Chapter 104. Clifford Algebras

Clifford algebras combine tensor algebra with quadratic forms. They are associative algebras generated by a vector space subject to a rule that relates multiplication to length.

The central relation is

v^2 = Q(v)1.

Here $V$ is a vector space, $Q$ is a quadratic form on $V$ , and $1$ is the multiplicative identity of the algebra. This relation forces vectors to multiply in a way that remembers the geometry of $Q$ . Clifford algebras generalize real numbers, complex numbers, quaternions, exterior algebras, and geometric algebras. They are closely tied to quadratic forms and orthogonal transformations.

104.1 Motivation

Exterior algebra gives a product with the rule

v\wedge v=0.

This is useful for encoding oriented area, volume, and alternating structure. But it discards the length of a vector.

Clifford algebra modifies this idea. Instead of forcing the square of every vector to vanish, it forces the square of a vector to equal its quadratic length:

v^2=Q(v)1.

Thus Clifford multiplication contains both alternating information and metric information.

This makes Clifford algebras useful in geometry. They can represent reflections, rotations, orthogonal transformations, spinors, and metric-dependent geometric operations.

104.2 Quadratic Forms

Let $V$ be a vector space over a field $F$ .

A quadratic form is a function

Q:V\to F

such that

Q(cv)=c^2Q(v)

and such that the associated polar form

B(u,v)=\frac{1}{2}\bigl(Q(u+v)-Q(u)-Q(v)\bigr)

is bilinear, assuming the field has characteristic not equal to $2$ .

For real vector spaces, a common example is

Q(x_1,\ldots,x_n) = x_1^2+\cdots+x_p^2 - x_{p+1}^2-\cdots-x_{p+q}^2.

The pair $(p,q)$ is the signature of the quadratic form.

Clifford algebra depends on this quadratic form. Changing $Q$ changes the multiplication rules.

104.3 Definition

Let $V$ be a vector space over $F$ , and let

Q:V\to F

be a quadratic form.

The Clifford algebra of $(V,Q)$ , denoted

\operatorname{Cl}(V,Q),

is a unital associative algebra generated by $V$ subject to the relation

v^2=Q(v)1

for every $v\in V$ .

The word generated means that every element of $\operatorname{Cl}(V,Q)$ can be expressed as a finite linear combination of products of vectors from $V$ .

Thus typical elements look like

a_0 + a_i v_i + a_{ij}v_iv_j + a_{ijk}v_iv_jv_k +\cdots.

Clifford algebra is therefore built from scalar parts, vector parts, bivector parts, and higher-grade parts.

104.4 The Fundamental Relation

The defining relation

v^2=Q(v)1

implies a relation between products of two different vectors.

Using

(u+v)^2=Q(u+v)1,

expand the left side:

(u+v)^2 = u^2+uv+vu+v^2.

Using the defining relation,

u^2=Q(u)1, \qquad v^2=Q(v)1.

Therefore,

uv+vu = \bigl(Q(u+v)-Q(u)-Q(v)\bigr)1.

If $B$ is the polar bilinear form, then

uv+vu=2B(u,v)1.

This identity is often the working rule of Clifford algebra.

It says that the symmetric part of Clifford multiplication is controlled by the bilinear form.

104.5 Orthogonal Bases

Suppose $e_1,\ldots,e_n$ is an orthogonal basis for $V$ with respect to $B$ .

Then

B(e_i,e_j)=0

for $i\ne j$ .

The Clifford relation gives

e_i e_j + e_j e_i = 0

for $i\ne j$ . Hence

e_i e_j = -e_j e_i.

Also,

e_i^2=Q(e_i)1.

Thus orthogonal basis vectors anticommute, and their squares are determined by the quadratic form. This is the basic computational rule for Clifford algebras.

104.6 Basis and Dimension

If $V$ has dimension $n$ , then $\operatorname{Cl}(V,Q)$ has dimension

2^n

when $Q$ is nondegenerate over the usual finite-dimensional setting.

For an orthogonal basis

e_1,\ldots,e_n,

a basis of the Clifford algebra is given by products

e_{i_1}e_{i_2}\cdots e_{i_k}

where

1\le i_1<i_2<\cdots<i_k\le n.

The empty product is the scalar $1$ .

Thus the basis consists of

Grade	Basis elements
0	$1$
1	$e_i$
2	$e_ie_j$ , $i<j$
3	$e_ie_je_k$ , $i<j<k$
$k$	$e_{i_1}\cdots e_{i_k}$ , increasing indices

There are

\binom{n}{k}

basis elements of grade $k$ . Summing over all $k$ gives

\sum_{k=0}^n \binom{n}{k}=2^n.

104.7 Comparison with Exterior Algebra

As vector spaces, Clifford algebra and exterior algebra have the same dimension in the finite-dimensional nondegenerate case:

\dim \operatorname{Cl}(V,Q)=\dim \Lambda(V)=2^n.

But their products differ.

In exterior algebra,

v\wedge v=0.

In Clifford algebra,

v^2=Q(v)1.

Thus exterior algebra records alternating structure, while Clifford algebra records alternating structure plus metric structure.

For orthogonal vectors $u$ and $v$ ,

uv=-vu.

This resembles the exterior product. But for a single vector,

vv=Q(v)1,

which has no exterior-algebra analogue unless $Q(v)=0$ .

104.8 Clifford Product

The multiplication in $\operatorname{Cl}(V,Q)$ is called the Clifford product or geometric product.

For vectors $u,v$ , the Clifford product decomposes into symmetric and alternating parts:

uv = B(u,v) + u\wedge v.

This formula is commonly used in geometric algebra notation.

The scalar part

B(u,v)

measures the metric pairing.

The bivector part

u\wedge v

measures oriented area.

Thus the Clifford product combines dot-product-like and wedge-product-like information in one operation.

104.9 Low-Dimensional Examples

Let $V=\mathbb{R}$ with basis $e$ .

Q(e)=1,

then

e^2=1.

The algebra has basis

1,e.

Every element has the form

a+be.

If instead

Q(e)=-1,

then

e^2=-1.

The algebra generated by $e$ behaves like the complex numbers, with $e$ playing the role of $i$ .

Thus complex numbers can be viewed as a Clifford algebra in one negative direction.

104.10 The Plane

Let $V=\mathbb{R}^2$ with orthonormal basis $e_1,e_2$ , and suppose

e_1^2=e_2^2=1.

Since $e_1$ and $e_2$ are orthogonal,

e_1e_2=-e_2e_1.

The Clifford algebra has basis

1,\quad e_1,\quad e_2,\quad e_1e_2.

Let

I=e_1e_2.

Then

I^2=e_1e_2e_1e_2.

Using anticommutation,

e_2e_1=-e_1e_2.

Therefore,

I^2 = e_1(e_2e_1)e_2 = -e_1e_1e_2e_2 = -1.

Thus the bivector $I$ behaves like an imaginary unit.

This explains why rotations in the plane can be represented using exponentials of bivectors.

104.11 Reflections

Clifford algebras encode reflections efficiently.

Let $v$ be a nonzero vector with

Q(v)\ne 0.

The reflection of a vector $x$ across the hyperplane orthogonal to $v$ can be written as

x' = -xvx^{-1}

only when the reflecting vector is placed correctly. The standard reflection in the hyperplane normal to $v$ is

x' = -vxv^{-1}.

Here

v^{-1}=\frac{v}{Q(v)}.

This formula expresses a geometric reflection using algebra multiplication.

Reflections generate orthogonal transformations. Therefore Clifford algebras give algebraic control over rotations and orthogonal groups.

104.12 Rotations and Spin Groups

A rotation can be expressed as a product of two reflections.

In Clifford algebra, this means rotations can be represented by products of unit vectors.

Elements generated by products of an even number of unit vectors form the spin group.

The spin group maps onto the special orthogonal group. It gives a double cover:

\operatorname{Spin}(n)\to SO(n).

This means two spin elements correspond to the same rotation.

Spin groups and spinors are central in geometry, topology, and quantum physics.

104.13 Even Subalgebra

The Clifford algebra has a natural parity decomposition:

\operatorname{Cl}(V,Q) = \operatorname{Cl}^{0}(V,Q) \oplus \operatorname{Cl}^{1}(V,Q).

The even part contains sums of products of an even number of vectors.

The odd part contains sums of products of an odd number of vectors.

The even part

\operatorname{Cl}^{0}(V,Q)

is a subalgebra.

The odd part is not generally a subalgebra, because the product of two odd elements is even.

The even subalgebra is especially important for rotations and spin groups.

104.14 Real Clifford Algebras

For real vector spaces, Clifford algebras are classified by signature.

Let

Q(x) = x_1^2+\cdots+x_p^2 - x_{p+1}^2-\cdots-x_{p+q}^2.

The corresponding real Clifford algebra is often denoted

\operatorname{Cl}_{p,q}(\mathbb{R}).

Here $p$ is the number of positive square directions, and $q$ is the number of negative square directions.

These algebras include matrix algebras over

\mathbb{R},\quad \mathbb{C},\quad \mathbb{H},

and their direct sums in certain cases. Real Clifford algebras are commonly classified by this signature.

104.15 Complex Clifford Algebras

Over the complex numbers, the classification is simpler because signs can be absorbed by complex scalars.

The distinction between positive and negative squares largely disappears.

Complex Clifford algebras depend mainly on the dimension of the vector space.

They are closely related to matrix algebras and spinor representations.

This is why complex Clifford algebras are prominent in representation theory and quantum mechanics.

104.16 Clifford Algebra and Quaternions

The quaternions

\mathbb{H}

are generated by elements $i,j,k$ satisfying

i^2=j^2=k^2=ijk=-1.

Certain real Clifford algebras are isomorphic to quaternionic algebras.

For example, in a negative definite two-dimensional space with basis $e_1,e_2$ ,

e_1^2=e_2^2=-1

and

e_1e_2=-e_2e_1.

Let

i=e_1, \qquad j=e_2, \qquad k=e_1e_2.

Then

i^2=j^2=k^2=-1

and the quaternionic multiplication rules appear.

Thus quaternions are naturally connected to Clifford algebra.

104.17 Clifford Algebra and Geometry

Clifford algebra provides a compact language for Euclidean and pseudo-Euclidean geometry.

It represents:

Object	Clifford-algebra interpretation
Scalar	Grade $0$ element
Vector	Grade $1$ element
Oriented plane element	Grade $2$ element
Oriented volume element	Top-grade element
Reflection	Conjugation by a vector
Rotation	Product of two reflections
Spinor	Element representing rotation in spin space

The product keeps track of both metric and orientation.

This makes the algebra effective for computations involving rotations, rigid motion, and geometric transformations.

104.18 Clifford Algebra in Physics

Clifford algebras appear throughout theoretical physics.

Examples include:

Area	Role
Quantum mechanics	Spinors and Pauli matrices
Relativity	Spacetime algebra
Dirac equation	Gamma matrices
Electromagnetism	Field bivectors
Classical mechanics	Rotors and transformations

The gamma matrices used in the Dirac equation satisfy Clifford relations. Their anticommutation rules encode the spacetime metric.

This is a direct use of the identity

\gamma_i\gamma_j+\gamma_j\gamma_i = 2g_{ij}1.

104.19 Example

Let $V=\mathbb{R}^2$ with basis $e_1,e_2$ , and let

e_1^2=1, \qquad e_2^2=1, \qquad e_1e_2=-e_2e_1.

Compute

(e_1+2e_2)^2.

Expand:

(e_1+2e_2)^2 = e_1^2 + 2e_1e_2 + 2e_2e_1 + 4e_2^2.

Since

e_2e_1=-e_1e_2,

the mixed terms cancel:

2e_1e_2+2e_2e_1=0.

Therefore,

(e_1+2e_2)^2 = 1+4 = 5.

This agrees with the quadratic form:

Q(e_1+2e_2)=1^2+2^2=5.

The example shows how the Clifford relation enforces metric length algebraically.

104.20 Summary

Clifford algebras are associative algebras generated by a vector space with a quadratic form.

Their defining relation is

v^2=Q(v)1.

Key structures include:

Concept	Meaning
Quadratic form $Q$	Determines vector squares
Clifford product	Multiplication combining metric and exterior structure
Orthogonal basis rule	$e_ie_j=-e_je_i$ , $e_i^2=Q(e_i)$
Even subalgebra	Algebra of even-grade elements
Spin group	Products of unit vectors representing rotations
Bivector	Oriented plane element

Clifford algebra extends exterior algebra by adding metric information. It gives a unified algebraic language for length, angle, orientation, reflection, rotation, spinors, and orthogonal geometry.