SO(3)

Prerequisites: Lie Groups. Full path (3 pages)

Intuitive
Concrete
Abstract

Intuitive

The Lie group $SO(3)$ describes all possible rotations of an object in 3-dimensional Euclidean space. It thus describes an important symmetry of the physical space we live in. (Other important spacetime symmetries are translations and boosts.)

$SO(3)$ is closely related to the groups $SU(2)$ and $Sp(1)$ . They are all locally isomorphic, that is, they have the same Lie algebra.

Concrete

A general element of $SO(3)$ can be written as the product of three rotation matrices, one about the $x$ , $y$ , and $z$ axes: $R(\theta_x,\theta_y,\theta_z) = R_x(\theta_x) \cdot R_y(\theta_y) \cdot R_z(\theta_z)$ , where

$\begin{array}{rcl} & & R_x(\theta_x) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta_x & -\sin\theta_x \\ 0 & \sin\theta_x & \cos\theta_x \end{pmatrix}, \\ & & R_y(\theta_y) = \begin{pmatrix} \cos \theta_y & 0 & \sin\theta_y \\ 0 & 1 & 0 \\ -\sin\theta_y & 0 & \cos\theta_y \end{pmatrix}, \\ & & R_z(\theta_z) = \begin{pmatrix} \cos \theta_z & -\sin \theta_z & 0 \\\sin\theta_z & \cos\theta_z & 0 \\ 0 & 0 & 1 \end{pmatrix}. \end{array}$

Representations

The diagram below shows the defining (3-dimensional) representation of $SO(3)$ in its upper branch and a 5-dimensional representation of the same group in its lower branch.

Instead of using 3x3 matrices for the Lie-algebra elements of the defining representation, we can also use 3-dimensional vectors (red box in the diagram below). Then, the Lie-algebra elements act on the representation space by means of the cross product (lower branch of the diagram).

For a more detailed explanation of these diagrams and additional representations of $SO(3)$ , see Fun with Symmetry.

Abstract

$SO(3) = \{R\in\mathbb{R}^{3\times 3} : R^TR = 1,\ \det R = 1\}$ is a compact, connected Lie group of dimension three. Every rotation is a rotation by an angle $\theta\in[0,\pi]$ about some axis $\hat n$ . Representing it by the vector $\theta\hat n$ in a ball of radius $\pi$ , with antipodal points on the boundary identified, shows that $SO(3)$ is diffeomorphic to the real projective space $\mathbb{RP}^3$ . Its fundamental group is $\pi_1(SO(3)) = \mathbb{Z}_2$ , and its universal cover is $SU(2)\cong Sp(1)\cong S^3$ , with $SO(3) = SU(2)/\{\pm 1\}$ .

Lie algebra. $\mathfrak{so}(3)$ consists of the real antisymmetric $3\times 3$ matrices. The basis $(L_i)_{jk} = -\epsilon_{ijk}$ acts on vectors by $L_i\vec v = \hat e_i\times\vec v$ , and the Jacobi identity of the cross product gives $[L_i, L_j] = \epsilon_{ijk}L_k .$ With this basis $R_z(\theta) = \mathrm{e}^{\theta L_3}$ is the matrix above, and every element is $R = \mathrm{e}^{\theta\,\hat n\cdot\vec L}$ , so the exponential map is surjective. The Hermitian generators $J_k = iL_k$ obey $[J_i,J_j] = i\epsilon_{ijk}J_k$ . As a Lie algebra $\mathfrak{so}(3)\cong\mathfrak{su}(2)\cong\mathfrak{sp}(1)$ , which is $(\mathbb{R}^3,\times)$ .

Representations. The irreducible representations of $\mathfrak{so}(3)$ are labelled by $j\in\{0,\frac12,1,\dots\}$ , have dimension $2j+1$ , and $\vec J^{\,2} = j(j+1)$ on them. Only those with integer $j=\ell$ exponentiate to representations of $SO(3)$ , because the rotation by $2\pi$ acts as $(-1)^{2j}$ . Half-integer $j$ give projective representations of $SO(3)$ , which are true representations of $SU(2)$ . The integer-spin representations can be realised on traceless symmetric tensors of rank $\ell$ , or on the spherical harmonics $Y_{\ell m}$ . The sphere is the homogeneous space $S^2 = SO(3)/SO(2)$ , and the Peter-Weyl theorem gives $L^2(S^2) = \bigoplus_{\ell=0}^{\infty} V_\ell ,$ with each integer spin appearing exactly once. This is the group-theoretic content of the expansion of functions on the sphere in spherical harmonics. The matrix elements $D^{(\ell)}_{mm'}(R)$ of the irreducible representations are the Wigner $D$ -functions.

The defining representation of $SO(3)$ is the $\ell=1$ representation. It is equivalent to the adjoint representation of $SO(3)$ , and composed with the covering map it is the adjoint representation $\mathrm{Ad}: SU(2)\to SO(3)$ .

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