Spin and the $SU(2)$ Representation

Spin is an intrinsic angular momentum carried by particles that has no classical or spatial analog: it is not $\hat{\mathbf{r}}\times \hat{\mathbf{p}}$ and does not arise from orbital motion. It obeys the same angular-momentum algebra as $\hat{\mathbf{L}}$, but realizes the half-integer representations that orbital angular momentum forbids. Spin is where the double-cover relationship between the rotation group $SO(3)$ and $SU(2)$ becomes physical.

1. The Spin Algebra

The spin operators satisfy

$$[{\hat{S}}_i,{\hat{S}}_j]=i\hslash {ϵ}_{ijk}{\hat{S}}_k,$$

identical in form to the orbital case. By the same ladder argument, simultaneous eigenstates $|s,m_s⟩$ have

$${\hat{S}}^2|s,m_s⟩={\hslash }^{2}s(s+1)|s,m_s⟩,{\hat{S}}_z|s,m_s⟩=\hslash m_s|s,m_s⟩,$$

with $m_s=-s,\dots ,+s$. Now, however, both integer and half-integer $s=0,\frac{1}{2},1,\frac{3}{2},\dots$ occur. The spin $s$ is a fixed property of a particle (electron $s=\frac{1}{2}$, photon $s=1$, Higgs $s=0$); the state space is the finite-dimensional ${\mathbb{C}}^{2s+1}$, tensored with the spatial $L^2$.

2. The Stern–Gerlach Experiment

A beam of silver atoms passed through an inhomogeneous magnetic field splits into two discrete spots, not a continuous smear. This is the empirical signature of a two-valued angular momentum: $s=\frac{1}{2}$, $m_s=\pm \frac{1}{2}$. Sequential Stern–Gerlach devices along different axes demonstrate the non-commutativity of ${\hat{S}}_x,{\hat{S}}_y,{\hat{S}}_z$ directly — measuring $S_x$ destroys prior knowledge of $S_z$ — and provide the canonical concrete two-level (qubit) system.

3. Spin-$\frac{1}{2}$ and the Pauli Matrices

For $s=\frac{1}{2}$ the space is ${\mathbb{C}}^2$, spanned by $|\uparrow ⟩,|\downarrow ⟩$ (eigenstates of ${\hat{S}}_z$ with $m_s=\pm \frac{1}{2}$). Writing $\hat{\mathbf{S}}=\frac{\hslash }{2}\mathbf{\sigma }$, the Pauli matrices are

$${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$$

They satisfy ${\sigma }_i{\sigma }_j={\delta }_{ij}+i{ϵ}_{ijk}{\sigma }_k$, hence both the commutator $[{\sigma }_i,{\sigma }_j]=2i{ϵ}_{ijk}{\sigma }_k$ (the algebra) and the anticommutator $\{{\sigma }_i,{\sigma }_j\}=2{\delta }_{ij}$ (Clifford relation — the bridge to the Dirac equation). A general pure spin state is a point on the Bloch sphere,

$$|\psi ⟩=\cos \frac{\theta }{2}|\uparrow ⟩+e^{i\varphi }\sin \frac{\theta }{2}|\downarrow ⟩.$$

4. Rotations and the $4\pi$ Sign

A rotation by angle $\theta$ about axis $\hat{\mathbf{n}}$ is represented by

$$\hat{U}(\theta ,\hat{\mathbf{n}})=e^{-i\theta \hat{\mathbf{n}}\cdot \hat{\mathbf{S}}/\hslash }.$$

For spin-$\frac{1}{2}$ this exponentiates to $\hat{U}=\cos \frac{\theta }{2}-i(\hat{\mathbf{n}}\cdot \mathbf{\sigma })\sin \frac{\theta }{2}$. A full $2\pi$ rotation gives $\hat{U}=-\hat{1}$, not $+\hat{1}$: the state picks up a minus sign, and only a $4\pi$ rotation returns it to itself. This is the statement that spin-$\frac{1}{2}$ transforms under $SU(2)$, the double cover of the rotation group $SO(3)$ — a single physical rotation corresponds to two $SU(2)$ elements $\pm \hat{U}$. The sign is observable via interference (neutron interferometry) and is the group-theoretic root of the spin–statistics connection.

5. Higher Spin and the General Representation

For general $s$, the $(2s+1)$-dimensional irreducible representation of $SU(2)$ is built by the same ladder construction, with ${\hat{S}}_{\pm }|s,m_s⟩=\hslash \sqrt{s(s+1)-m_s(m_s\pm 1)}|s,m_s\pm 1⟩$. Integer $s$ representations descend to genuine $SO(3)$ representations (they are single-valued under $2\pi$); half-integer $s$ are spinorial (double-valued). The full representation theory — Casimir ${\hat{S}}^2$, weights $m_s$, tensor products — is the subject of math/group-theory/00-README.md, and its relativistic extension classifies particles in QFT.

See also

• Orbital angular momentum — the same algebra, integer case.
• Addition of angular momenta — coupling spin and orbit.
• Identical particles — spin–statistics.
• math/group-theory/00-README.md — $SU(2)$, $SO(3)$, and reps.