SU(2) and SO(3)

Intuition

SU(2) is the double cover of SO(3): two SU(2) matrices correspond to one SO(3) rotation. Spinors live in SU(2).

Formal Definition

SU(2): $2\times2$ unitary matrices with det=1. SO(3): $3\times3$ orthogonal matrices with det=1.

Mathematical Formulation

$$U=\exp\left(-i\frac{\vec{\theta}\cdot\vec{\sigma}}{2}\right)$$ where $\vec{\sigma}$ are Pauli matrices. Mapping: double cover because $U$ and $-U$ give the same SO(3) rotation.

Derivation

Parameterize SU(2) by Euler angles; compose two rotations. The kernel $\{I,-I\}$ shows SU(2) covers SO(3) twice.

Worked Example

A $2\pi$ rotation in SO(3) corresponds to $-I$ in SU(2), not $I$.

Common Mistakes

Confusing SU(2) and SO(3) as the same group.
Forgetting the global phase in SU(2).

Quiz

Q1. How many preimages does one SO(3) rotation have in SU(2)?

Answer

Two.