In mathematics, a Clifford algebra 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 -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 (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.
Introduction and basic properties A Clifford algebra is a unital associative algebra that contains and is generated by a vector space over a field , where is equipped with a quadratic form . The Clifford algebra is the "freest" unital associative algebra generated by subject to the condition
where the product on the left is that of the algebra, and the on the right is the algebra's multiplicative identity (not to be confused with the multiplicative identity of ). 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 is a finite-dimensional real vector space and is nondegenerate, may be identified by the label , indicating that has an orthogonal basis with elements with , with , and where indicates that this is a Clifford algebra over the reals; i.e. coefficients of elements of the algebra are real numbers. Such a basis may be found by orthogonal diagonalization.
The free algebra generated by may be written as the tensor algebra , that is, the direct sum of the tensor product of copies of over all . Therefore one obtains a Clifford algebra as 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 (e.g. ). Its associativity follows from the associativity of the tensor product.
The Clifford algebra has a distinguished subspace , being the image of the embedding map. Such a subspace cannot in general be uniquely determined given only a -algebra that is isomorphic to the Clifford algebra.
If is invertible in the ground field , then one can rewrite the fundamental identity above in the form
where
is the symmetric bilinear form associated with , via the polarization identity.
Quadratic forms and Clifford algebras in characteristic form an exceptional case in this respect. In particular, if it is not true that a quadratic form necessarily or uniquely determines a symmetric bilinear form that satisfies , Many of the statements in this article include the condition that the characteristic is not , 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 . Whenever is invertible in the ground field , there exists a canonical linear isomorphism between and . 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 .
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 be a vector space over a field , and let be a quadratic form on . In most cases of interest the field is either the field of real numbers , or the field of complex numbers , or a finite field.
A Clifford algebra is a pair , where is a unital associative algebra over and is a linear map that satisfies for all in , defined by the following universal property: given any unital associative algebra over and any linear map such that
(where denotes the multiplicative identity of ), there is a unique algebra homomorphism such that the following diagram commutes (i.e. such that ):
The quadratic form may be replaced by a (not necessarily symmetric) bilinear form that has the property , in which case an equivalent requirement on is
When the characteristic of the field is not , this may be replaced by what is then an equivalent requirement,
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 , namely the tensor algebra , and then enforce the fundamental identity by taking a suitable quotient. In our case we want to take the two-sided ideal in generated by all elements of the form for all 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 and satisfies the above universal property, so that is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra . It also follows from this construction that is injective. One usually drops the and considers as a linear subspace of .
The universal characterization of the Clifford algebra shows that the construction of is in nature. Namely, 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 comes equipped with a quadratic form , in characteristic not equal to there exist bases for that are orthogonal. An orthogonal basis is one such that for a symmetric bilinear form for , and
The fundamental Clifford identity implies that for an orthogonal basis for , and
This makes manipulation of orthogonal basis vectors quite simple. Given a product of distinct orthogonal basis vectors of , 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 over is and is an orthogonal basis of , then is free over with a basis
The empty product () is defined as being the multiplicative identity element. For each value of there are 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 and is isomorphic to or , where is a full matrix ring with entries from , , or . For a complete classification of these algebras see Classification of Clifford algebras.
Real numbers
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:
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 The Clifford algebra on is denoted The symbol means either or , depending on whether the author prefers positive-definite or negative-definite spaces.
A standard basis for consists of mutually orthogonal vectors, of which square to and of which square to . Of such a basis, the algebra will therefore have vectors that square to and vectors that square to .
A few low-dimensional cases are: is naturally isomorphic to since there are no nonzero vectors. is a two-dimensional algebra generated by that squares to , and is algebra-isomorphic to , the field of complex numbers. is a two-dimensional algebra generated by that squares to , and is algebra-isomorphic to the split-complex numbers. is a four-dimensional algebra spanned by . The latter three elements all square to and anticommute, and so the algebra is isomorphic to the quaternions . is isomorphic to the algebra of split-quaternions. is an 8-dimensional algebra isomorphic to the direct sum , the split-biquaternions. is isomorphic to the algebra of biquaternions and to the Pauli algebra.
Complex numbers One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space of dimension is equivalent to the standard diagonal form
Thus, for each dimension , 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 with the standard quadratic form by .
For the first few cases one finds that , the complex numbers , the bicomplex numbers , the biquaternions where denotes the algebra of matrices over .
Examples: constructing quaternions and dual quaternions Quaternions In this section, Hamilton's quaternions are constructed as the even subalgebra of the Clifford algebra .
Let the vector space be real three-dimensional space , and the quadratic form be Euclidean. Then, for in we have the bilinear form (or scalar product)
Now introduce the Clifford product of vectors and given by
Denote a set of orthogonal unit vectors of as , then the Clifford product yields the relations
and
The general element of the Clifford algebra is given by
The linear combination of the even degree elements of defines the even subalgebra with the general element
The basis elements can be identified with the quaternion basis elements as
which shows that the even subalgebra is Hamilton's real quaternion algebra.
To see this, compute
and
Finally,
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.
Let the vector space be real four-dimensional space and let the quadratic form be a degenerate form derived from the Euclidean metric on For in introduce the degenerate bilinear form
This degenerate scalar product projects distance measurements in onto the hyperplane.
The Clifford product of vectors and is given by
Note the negative sign is introduced to simplify the correspondence with quaternions.
Denote a set of mutually orthogonal unit vectors of as , 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 and the dual unit as
This provides the correspondence of with dual quaternion algebra.
To see this, compute
and
The exchanges of and alternate signs an even number of times, and show the dual unit commutes with the quaternion basis elements .
Examples: in small dimension Let be any field of characteristic not .
Dimension 1