6. Rotation Group SO(3) & SU(2)

The mathematical structure of angular momentum is intimately connected to the rotation group SO(3) and its universal cover SU(2). Understanding this connection reveals why angular momentum is quantized and why spinors behave differently from vectors.

Rotation Group SO(3)

The rotation group SO(3) consists of all 3×3 orthogonal matrices with determinant +1:

$$\text{SO}(3) = \{R \in \mathbb{R}^{3\times 3} : R^T R = \mathbb{I}, \det R = 1\}$$

Properties:

Group operation: matrix multiplication
Non-abelian: $R_1 R_2 \neq R_2 R_1$ in general
Compact: bounded manifold
3-dimensional: parametrized by 3 Euler angles (or axis + angle)

Rotation by angle $\theta$ about axis $\hat{n}$ (Rodrigues formula):

$$R_{\hat{n}}(\theta) = \mathbb{I} + \sin\theta \, [\hat{n} \times] + (1-\cos\theta) \, [\hat{n} \times]^2$$

Where $[\hat{n} \times]$ is the matrix representation of cross product with $\hat{n}$

Lie Algebra so(3)

Infinitesimal rotations form the Lie algebra so(3):

$$R_i(\epsilon) = \mathbb{I} + i\epsilon L_i + O(\epsilon^2)$$

Generators $L_i$ (3×3 antisymmetric matrices):

$$L_1 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}, \quad L_2 = \begin{pmatrix} 0 & 0 & i \\ 0 & 0 & 0 \\ -i & 0 & 0 \end{pmatrix}, \quad L_3 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$

Commutation relations:

$$[L_i, L_j] = i\epsilon_{ijk} L_k$$

Identical to angular momentum algebra! (with $\hbar = 1$)

SU(2) Group

The special unitary group SU(2):

$$\text{SU}(2) = \{U \in \mathbb{C}^{2\times 2} : U^\dagger U = \mathbb{I}, \det U = 1\}$$

General element:

$$U = \begin{pmatrix} \alpha & \beta \\ -\beta^* & \alpha^* \end{pmatrix}, \quad |\alpha|^2 + |\beta|^2 = 1$$

Key properties:

Topologically equivalent to 3-sphere $S^3$
Simply connected (no holes)
Compact and connected
Same dimension as SO(3): 3 parameters

Lie Algebra su(2)

Generators are Pauli matrices (divided by 2):

$$J_i = \frac{\sigma_i}{2}, \quad \text{where } \sigma_i \text{ are Pauli matrices}$$

Explicit form:

$$J_1 = \frac{1}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad J_2 = \frac{1}{2}\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad J_3 = \frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$

Commutation relations:

$$[J_i, J_j] = i\epsilon_{ijk} J_k$$

Same algebra as so(3)! The groups have isomorphic Lie algebras

The 2-to-1 Homomorphism

There exists a surjective homomorphism $\phi: \text{SU}(2) \to \text{SO}(3)$:

$$\phi(U) = R, \quad \text{where } U\vec{\sigma}U^\dagger = R\vec{\sigma}$$

Key property: $\phi$ is 2-to-1:

$$\phi(U) = \phi(-U)$$

Both $U$ and $-U$ map to the same rotation $R$

Kernel: $\text{ker}(\phi) = \{\mathbb{I}, -\mathbb{I}\} \cong \mathbb{Z}_2$

Consequence: SU(2) is the double cover of SO(3):

$$\text{SU}(2) / \mathbb{Z}_2 \cong \text{SO}(3)$$

Spin-1/2 and Spinors

Rotation of spin-1/2 state by angle $\theta$ about $\hat{n}$:

$$|\psi\rangle \to e^{-i\theta \hat{n} \cdot \vec{J}}|\psi\rangle = e^{-i\theta \hat{n} \cdot \vec{\sigma}/2}|\psi\rangle$$

Crucial observation: $2\pi$ rotation gives minus sign!

$$e^{-i(2\pi) \hat{n} \cdot \vec{\sigma}/2} = e^{-i\pi \hat{n} \cdot \vec{\sigma}} = -\mathbb{I}$$

$|\psi\rangle \to -|\psi\rangle$ after full rotation

Need $4\pi$ rotation to return to original state:

$$e^{-i(4\pi) \hat{n} \cdot \vec{\sigma}/2} = +\mathbb{I}$$

Interpretation: Spin-1/2 objects are spinors, transforming under SU(2), not vectors transforming under SO(3)

Representations

SO(3) representations: Integer angular momentum only

$j = 0$: scalars (1D)
$j = 1$: vectors (3D) - the defining representation
$j = 2$: traceless symmetric tensors (5D)
General: $(2j+1)$-dimensional for integer $j$

SU(2) representations: Integer AND half-integer

$j = 0$: singlet (1D)
$j = 1/2$: fundamental/spinor (2D) - what electrons have!
$j = 1$: adjoint (3D) - equivalent to SO(3) vectors
$j = 3/2$: (4D) - delta baryons
General: $(2j+1)$-dimensional for integer or half-integer $j$

Key difference: Half-integer representations don't come back to themselves under $2\pi$ rotation, so they can't be SO(3) representations. They're representations of the double cover SU(2).

Wigner D-Matrices

Matrix elements of rotation operator in angular momentum basis:

$$D^{(j)}_{m'm}(\alpha, \beta, \gamma) = \langle j, m' | e^{-i\alpha J_z}e^{-i\beta J_y}e^{-i\gamma J_z} | j, m \rangle$$

Parametrized by Euler angles $(\alpha, \beta, \gamma)$

Properties:

Unitary: $\sum_k D^{(j)}_{mk} D^{(j)^*}_{nk} = \delta_{mn}$
Group property: $D(R_1)D(R_2) = D(R_1 R_2)$
Complete set of functions on SO(3) or SU(2)

Reduced d-matrices (rotation about y-axis):

$$d^{(j)}_{m'm}(\beta) = \langle j, m' | e^{-i\beta J_y} | j, m \rangle$$

Related to Jacobi polynomials; tabulated for common values

Physical Implications

1. Spin-statistics theorem:

Half-integer spin (fermions) must be antisymmetric under exchange

Integer spin (bosons) must be symmetric

Connected to topology of configuration space in 3D

2. Spinor structure:

Electron wave function has extra $\mathbb{Z}_2$ phase freedom

Neutron interferometry experiments directly observe $2\pi$ sign change!

3. Gauge theories:

Electroweak theory based on SU(2) × U(1)

Quarks transform under SU(3) (color charge)

Understanding SU(2) is gateway to Standard Model

4. Geometric phase:

Berry phase for spin-1/2 in rotating magnetic field = solid angle/2

Factor of 1/2 comes from double cover structure

Why Nature Uses SU(2), Not SO(3)

Question: Why does nature allow half-integer spin?

Answer: Quantum mechanics requires projective representations

States defined up to phase: $|\psi\rangle \sim e^{i\phi}|\psi\rangle$
Observables unchanged by overall phase
Group action only needs to be defined up to phase

Mathematical fact: Projective representations of SO(3) are ordinary representations of SU(2)

Since SO(3) is not simply connected, it has "extra" representations when we allow phases → these are precisely the half-integer spin representations!

Deep insight: Topology of rotation group determines possible spins in quantum mechanics

Profound Connection: The existence of spin-1/2 particles is not an ad hoc feature of quantum mechanics - it's a mathematical necessity arising from the topology of 3D space! The fact that SO(3) has a double cover (SU(2)) that is simply connected means nature can choose representations from this larger group. This beautiful interplay between geometry, topology, and quantum mechanics is one of the deepest insights of 20th century physics.

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