

In this section we assume that V is finite-dimensional and its bilinear form is non-singular.
The Pin group PinV is the subgroup of the Lipschitz group Γ of elements of spinor norm 1, and similarly the Spin group SpinV is the subgroup of elements of Dickson invariant 0 in PinV. When the characteristic is not 2, these are the elements of determinant 1. The Spin group usually has index 2 in the Pin group.
Recall from the previous section that there is a homomorphism from the Clifford group onto the orthogonal group. We define the special orthogonal group to be the image of Γ0. If K does not have characteristic 2 this is just the group of elements of the orthogonal group of determinant 1. If K does have characteristic 2, then all elements of the orthogonal group have determinant 1, and the special orthogonal group is the set of elements of Dickson invariant 0.
There is a homomorphism from the Pin group to the orthogonal group. The image consists of the elements of spinor norm. The kernel consists of the elements +1 and −1, and has order 2 unless K has characteristic 2. Similarly there is a homomorphism from the Spin group to the special orthogonal group of V.
In the common case when V is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected when V has dimension at least 3. Further the kernel of this homomorphism consists of 1 and −1. So in this case the spin group, Spin, is a double cover of SO. Please note, however, that the simple connectedness of the spin group is not true in general: if V is Rp,q for p and q both at least 2 then the spin group is not simply connected. In this case the algebraic group Spinp,q is simply connected as an algebraic group, even though its group of real valued points Spinp,q is not simply connected. This is a rather subtle point, which completely confused the authors of at least one standard book about spin groups.

Spinors

Clifford algebras Cl, with even, are matrix algebras which have a complex representation of dimension 2n. By restricting to the group Pinp,q 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 Spinp,q then it splits as the sum of two half spin representations of dimension 2n−1.
If is odd then the Clifford algebra Cl is a sum of two matrix algebras, each of which has a representation of dimension 2n, and these are also both representations of the Pin group Pinp,q. On restriction to the spin group Spinp,q these become isomorphic, so the spin group has a complex spinor representation of dimension 2n.
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

To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The Pin group, Pinp,q is the set of invertible elements in Cl that can be written as a product of unit vectors:
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. The Spin group consists of those elements of Pinp, q which 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.
Let be the automorphism which is given by the mapping acting on pure vectors. Then in particular, Spinp,q is the subgroup of Pinp,q whose elements are fixed by α. Let
Then the spin group lies within Cl.
The irreducible representations of Cl 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 Cl.
To classify the pin representations, one need only appeal to the classification of Clifford algebras. To find the spin representations, one can first make use of either of the isomorphisms
and realize a spin representation in signature as a pin representation in either signature or.

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 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 importantly is the link to a spin manifold, its associated spinor bundle and spinc 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 called Dirac matrices which have the property that
where is the matrix of a quadratic form of signature . These are exactly the defining relations for the Clifford algebra, whose complexification is which, by the classification of Clifford algebras, is isomorphic to the algebra of complex matrices. However, it is best to retain the notation, 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. 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 sitting inside the Clifford algebra conventionally requires a complex Clifford algebra. For reference, the spin Lie algebra is given by
This is in the convention, hence fits in.
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, 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.

Computer vision

Clifford algebras have been applied in the problem of action recognition and classification in computer vision. Rodriguez et al. propose a Clifford embedding to generalize traditional MACH filters to video, 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