Section9.10

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## 9.10 CliffordAlgebra

CliffordAlgebra(n,K,Q) defines a vector space of dimension 2 n over the field K with a given quadratic form Q. If {e 1,…,e n} is a basis for K n then

{ 1,
e(i) 1 <= i <= n,
e(i1)*e(i2) 1 <= i1 < i2 <=n,
...,
e(1)*e(2)*...*e(n) }

is a basis for the Clifford algebra. The algebra is defined by the relations

e(i)*e(i) = Q(e(i))
e(i)*e(j) = -e(j)*e(i), i ^= j

Examples of Clifford Algebras are gaussians (complex numbers), quaternions, exterior algebras and spin algebras.

### 9.10.1 The Complex Numbers as a Clifford Algebra

This is the field over which we will work, rational functions with integer coefficients.

K := Fraction Polynomial Integer

FractionPolynomialInteger

Type: Domain

We use this matrix for the quadratic form.

m := matrix [ [-1] ]

[-1]

Type: Matrix Integer

We get complex arithmetic by using this domain.

C := CliffordAlgebra(1, K, quadraticForm m)

CliffordAlgebra(1,FractionPolynomialInteger,MATRIX)

Type: Domain

Here is i, the usual square root of -1.

i: C := e(1)

e 1

Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)

Here are some examples of the arithmetic.

x := a + b * i

a+b e 1

Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)
y := c + d * i

c+d e 1

Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)

See ComplexXmpPage for examples of Axiom's constructor implementing complex numbers.

x * y

-b d+a c+(a d+b c)e 1

Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)

### 9.10.2 The Quaternion Numbers as a Clifford Algebra

This is the field over which we will work, rational functions with integer coefficients.

K := Fraction Polynomial Integer

FractionPolynomialInteger

Type: Domain

We use this matrix for the quadratic form.

m := matrix [ [-1,0],[0,-1] ]

[-1 0 0-1]

Type: Matrix Integer

The resulting domain is the quaternions.

H := CliffordAlgebra(2, K, quadraticForm m)

CliffordAlgebra(2,FractionPolynomialInteger,MATRIX)

Type: Domain

We use Hamilton's notation for i,j,k.

i: H := e(1)

e 1

Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
j: H := e(2)

e 2

Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
k: H := i * j

e 1 e 2

Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
x := a + b * i + c * j + d * k

a+b e 1+c e 2+d e 1 e 2

Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
y := e + f * i + g * j + h * k

e+f e 1+g e 2+h e 1 e 2

Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
x + y

e+a+(f+b)e 1+(g+c)e 2+(h+d)e 1 e 2

Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
x * y

-d h-c g-b f+a e+(c h-d g+a f+b e)e 1+(-b h+a g+d f+c e)e 2+(a h+b g-c f+d e)e 1 e 2

Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)

See QuaternionXmpPage for examples of Axiom's constructor implementing quaternions.

y * x

-d h-c g-b f+a e+(-c h+d g+a f+b e)e 1+(b h+a g-d f+c e)e 2+(a h-b g+c f+d e)e 1 e 2

Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)

### 9.10.3 The Exterior Algebra on a Three Space

This is the field over which we will work, rational functions with integer coefficients.

K := Fraction Polynomial Integer

FractionPolynomialInteger

Type: Domain

If we chose the three by three zero quadratic form, we obtain the exterior algebra on e(1),e(2),e(3).

Ext := CliffordAlgebra(3, K, quadraticForm 0)

CliffordAlgebra(3,FractionPolynomialInteger,MATRIX)

Type: Domain

This is a three dimensional vector algebra. We define i, j, k as the unit vectors.

i: Ext := e(1)

e 1

Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
j: Ext := e(2)

e 2

Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
k: Ext := e(3)

e 3

Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)

Now it is possible to do arithmetic.

x := x1*i + x2*j + x3*k

x1 e 1+x2 e 2+x3 e 3

Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
y := y1*i + y2*j + y3*k

y1 e 1+y2 e 2+y3 e 3

Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
x + y

(y1+x1)e 1+(y2+x2)e 2+(y3+x3)e 3

Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
x * y + y * x

0

Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)

On an n space, a grade p form has a dual n-p form. In particular, in three space the dual of a grade two element identifies e1*e2->e3, e2*e3->e1, e3*e1->e2.

dual2 a == coefficient(a,[2,3]) * i + coefficient(a,[3,1]) * j + coefficient(a,[1,2]) * k
Type: Void

The vector cross product is then given by this.

dual2(x*y)
Compiling function dual2 with type CliffordAlgebra(3,Fraction
Polynomial Integer,MATRIX) -> CliffordAlgebra(3,Fraction
Polynomial Integer,MATRIX)

(x2 y3-x3 y2)e 1+(-x1 y3+x3 y1)e 2+(x1 y2-x2 y1)e 3

Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)

### 9.10.4 The Dirac Spin Algebra

In this section we will work over the field of rational numbers.

K := Fraction Integer

FractionInteger

Type: Domain

We define the quadratic form to be the Minkowski space-time metric.

g := matrix [ [1,0,0,0], [0,-1,0,0], [0,0,-1,0], [0,0,0,-1] ]

[1 0 0 0 0-1 0 0 0 0-1 0 0 0 0-1]

Type: Matrix Integer

We obtain the Dirac spin algebra used in Relativistic Quantum Field Theory.

D := CliffordAlgebra(4,K, quadraticForm g)

CliffordAlgebra(4,FractionInteger,MATRIX)

Type: Domain

The usual notation for the basis is γ with a superscript. For Axiom input we will use gam(i):

gam := [e(i)$D for i in 1..4]

[e 1, e 2, e 3, e 4]

Type: List CliffordAlgebra(4,Fraction Integer,MATRIX)

There are various contraction identities of the form

g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) =
2*(gam(s)gam(m)gam(n)gam(r) + gam(r)*gam(n)*gam(m)*gam(s))

where a sum over l and t is implied.

Verify this identity for particular values of m,n,r,s.

m := 1; n:= 2; r := 3; s := 4;
Type: PositiveInteger
lhs := reduce(+, [reduce(+, [ g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) for l in 1..4]) for t in 1..4])

-4 e 1 e 2 e 3 e 4

Type: CliffordAlgebra(4,Fraction Integer,MATRIX)
rhs := 2*(gam s * gam m*gam n*gam r + gam r*gam n*gam m*gam s)

-4 e 1 e 2 e 3 e 4

Type: CliffordAlgebra(4,Fraction Integer,MATRIX)
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