Mathematical Background

This section provides the mathematical foundations for understanding Clifford algebras and their implementation in CliffordAlgebras.jl.

What is a Clifford Algebra?

A Clifford algebra, also known as a geometric algebra, is an associative algebra that extends the concept of complex numbers to higher dimensions while preserving the geometric interpretation of multiplication.

Formal Definition

Given a vector space V over the real numbers ℝ with a symmetric bilinear form Q (quadratic form), the Clifford algebra Cl(V,Q) is the quotient of the tensor algebra T(V) by the two-sided ideal I generated by elements of the form:

v ⊗ v - Q(v,v) · 1

for all v ∈ V.

Signature Notation

A Clifford algebra is often denoted by its signature (p,q,r) where:

p = number of basis vectors that square to +1
q = number of basis vectors that square to -1
r = number of basis vectors that square to 0

The total dimension of the underlying vector space is n = p + q + r.

The Clifford Relation

The fundamental relation in a Clifford algebra is:

eᵢeⱼ + eⱼeᵢ = 2ηᵢⱼ

where ηᵢⱼ is the metric tensor. For orthonormal basis vectors:

eᵢ² = +1 if i ≤ p (positive signature)
eᵢ² = -1 if p < i ≤ p+q (negative signature)
eᵢ² = 0 if i > p+q (null signature)

Multivectors and Grades

Grade Structure

A general element (multivector) in Cl(p,q,r) can be decomposed by grade:

M = M₀ + M₁ + M₂ + ... + Mₙ

where Mₖ contains only k-vectors (elements of grade k).

Grade 0: Scalars
Grade 1: Vectors
Grade 2: Bivectors (oriented areas)
Grade 3: Trivectors (oriented volumes)
Grade n: Pseudoscalars

Basis Elements

For an n-dimensional space, there are 2ⁿ basis elements:

1 scalar: 1
(n choose 1) vectors: eᵢ
(n choose 2) bivectors: eᵢeⱼ (i < j)
...
1 pseudoscalar: e₁e₂...eₙ

Example: Cl(2,0,0)

In 2D Euclidean space:

Basis: {1, e₁, e₂, e₁e₂}
Relations: e₁² = e₂² = 1, e₁e₂ = -e₂e₁

Products in Clifford Algebra

Geometric Product

The geometric product is the fundamental operation, combining the inner and outer products:

ab = a·b + a∧b

Properties:

Associative: (ab)c = a(bc)
Distributive: a(b+c) = ab + ac
Generally non-commutative

Exterior Product (Wedge Product)

The exterior product a∧b creates higher-grade elements:

a∧b = ½(ab - ba)

Properties:

Anti-commutative: a∧b = -b∧a
Associative: (a∧b)∧c = a∧(b∧c)
a∧a = 0

Interior Product (Contraction)

Several types of contraction exist:

Left Contraction (⨼)

a⨼b = Σₖ ⟨ab⟩ₖ₋ᵢ where a has grade i

Right Contraction (⨽)

a⨽b = Σₖ ⟨ab⟩ⱼ₋ₖ where b has grade j

Fat Dot Product (⋅)

a⋅b = Σₖ ⟨ab⟩|ᵢ₋ⱼ|

Scalar Product (⋆)

a⋆b = ⟨ab⟩₀ (grade 0 part only)

Involutions and Operations

Grade Involution

α̃ = Σₖ (-1)ᵏ ⟨α⟩ₖ

Reverse (Reversion)

α† = Σₖ (-1)^(k(k-1)/2) ⟨α⟩ₖ

Clifford Conjugation

α̅ = Σₖ (-1)^(k(k+1)/2) ⟨α⟩ₖ

Duality

The Hodge dual with respect to the pseudoscalar I:

*α = αI⁻¹

Exponential and Logarithm

Exponential of Bivectors

For a bivector B, the exponential gives:

e^B = cos(|B|) + (B/|B|)sin(|B|)

This formula generates rotations (for positive signature) or hyperbolic rotations (for mixed signature).

Rotor Representation

A rotor R = e^(B/2) represents a rotation by angle |B| in the plane defined by B. The rotation is applied via:

v' = RvR†

Common Clifford Algebras

Cl(2,0,0) - 2D Euclidean

Used for 2D rotations and complex numbers
Isomorphic to ℝ + ℝ² (scalar + vector)

Cl(3,0,0) - 3D Euclidean

Standard 3D vector algebra with cross product
Quaternions live in the even subalgebra

Cl(1,3,0) - Spacetime Algebra

Used in relativistic physics
Signature matches Minkowski spacetime

Cl(3,0,1) - Projective Geometric Algebra (PGA)

Ideal for computer graphics and robotics
Includes points at infinity naturally

Cl(4,1,0) - Conformal Geometric Algebra (CGA)

Unified treatment of Euclidean geometry
Circles, spheres, and transformations

Doran, C. & Lasenby, A. (2003). Geometric Algebra for Physicists. Cambridge University Press.
Dorst, L., Fontijne, D., & Mann, S. (2007). Geometric Algebra for Computer Science. Morgan Kaufmann.
Hestenes, D. & Sobczyk, G. (1984). Clifford Algebra to Geometric Calculus. D. Reidel Publishing Company.
Perwass, C. (2009). Geometric Algebra with Applications in Engineering. Springer.