# Definition The [[Special Orthogonal Group]] in 3 dimensions, $SO(3)$, is the [[group]] of all [[Rotation|rotations]] in 3 spatial dimensions. It is a [[Lie group]] of [[dimension]] 3, and the [[Identity Component|identity component]] of the [[Orthogonal Group in 3 Dimensions|orthogonal group]] $O(3)$. The [[Special Unitary Group in 2 Dimensions|special unitary group]] $SU(2)$ is the [[Relationship Between SO(3) and SU(2)|double cover]] of $SO(3)$. # Elements We can write the group [[Group Element|elements]] as follows, where the subscript corresponds to the axis of rotation: $ \begin{align} R_x(\theta) &\equiv \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix}\\ R_{y}(\theta) &\equiv\left(\begin{array}{ccc} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \end{array}\right) \\ R_{z}(\theta) &\equiv\left(\begin{array}{ccc} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array}\right) \end{align} $ A general element in $SO(3)$ can be written in two ways. First, we use Euler angles. To rotate one orthonormal [[basis]] into another of the same orientation, we rotate by an angle $\phi$ around the z-axis, then an angle $\theta$ around the *new* x-axis, then an angle $\psi$ around the *new* z-axis. This finally gives the general form for $R \in SO(3)$ $ \left(\begin{array}{ccc} \cos \psi \cos \phi-\cos \theta \sin \phi \sin \psi & \cos \psi \sin \phi+\cos \theta \cos \phi \sin \psi & \sin \psi \sin \theta \\ -\sin \psi \cos \phi-\cos \theta \sin \phi \cos \psi & -\sin \psi \sin \phi+\cos \theta \cos \phi \cos \psi & \cos \psi \sin \theta \\ \sin \theta \sin \phi & -\sin \theta \cos \phi & \cos \theta \end{array}\right) \tag{1} \label{eq:rot_11} $ Alternatively, we can consider rotations by an angle $\theta$ about an arbitrary axis described by a *unit vector* $\pmb{n} = (n_x, n_y, n_z)$, and we get the general form: $ \left(\begin{array}{ccc} n_{x}^{2}(1-\cos \theta)+\cos \theta & n_{x} n_{y}(1-\cos \theta)-n_{z} \sin \theta & n_{x} n_{z}(1-\cos \theta)+n_{y} \sin \theta \\ n_{y} n_{x}(1-\cos \theta)+n_{z} \sin \theta & n_{y}^{2}(1-\cos \theta)+\cos \theta & n_{y} n_{z}(1-\cos \theta)-n_{x} \sin \theta \\ n_{z} n_{x}(1-\cos \theta)-n_{y} \sin \theta & n_{z} n_{y}(1-\cos \theta)+n_{x} \sin \theta & n_{z}^{2}(1-\cos \theta)+\cos \theta \end{array}\right) \tag{2} \label{eq:rot_22} $ # Infinitesimal Generators The elements of the [[Lie Algebra]] of $SO(3)$, denoted $\mathfrak{so}(3)$, are the infinitesimal generators shown below: $ \begin{aligned} &\left.L_{x} \equiv \frac{d R_{x}}{d \theta}\right|_{\theta=0}=\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{array}\right) \\ &L_{y}=\left(\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{array}\right) \\ &L_{z}=\left(\begin{array}{ccc} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) . \end{aligned} $ Observe that, up to a factor of $i$, these are just representations of the [[angular momentum|angular momentum]] operators. Observe that $ \text{span}\{L_x, L_y, L_z\} = \{\text{anti-symmetric 3$\times$3 matrices}\} \equiv \mathfrak{so}(3) $ where $\mathfrak{so}(3)$ denotes the [[Lie Algebra|Lie algebra]] of $SO(3)$. Note that we can write an arbitrary $3\times3$ antisymmetric matrix, i.e., an arbitrary generator of rotations, as $ \omega=\left(\begin{array}{ccc} 0 & -\omega_{z} & \omega_{y} \\ \omega_{z} & 0 & -\omega_{x} \\ -\omega_{y} & \omega_{x} & 0 \end{array}\right) = \omega_x L_x + \omega_y L_y + \omega_z L_z $ Now multiply $\omega$ by an arbitrary vector $\pmb{r} = (x,y,z)$ and we get: $ \omega \pmb{r} = \pmb{\omega} \times \pmb{r} $ where we have defined the [[Pseudovector|pseudovector]] $\pmb{\omega} = (\omega_x, \omega_y, \omega_z)$ associated with the infinitesimal generator $\omega$. Note that: $ \omega \pmb{\omega} = \pmb{\omega}\times\pmb{\omega} = \pmb{0} $ and we can conclude that, since $\pmb{\omega}$ is unchanged by the rotation, then it must lie along the axis of rotation. It is really just the familiar [[angular velocity]] [[vector]]. Finally, note that: $ \begin{align} [L_x, L_y] &= L_z,\\ [L_y, L_z] &= L_x,\\ [L_z, L_x] &= L_y \end{align} $ where $[\cdot,\cdot]$ is the [[commutator]]. This is one of the defining relations of a [[Lie Algebra]], i.e., closure under the commutator (Lie bracket). # Lie Algebra The Lie algebra of $SO(3)$, denoted $\mathfrak{so}(3)$, is the set of all $3\times3$ antisymmetric matrices, as noted in [[Special Orthogonal Group#Lie Algebra|here]]. We can pick a basis of this space: $ \begin{aligned} &L_{x}=\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{array}\right) \\ &L_{y}=\left(\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{array}\right) \\ &L_{z}=\left(\begin{array}{ccc} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) . \end{aligned} $ which are just the generators discussed [[#Infinitesimal Generators|above]]. Note that, labeling the generators by number ($x = 1, y = 2, z = 3$) we have: $ (L_i)_{jk} = -\epsilon_{ijk} $ and we can also rewrite the commutation relations noted above as: $ [L_i, L_j] = \sum_{k = 1}^3 \epsilon_{ijk} L_k $ An arbitrary element in $\mathfrak{so}(3)$ can be written as: $ X = \begin{pmatrix} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0 \end{pmatrix} = x L_x + y L_y + z L_z $ so we can write $[X]_{\{L_i\}} = (x,y,z)$. Via direct multiplication, one can show that, given $\pmb{v} = (v_x, v_y, v_z)$, $ X \pmb{v} = [X] \times v $ so that $X [X] = 0$ This implies that: $ e^{t X}[X] = I [X] + tX [X] + \frac{1}{2} t^2 X^2[X] + \ldots \implies e^{tX}[X] = [X], $ i.e., $e^{tX}$ is a rotation around the $[X]$ axis since it leaves $[X]$ invariant. Taking $[X]$ to be a unit vector $[X] = (n_x, n_y, n_z)$, and relabeling $t \rightarrow \theta$, we end up with: $ \left(\begin{array}{ccc} n_{x}^{2}(1-\cos \theta)+\cos \theta & n_{x} n_{y}(1-\cos \theta)-n_{z} \sin \theta & n_{x} n_{z}(1-\cos \theta)+n_{y} \sin \theta \\ n_{y} n_{x}(1-\cos \theta)+n_{z} \sin \theta & n_{y}^{2}(1-\cos \theta)+\cos \theta & n_{y} n_{z}(1-\cos \theta)-n_{x} \sin \theta \\ n_{z} n_{x}(1-\cos \theta)-n_{y} \sin \theta & n_{z} n_{y}(1-\cos \theta)+n_{x} \sin \theta & n_{z}^{2}(1-\cos \theta)+\cos \theta \end{array}\right) $ which is just Eq. $\eqref{eq:rot_11}$ above. ## The Commutator and the Cross Product Given two matrices $X, Y \in \mathfrak{so}(3)$, and the basis $\mathcal{B} = \{L_i\}$, we have the following relation: $ [[X,Y]]_\mathcal{B} = [X]_\mathcal{B} \times [Y]_\mathcal{B} $ The proof is straightforward via direct computation. # In Physics $SO(3)$ has a plethora of applications in physics. Note that, in physics, we [[Physicists' Definition of Lie Algebras|would define the generators Hermitian]], $L'_i \equiv i L_i$ so that we end up with: $ [L'_i, L'_j] = i\sum_{k = 1}^3 \epsilon_{ijk} L'_k $ which is the commutation relation we're most used to.