Rotation by 360°, spin-1/2 fermions and quaternions

Question

Rotating a spin-1/2 fermion by 360° multiplies the quantum state by -1.

Representing a continuous 360° rotation as a quaternion is also a multiplication by -1.

Is there a relationship between these two? What is the correspondence?

Answer 1

We know su(2) is isomorphic to unit quaternion, and the rotation group of spin-1/2 system is su(2) too. So in fact they are the same thing.

In fact, we can use Pauli matrix. Let $i = - \mathrm{i}\sigma_x$, $j = - \mathrm{i}\sigma_y$, $k = - \mathrm{i}\sigma_z$. Then we found $i^2 = j^2 = k^2 = -1$, and $a + bi + cj + dk$ is a representation of quaternion.

Answer 2

Yes, the 3D rotation group $SO(3)$ has double cover $$SPIN(3)~\cong~ SU(2)~\cong~ U(1,\mathbb{H}),$$ see e.g. this Phys.SE post for details. In particular, the fundamental spin 1/2 spinor is indeed a quaternionic/pseudoreal representation with vector space $$\mathbb{H}~\cong~ \mathbb{C}^2.$$