Abstract Algebra/Clifford Algebras
In mathematics, Clifford algebras are a type of associative algebra. They can be thought of as one of the possible generalizations of the complex numbers and quaternions. 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 and theoretical physics. They are named for the English geometer William Clifford.
- Some familiarity with the basics of multilinear algebra will be useful in reading this section.
Introduction and basic properties[edit]
Specifically, a Clifford algebra is a unital associative algebra which contains and is generated by a vector space V equipped with a quadratic form Q. The Clifford algebra Cℓ(V,Q) is the "freest" algebra generated by V subject to the condition^{1}
If the characteristic of the ground field K is not 2, then one can rewrite this fundamental identity in the form
where <u, v> = Q(u + v) − Q(u) − Q(v) is the symmetric bilinear form associated to Q. This idea of "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a universal property (see below).
Clifford algebras are closely related to exterior algebras. In fact, if Q = 0 then the Clifford algebra Cℓ(V,Q) is just the exterior algebra Λ(V). For nonzero Q there exists a canonical linear isomorphism between Λ(V) and Cℓ(V,Q) whenever the ground field K 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 is strictly richer than the exterior product since it makes use of the extra information provided by Q. More precisely, they 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.
Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case. In particular, if char K = 2 it is not true that a quadratic form is determined by its symmetric bilinear form, or 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.
Universal property and construction[edit]
Let V be a vector space over a field K, and let Q : V → K be a quadratic form on V. In most cases of interest the field K is either R or C (which have characteristic 0) or a finite field.
A Clifford algebra Cℓ(V,Q) is a unital associative algebra over K together with a linear map i : V → Cℓ(V,Q) defined by the following universal property: Given any associative algebra A over K and any linear map j : V → A such that
- j(v)^{2} = Q(v)1 for all v ∈ V
(where 1 denotes the multiplicative identity of A), there is a unique algebra homomorphism f : Cℓ(V,Q) → A such that the following diagram commutes (i.e. such that f o i = j):
Working with a symmetric bilinear form <·,·> instead of Q (in characteristic not 2), the requirement on j is
- j(v)j(w) + j(w)j(v) = <v, w> for all v, w ∈ V.
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(V), and then enforce the fundamental identity by taking a suitable quotient. In our case we want to take the two-sided ideal I_{Q} in T(V) generated by all elements of the form
- for all
and define Cℓ(V,Q) as the quotient
- Cℓ(V,Q) = T(V)/I_{Q.}
It is then straightforward to show that Cℓ(V,Q) contains V and satisfies the above universal property, so that Cℓ is unique up to isomorphism; thus one speaks of "the" Clifford algebra Cℓ(V, Q). It also follows from this construction that i is injective. One usually drops the i and considers V as a linear subspace of Cℓ(V,Q).
The universal characterization of the Clifford algebra shows that the construction of Cℓ(V,Q) is functorial in nature. Namely, Cℓ 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[edit]
If the dimension of V is n and {e_{1},…,e_{n}} is a basis of V, then the set
is a basis for Cℓ(V,Q). The empty product (k = 0) 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
Since V comes equipped with a quadratic form, there is a set of privileged bases for V: the orthogonal ones. An orthogonal basis in one such that
where <·,·> is the symmetric bilinear form associated to Q. 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, one can put them into standard order by including an overall sign corresponding to the number of flips needed to correctly order them (i.e. the signature of the ordering permutation).
If the characteristic is not 2 then an orthogonal basis for V exists, and one can easily extend the quadratic form on V to a quadratic form on all of Cℓ(V,Q) by requiring that distinct elements are orthogonal to one another whenever the {e_{i}}'s are orthogonal. Additionally, one sets
- .
The quadratic form on a scalar is just Q(λ) = λ^{2}. Thus, orthogonal bases for V extend to orthogonal bases for Cℓ(V,Q). The quadratic form defined in this way is actually independent of the orthogonal basis chosen (a basis-independent formulation will be given later).
Examples: Real and complex Clifford algebras[edit]
The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms.
Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:
where n = p + q is the dimension of the vector space. The pair of integers (p, q) is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted R^{p,q}. The Clifford algebra on R^{p,q} is denoted Cℓ_{p,q}(R). The symbol Cℓ_{n}(R) means either Cℓ_{n,0}(R) or Cℓ_{0,n}(R) depending on whether the author prefers positive definite or negative definite spaces.
A standard orthonormal basis {e_{i}} for R^{p,q} consists of n = p + q mutually orthogonal vectors, p of which have norm +1 and q of which have norm −1. The algebra Cℓ_{p,q}(R) will therefore have p vectors which square to +1 and q vectors which square to −1.
Note that Cℓ_{0,0}(R) is naturally isomorphic to R since there are no nonzero vectors. Cℓ_{0,1}(R) is a two-dimensional algebra generated by a single vector e_{1} which squares to −1, and therefore is isomorphic to C, the field of complex numbers. The algebra Cℓ_{0,2}(R) is a four-dimensional algebra spanned by {1, e_{1}, e_{2}, e_{1}e_{2}}. The latter three elements square to −1 and all anticommute, and so the algebra is isomorphic to the quaternions H. The next algebra in the sequence is Cℓ_{0,3}(R) is an 8-dimensional algebra isomorphic to the direct sum H ⊕ H called Clifford biquaternions.
One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space is equivalent to the standard diagonal form
where n = dim V, so there is essentially only one Clifford algebra in each dimension. We will denote the Clifford algebra on C^{n} with the standard quadratic form by Cℓ_{n}(C). One can show that the algebra Cℓ_{n}(C) may be obtained as the complexification of the algebra Cℓ_{p,q}(R) where n = p + q:
- .
Here Q is the real quadratic form of signature (p,q). Note that the complexification does not depend on the signature. The first few cases are not hard to compute. One finds that
- Cℓ_{0}(C) = C
- Cℓ_{1}(C) = C ⊕ C
- Cℓ_{2}(C) = M_{2}(C)
where M_{2}(C) denotes the algebra of 2×2 matrices over C.
It turns out that every one of the algebras Cℓ_{p,q}(R) and Cℓ_{n}(C) is isomorphic to a matrix algebra over R, C, or H or to a direct sum of two such algebras. For a complete classification of these algebras see classification of Clifford algebras.
Properties[edit]
Relation to the exterior algebra[edit]
Given a vector space V one can construct the exterior algebra Λ(V), whose definition is independent of any quadratic form on V. It turns out that if F does not have characteristic 2 then there is a natural isomorphism between Λ(V) and Cℓ(V,Q) 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 Q = 0. One can thus consider the Clifford algebra Cℓ(V,Q) as an enrichment (or more precisely, a quantization, cf. the Introduction) of the exterior algebra on V with a multiplication that depends on Q (one can still define the exterior product independent of Q).
The easiest way to establish the isomorphism is to choose an orthogonal basis {e_{i}} for V and extend it to an orthogonal basis for Cℓ(V,Q) as described above. The map Cℓ(V,Q) → Λ(V) is determined by
Note that this only works if the basis {e_{i}} 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 f_{k} : V × … × V → Cℓ(V,Q) by
where the sum is taken over the symmetric group on k elements. Since f_{k} is alternating it induces a unique linear map Λ^{k}(V) → Cℓ(V,Q). The direct sum of these maps gives a linear map between Λ(V) and Cℓ(V,Q). 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 Cℓ(V,Q). Recall that the tensor algebra T(V) has a natural filtration: F^{0} ⊂ F^{1} ⊂ F^{2} ⊂ … where F^{k} contains sums of tensors with rank ≤ k. Projecting this down to the Clifford algebra gives a filtration on Cℓ(V,Q). The associated graded algebra
is naturally isomorphic to the exterior algebra Λ(V). Since the associated graded algebra of a filtered algebra is always isomorphic to the filtered algebra as filtered vector spaces (by choosing complements of F^{k} in F^{k+1} for all k), this provides an isomorphism (although not a natural one) in any characteristic, even two.
Grading[edit]
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
- α : Cℓ(V,Q) → Cℓ(V,Q).
Since α is an involution (i.e. it squares to the identity) one can decompose Cℓ(V,Q) into positive and negative eigenspaces
where Cℓ^{i}(V,Q) = {x ∈ Cℓ(V,Q) | α(x) = (−1)^{i}x}. Since α is an automorphism it follows that
where the superscripts are read modulo 2. This means that Cℓ(V,Q) is a Z_{2}-graded algebra (also known as a superalgebra). Note that Cℓ^{0}(V,Q) forms a subalgebra of Cℓ(V,Q), called the even subalgebra. The piece Cℓ^{1}(V,Q) is called the odd part of Cℓ(V,Q) (it is not a subalgebra). This Z_{2}-grading plays an important role in the analysis and application of Clifford algebras. The automorphism α is called the main involution or grade involution.
Remark. In characteristic not 2 the algebra Cℓ(V,Q) inherits a Z-grading from the canonical isomorphism with the exterior algebra Λ(V). It is important to note, however, that this is a vector space grading only. That is, Clifford multiplication does not respect the Z-grading only the Z_{2}-grading. Happily, the gradings are related in the natural way: Z_{2} = Z/2Z. The degree of a Clifford number usually refers to the degree in the Z-grading. Elements which are pure in the Z_{2}-grading are simply said to be even or odd.
If the characteristic of F is not 2 then the even subalgebra Cℓ^{0}(V,Q) of a Clifford algebra is itself a Clifford algebra. If V is the orthogonal direct sum of a vector a of norm Q(a) and a subspace U, then Cℓ^{0}(V,Q) is isomorphic to Cℓ(U,−Q(a)Q), where −Q(a)Q is the form Q restricted to U and multiplied by −Q(a). In particular over the reals this implies that
- for q > 0, and
- for p > 0.
In the negative-definite case this gives an inclusion Cℓ_{0,n−1}(R) ⊂ Cℓ_{0, n}(R) which extends the sequence
- R ⊂ C ⊂ H ⊂ H⊕H ⊂ …
Likewise, in the complex case, one can show that the even subalgebra of Cℓ_{n}(C) is isomorphic to Cℓ_{n−1}(C).
Antiautomorphisms[edit]
In addition to the automorphism α, there are two antiautomorphisms which play an important role in the analysis of Clifford algebras. Recall that the tensor algebra T(V) comes with an antiautomorphism that reverses the order in all products:
- .
Since the ideal I_{Q} is invariant under this reversal, this operation descends to an antiautomorphism of Cℓ(V,Q) called the transpose or reversal operation, denoted by x^{t}. The transpose is an antiautomorphism: . The transpose operation makes no use of the Z_{2}-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.^{3}
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 4 | 0 | 1 | 2 | 3 | |
---|---|---|---|---|---|
+ | − | + | − | (−1)^{k} | |
+ | + | − | − | (−1)^{k(k−1)/2} | |
+ | − | − | + | (−1)^{k(k+1)/2} |
The Clifford scalar product[edit]
When the characteristic is not 2 the quadratic form Q on V can be extended to a quadratic form on all of Cℓ(V,Q) as explained earlier (which we also denoted by Q). A basis independent definition is
where <a> denotes the scalar part of a (the grade 0 part in the Z-grading). One can show that
where the v_{i} are elements of V — this identity is not true for arbitrary elements of Cℓ(V,Q).
The associated symmetric bilinear form on Cℓ(V,Q) is given by
One can check that this reduces to the original bilinear form when restricted to V. The bilinear form on all of Cℓ(V,Q) is nondegenerate if and only it is nondegenerate on V.
It is not hard to verify that the transpose is the adjoint of left/right Clifford multiplication with respect to this inner product. That is,
- and
Structure of Clifford algebras[edit]
In this section we assume that the vector space V is finite dimensional and that the bilinear form of Q is non-singular. A central simple algebra over K is a matrix algebra over a (finite dimensional) division algebra with center K. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions.
- If V has even dimension then Cℓ(V,Q) is a central simple algebra over K.
- If V has even dimension then Cℓ^{0}(V,Q) is a central simple algebra over a quadratic extension of K or a sum of two isomorphic central simple algebras over K.
- If V has odd dimension then Cℓ(V,Q) is a central simple algebra over a quadratic extension of K or a sum of two isomorphic central simple algebras over K.
- If V has odd dimension then Cℓ^{0}(V,Q) is a central simple algebra over K.
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 U+V is isomorphic to the tensor product of the Clifford algebras of U and (−1)^{dim(U)/2}dV, which is the space V with its quadratic form multiplied by (−1)^{dim(U)/2}d. Over the reals, this implies in particular that
These formulas can be used to find the structure of all real Clifford algebras.
The Clifford group Γ[edit]
In this section we assume that V is finite dimensional and the bilinear form of Q is non-singular.
The Clifford group Γ is defined to be the set of invertible elements x of the Clifford algebra such that
for all v in V. This formula also defines an action of the Clifford group on the vector space V that preserves the norm Q, and so gives a homomorphism from the Clifford group to the orthogonal group. The Clifford group contains all elements r of V of nonzero norm, and these act on V by the corresponding reflections that take v to v − <v,r>r/Q(r) (In characteristic 2 these are called orthogonal transvections rather than reflections.)
Many authors define the Clifford group slightly differently, by replacing the action xvα(x)^{−1} by xvx^{−1}. This produces the same Clifford group, but the action of the Clifford group on V is changed slightly: the action of the odd elements Γ^{1} of the Clifford group is multiplied by an extra factor of −1. This action used here has several minor advantages: it is consistent with the usual superalgebra sign conventions, elements of V correspond to reflections, and in odd dimensions the map from the Clifford group to the orthogonal group is onto, and the kernel is no larger than K^{*}. Using the action α(x)vx^{−1} instead of xvα(x)^{−1} makes no difference: it produces the same Clifford group with the same action on V.
The Clifford group Γ is the disjoint union of two subsets Γ^{0} and Γ^{1}, where Γ^{i} is the subset of elements of degree i. The subset Γ^{0} is a subgroup of index 2 in Γ.
If V is finite dimensional with nondegenerate bilinear form then the Clifford group maps onto the orthogonal group of V and the kernel consists of the nonzero elements of the field K. This leads to exact sequences
In arbitrary characteristic, the spinor norm Q is defined on the Clifford group by
It is a homomorphism from the Clifford 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 Γ^{1}. The difference is not very important.
The nonzero elements of K have spinor norm in the group K^{*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^{*}/K^{*2}, also called the spinor norm. The spinor norm of the reflection of a vector r has image Q(r) in K^{*}/K^{*2}, and this property uniquely defines it on the orthogonal group. This gives exact sequences:
Note that in characteristic 2 the group {±1} has just one element.
Spin and Pin groups[edit]
In this section we assume that V is finite dimensional and its bilinear form is non-singular. (If K has characteristic 2 this implies that the dimension of V is even.)
The Pin group Pin_{V}(K) is the subgroup of the Clifford group Γ of elements of spinor norm 1, and similarly the Spin group Spin_{V}(K) is the subgroup of elements of Dickson invariant 0 in Pin_{V}(K). 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 Γ^{0}. 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 1 ∈ K^{*}/K^{*2}. 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. Warning: This is not true in general: if V is R^{p,q} for p and q both at least 2 then the spin group is not simply connected and does not map onto the special orthogonal group. In this case the algebraic group Spin_{p,q} is simply connected as an algebraic group, even though its group of real valued points Spin_{p,q}(R) 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[edit]
Suppose that p+q=2n is even. Then the Clifford algebra Cℓ_{p,q}(C) is a matrix algebra, and so has a complex representation of dimension 2^{n}. By restricting to the group Pin_{p,q}(R) we get a complex representation of the Pin group of the same dimension, called the spinor representation. If we restrict this to the spin group Spin_{p,q}(R) then it splits as the sum of two half spin representations (or Weyl representations) of dimension 2^{n-1}.
If p+q=2n+1 is odd then the Clifford algebra Cℓ_{p,q}(C) is a sum of two matrix algebras, each of which has a representation of dimension 2^{n}, and these are also both representations of the Pin group Pin_{p,q}(R). On restriction to the spin group Spin_{p,q}(R) these become isomorphic, so the spin group has a complex spinor representation of dimension 2^{n}.
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.
Applications[edit]
Differential geometry[edit]
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.
Physics[edit]
Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra spanned by matrices γ_{1},…,γ_{n} called Dirac matrices which have the property that
where η is the matrix of a quadratic form of signature (p,q) — typically (1,3) when working in Minkowski space. These are exactly the defining relations for the Clifford algebra Cl_{1,3}(C) (up to an unimportant factor of 2), which by the classification of Clifford algebras is isomorphic to the algebra of 4 by 4 complex matrices.
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. The entire Clifford algebra shows up in quantum field theory in the form of Dirac field bilinears.
Footnotes[edit]
- Mathematicians who work with real Clifford algebras and prefer positive definite quadratic forms (especially those working in index theory) sometimes use a different choice of sign in the fundamental Clifford identity. That is, they take v^{2} = −Q(v). One must replace Q with −Q in going from one convention to the other.
- The opposite is true when uses the alternate (−) sign convention for Clifford algebras: it is the conjugate which is more important. In general, the meanings of conjugation and transpose are interchanged when passing from one sign convention to the other. For example, in the convention used here the inverse of a vector is given by while in the (−) convention it is given by .
References[edit]
- Carnahan, S. Borcherds Seminar Notes, Uncut. Week 5, "Spinors and Clifford Algebras".
- Lawson and Michelsohn, Spin Geometry, Princeton University Press. 1989. ISBN 0-691-08542-0. An advanced textbook on Clifford algebras and their applications to differential geometry.
- Lounesto, P., Clifford Algebras and Spinors, Cambridge University Press. 2001. ISBN 0-521-00551-5.
- Porteous, I., Clifford Algebras and the Classical Groups, Cambridge University Press. 1995. ISBN 0-521-55177-3.