Clifford algebra

Clifford algebras are a type of associative algebra in mathematics. 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 article.

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 condition¹

${\displaystyle v^{2}=Q(v)\,}$ for all ${\displaystyle v\in V.}$

If the characteristic of the ground field is not 2, then one can rewrite this fundamental identity in the form

${\displaystyle uv+vu=2\langle u,v\rangle }$ for all ${\displaystyle u,v\in V}$

where <·,·> 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). That is they are naturally isomorphic as vector spaces but with different multiplications. Clifford multiplication is strictly richer than the exterior product since it makes use of the extra information provided by Q.

Universal property and construction

Let V be a vector space over a field K, and let Q : V → K be a quadratic form on V. We will assume for simplicity that the characteristic of K is not two.² In most cases of interest the field K is either R or C (which have characteristic 0).

The 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, the requirement on j is

j(v)j(w) + j(w)j(v) = 2<v,w> for all v,w ∈ V.

The Clifford algebra 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

${\displaystyle v\otimes v-Q(v)1}$  for all ${\displaystyle v\in V}$ 

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. It 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

If the dimension of V is n and {e₁,…,e_n} is a basis of V, then the set

${\displaystyle \{e_{i_{1}}e_{i_{2}}\cdots e_{i_{k}}\mid 1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n{\mbox{ and }}0\leq k\leq n\}}$ 

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

${\displaystyle \dim C\ell (V,Q)=\sum _{k=0}^{n}{\begin{pmatrix}n\\k\end{pmatrix}}=2^{n}.}$ 

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

${\displaystyle \langle e_{i},e_{j}\rangle =0\qquad i\neq j}$ .

where <·,·> is the symmetric bilinear form associated to Q. The fundamental Clifford identity implies that for an orthogonal basis

${\displaystyle e_{i}e_{j}=-e_{j}e_{i}\qquad i\neq j}$ .

This makes manipulation of orthogonal basis vectors quite simple. Given a product ${\displaystyle e_{i_{1}}e_{i_{2}}\cdots e_{i_{k}}}$  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).

One can easily extend the quadratic form on V to a quadratic form on all of Cℓ(V,Q) by requiring that distinct elements ${\displaystyle e_{i_{1}}e_{i_{2}}\cdots e_{i_{k}}}$  are orthogonal to one another whenever the {e_i}'s are orthogonal. Additionally, one sets

${\displaystyle Q(e_{i_{1}}e_{i_{2}}\cdots e_{i_{k}})=Q(e_{i_{1}})Q(e_{i_{2}})\cdots Q(e_{i_{k}})}$ .

The quadratic form on a scalar is just Q(λ) = λ². 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

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:

${\displaystyle Q(v)=v_{1}^{2}+\cdots +v_{p}^{2}-v_{p+1}^{2}-\cdots -v_{p+q}^{2}}$ 

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 e₁ 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, e₁, e₂, e₁e₂}. 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. This is the algebra of biquaternions first studied by Clifford.

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

${\displaystyle Q(z)=z_{1}^{2}+z_{2}^{2}+\cdots +z_{n}^{2}}$ 

where n = dim V. So there is essentially only one Clifford algebra in each dimension. We will denote the Clifford algebra on Cⁿ 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:

${\displaystyle C\ell _{n}(\mathbb {C} )\cong C\ell _{p,q}(\mathbb {R} )\otimes \mathbb {C} \cong C\ell (\mathbb {C} ^{p+q},Q\otimes \mathbb {C} )}$ .

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

Cℓ₀(C) = C

Cℓ₁(C) = C ⊕ C

Cℓ₂(C) = M₂(C)

where M₂(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

Relation to the exterior algebra

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 there is a natural isomorphism between Λ(V) and Cℓ(V,Q) considered as vector spaces. This is an algebra isomorphism iff Q = 0. One can thus consider the Clifford algebra Cℓ(V,Q) as an enrichment 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

${\displaystyle e_{i_{1}}e_{i_{2}}\cdots e_{i_{k}}\mapsto e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}.}$ 

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

${\displaystyle f_{k}(v_{1},\cdots ,v_{k})={\frac {1}{k!}}\sum _{\sigma \in S_{k}}{\rm {sgn}}(\sigma )\,v_{\sigma (1)}\cdots v_{\sigma (k)}}$ 

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.

Yet another way to see the relation is to construct a filtration on Cℓ(V,Q). Recall that the tensor algebra T(V) has a natural filtration: F⁰ ⊂ F¹ ⊂ F² ⊂ … 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

${\displaystyle \bigoplus _{k}F^{k}/F^{k-1}}$ 

is naturally isomorphic to the exterior algebra Λ(V).

Grading

The linear map on V defined by ${\displaystyle v\mapsto -v}$  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

${\displaystyle C\ell (V,Q)=C\ell ^{0}(V,Q)\oplus C\ell ^{1}(V,Q)}$ 

where Cℓⁱ(V,Q) = {x ∈ Cℓ(V,Q) | α(x) = (−1)ⁱx}. Since α is an automorphism it follows that

${\displaystyle C\ell ^{\,i}(V,Q)C\ell ^{\,j}(V,Q)=C\ell ^{\,i+j}(V,Q)}$ 

where the superscripts are read modulo 2. This means that Cℓ(V,Q) is a Z₂-graded algebra (also known as a superalgebra). Note that Cℓ⁰(V,Q) forms a subalgebra of Cℓ(V,Q), called the even subalgebra. The piece Cℓ¹(V,Q) is called the odd part of Cℓ(V,Q) (it is not a subalgebra). 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.

Remark. 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 Z₂-grading. Happily, the gradings are related in the natural way: Z₂ = Z/2Z. The degree of a Clifford number usually refers to the degree in the Z-grading. Elements which are pure in the Z₂-grading are simply said to be even or odd.

The even subalgebra of Cℓ_{p,q(R) is itself a Clifford algebra. One can show that}

${\displaystyle C\ell _{p,q}^{0}(\mathbb {R} )\cong C\ell _{p,q-1}(\mathbb {R} )}$  for q > 0, and

${\displaystyle C\ell _{q,0}^{0}(\mathbb {R} )\cong C\ell _{0,q-1}(\mathbb {R} ).}$ 

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

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 a antiautomorphism that reverses the order in all products:

${\displaystyle v_{1}\otimes v_{2}\otimes \cdots \otimes v_{k}\mapsto v_{k}\otimes \cdots \otimes v_{2}\otimes v_{1}}$ .

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: ${\displaystyle (xy)^{t}=y^{t}x^{t}}$ . 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 ${\displaystyle {\bar {x}}}$ 

${\displaystyle {\bar {x}}=\alpha (x^{t})=\alpha (x)^{t}.}$ 

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

${\displaystyle \alpha (x)=\pm x\qquad x^{t}=\pm x\qquad {\bar {x}}=\pm x}$ 

where the signs are given by the following table:

k mod 4	0	1	2	3
${\displaystyle \alpha (x)\,}$	+	−	+	−	(−1)k
${\displaystyle x^{t}\,}$	+	+	−	−	(−1)k(k−1)/2
${\displaystyle {\bar {x}}}$	+	−	−	+	(−1)k(k+1)/2

The Clifford scalar product

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

${\displaystyle Q(x)=\langle {\bar {x}}x\rangle }$ 

where <a> denotes the scalar part of a (the grade 0 part in the Z-grading). One can show that

${\displaystyle Q(v_{1}v_{2}\cdots v_{k})=Q(v_{1})Q(v_{2})\cdots Q(v_{k})}$ 

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

${\displaystyle \langle x,y\rangle =\langle {\bar {x}}y\rangle }$ .

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 Clifford conjugation is the adjoint of left/right Clifford multiplication with respect to this inner product. That is,

${\displaystyle \langle ax,y\rangle =\langle x,{\bar {a}}y\rangle }$ , and

${\displaystyle \langle xa,y\rangle =\langle x,y{\bar {a}}\rangle }$ .

The Clifford group Γ

The Clifford group Γ is defined to be the set of invertible elements x of the Clifford algebra such that

xvα(x)⁻¹ ∈ V

for all v in V. the Clifford algebra. 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 a slightly different Clifford group, by replacing the action xvα(x)⁻¹ by xvx⁻¹. This alternative definition has several minor disadvantages in odd dimensions: for example, the map from the Clifford group to the orthogonal group is no longer onto, and the kernel is a little more complicated to describe.

The Clifford group Γ is the disjoint union of two subsets Γ₀ and Γ_i, where Γ_i is the subset of elements of degree i. The subset Γ₀ is a subgroup of index 2 in Γ.

If the bilinear form of V is non-degenerate and V is finite dimensional then the Clifford group maps onto the orthogonal group of V and the kernel consists of the nonzero elements of the field F.

The spinor norm Q is defined on the Clifford algebra by

Q(a)= aa^T.

It is not a homomorphism on the Clifford algebra. However when restricted to the Clifford group, it is a homomorphism from the Clifford group to the group F^* of non-zero elements of F. (It coincides with the quadratic form Q of V when V is identified with a subspace of the Clifford algebra.)

The nonzero elements of F have spinor norm in the group F^*2 of squares of nonzero elements of the field F. So when V is finite dimensional and non-singular we get an induced map from the orthogonal group of V to the group F^*/F^*2, also called the spinor norm. The spinor norm of the reflection of a vector r has image Q(r) in F^*/F^*2, and this property uniquely defines it on the orthogonal group.

Spin and Pin groups

In this section we assume that V is finite dimensional and its bilinear form is non-singular. (If F has characteristic 2 this implies that the dimension of V is even.)

The Pin group Pin_V(F) is the subgroup of the Clifford group Γof elements of spinor norm 1, and similarly the Spin group Spin_V(F) is the subgroup of the group Γ₀ of elements of spinor norm 1. It 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 F does not have characteristic 2 this is just the group of elements of the orthogonal group of determinant 1. If F 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 ∈ F^*/F^*2. The kernel consists of the elements +1 and −1, and has order 2 unless F 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.

See spinor group, spinor.

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 defined a Clifford bundle in analogy with the exterior bundle. This has a number of important applications in Riemannian geometry.

Physics

Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra spanned by matrices γ₁,…,γ_n called Dirac matrices which have the property that

${\displaystyle \gamma _{i}\gamma _{j}+\gamma _{j}\gamma _{i}=2\eta _{ij}\,}$ 

where η is the matrix of a quadratic form of signature (p,q) — typically (3,1) or (1,3) when working in Minkowski space.

The Dirac matrices where first written down by Paul Dirac when he was trying to write a relativistic first-order wave equation for the electron. The result was the Dirac equation. The entire Clifford algebra shows up in quantum field theory in the form of Dirac field bilinears.

See also

• classification of Clifford algebras
• representations of Clifford algebras
• exterior algebra
• geometric algebra
• spinor group
• spinor

Footnotes

1. Mathematicians who work with real Clifford algebras and prefer positive definite quadratic forms sometimes use a different choice of sign in the fundamental Clifford identity. One must replace Q with −Q in going from one convention to the other.
2. Quadratic forms in characteristic 2 form an exceptional case. In particular, if char K = 2 it is not true that every quadratic form has an associated symmetric bilinear form, or that every quadratic form admits an orthogonal basis. Many of the statements in this article do not hold in this case. For example, there need not be a canonical linear isomorphism between Cℓ(V,Q) and Λ(V).
3. The opposite is true when uses the alternate (-) sign convention for Clifford algebras: it is the transpose 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 ${\displaystyle v^{-1}=v^{t}/Q(v)}$  while in the (-) convention it is given by ${\displaystyle v^{-1}={\bar {v}}/Q(v)}$ .

References

• 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.

External links

• A history of Clifford algebras (unverified)