Paired elements

Consider an element a ∈ ℒ that has a σ-image σ(a) ∈ 𝒟.

This is a paired element. It participates fully in the ℤ₂ symmetry.

Under the involution: σ(a) = a′ ∈ 𝒟, and σ(a′) = a ∈ ℒ.

The ℤ₂ action cycles between a and a′. Two applications return to the start: σ²(a) = a.

On the Hilbert space of the embedded manifold, this paired element transforms under a representation of SU(2) where the ℤ₂ kernel acts trivially.

The swap completes in one step.

The element sees the full rotation. 360° returns it to itself.

This is a representation of SO(3) — which means it is a representation of SU(2) with integer spin.

Paired elements have integer spin. They are bosons.

You have just seen the first half of the classification.