Complex Numbers, Quaternions, and Spinors: One Family - The Unifying Power of Geometric Algebra

> "If you think complex numbers, quaternions, and spinors are three completely different things, it's like thinking cats, tigers, and lions aren't related. They come from the same family - they just live in different rooms."

This long-form forum post argues that complex numbers, quaternions, and spinors are not isolated inventions but branches of a single tree: Clifford (geometric) algebra, introduced by William Clifford in 1878. Below is a structured summary of its main content.

Key points

• Historical framing: Hamilton carved \(i^2 = j^2 = k^2 = ijk = -1\) on Broom Bridge, Dublin, in 1843; Schrödinger's 1926 wave equation introduced spinors, whose 720-degree rotation behavior seemed absurd. The post claims both come from the same structure.
• Complex numbers = Cl⁺(2,0): In the geometric algebra of the plane with basis \(e_1, e_2\), the bivector \(e_{12} = e_1 e_2\) satisfies
\[e_{12}^2 = (e_1 e_2)(e_1 e_2) = -e_1^2 e_2^2 = -1\]

The even subalgebra \(\text{Cl}^+(2,0) = \{1, e_{12}\} \cong \mathbb{C}\). The imaginary unit is not "imaginary" - it is the oriented area element of the plane, and \(i^2 = -1\) geometrically encodes a 180° rotation. Euler's formula becomes \(e^{e_{12}\theta} = \cos\theta + e_{12}\sin\theta\), describing planar rotation.

• Quaternions = Cl⁺(3,0): The even subalgebra of 3D geometric algebra is \(\{1, e_{12}, e_{13}, e_{23}\} \cong \mathbb{H}\), with \(i = e_{23}, j = e_{13}, k = e_{12}\). The post verifies Hamilton's relations explicitly, e.g.:
\[i^2 = e_{23}^2 = e_2 e_3 e_2 e_3 = -e_2^2 e_3^2 = -1\]

Quaternion multiplication is just the geometric product of bivectors (non-commutative because plane rotations do not commute), and the rotation formula \(\vec{v}' = q\vec{v}q^{-1}\) is the rotor sandwich product \(v' = Rv\tilde{R}\). Quaternions work because they live in Spin(3), the double cover of SO(3) - which also explains the absence of gimbal lock and enables slerp interpolation.

• Spinors and the 720° mystery: SO(3) is not simply connected (\(\pi_1(SO(3)) = \mathbb{Z}_2\)); its double cover is SU(2):
\[SU(2) \cong \text{Spin}(3) \cong \{\text{unit quaternions}\} \cong S^3\]

Because rotors use half-angles, \(R(\theta) = e^{B\theta/2}\) gives \(R(2\pi) = -1\) but \(R(4\pi) = +1\) - hence a spin-1/2 particle needs 720° to return to its state (demonstrated physically by the Dirac belt trick). Pauli matrices are a matrix representation of Cl(3,0); Dirac gamma matrices represent Cl(3,1).

• Unified rotor formula: In every dimension, rotations use \(v' = Rv\tilde{R}\) with \(R = e^{B\theta/2}\), where \(B\) is the unit bivector of the rotation plane. Exceptional isomorphisms: Spin(2) ≅ U(1), Spin(3) ≅ SU(2) ≅ Sp(1), Spin(4) ≅ SU(2)×SU(2), Spin(5) ≅ Sp(2), Spin(6) ≅ SU(4).
• Applications: quaternion slerp \(\text{slerp}(q_1,q_2,t) = q_1(q_1^{-1}q_2)^t\) in graphics and robotics; strapdown inertial navigation; qubits as spinors on the Bloch sphere; Lorentz boosts and the Dirac equation \(\nabla\psi I = m\psi\) in spacetime algebra Cl(3,1); Maxwell's equations collapse to \(\nabla F = J\).
• Deeper structure: Real Bott periodicity \(Cl_{n+8} \cong Cl_n \otimes \mathbb{R}(16)\) and complex periodicity \(Cl^{\mathbb{C}}_{n+2} \cong Cl^{\mathbb{C}}_n \otimes \mathbb{C}(2)\), connecting to K-theory via the Atiyah-Bott-Shapiro theorem. Spinor bundles on manifolds require the spin condition (\(w_2 = 0\)), relevant to fermions in curved spacetime.

Bottom line

Complex numbers are the language of 2D rotation, quaternions of 3D rotation, and spinors of rotation in any dimension - all living in even subalgebras, all built on the same double-cover structure \(\text{Spin}(n) \to SO(n)\), and all expressible through one rotor formula.

*References cited include Hestenes, Doran & Lasenby, Lounesto, and the original works of Hamilton (1844) and Dirac (1928).*