SU(2) — the double cover

The double cover of SO(3) is SU(2). The special unitary group in two dimensions.

The covering map sends every pair of elements ±U in SU(2) to a single rotation R in SO(3).

Standard mathematics.

The physical significance is what matters.

Quantum systems can transform under representations of SU(2), not just SO(3).

A representation of SO(3) returns to itself after a 360° rotation. Integer spin.

A representation of SU(2) that is NOT a representation of SO(3) picks up a phase of −1 under 360° rotation, and returns to itself only after 720°. Half-integer spin.

You have just met the mathematical structure that makes two kinds of particles possible.

Whether a particle has integer or half-integer spin depends on whether it transforms under a representation that respects the ℤ₂ kernel (integer) or does not (half-integer).

The question is: what determines which particles are which?

You have been told it is an empirical fact. You measure spin and accept it.

It is not an empirical fact.

It is a consequence of the axioms.