# Definition The [[Special Unitary Group]] in 2 dimensions is denoted $SU(2)$. It is the group of all complex $2\times2$ [[Unitary Operator|unitary matrices]] with [[determinant]] $+1$. This group implements rotations on [[spin-half|spin-1/2]] particles in quantum mechanics. $SU(2)$ is the [[Relationship Between SO(3) and SU(2)|double cover]] of the [[Special Orthogonal Group in 3 Dimensions|special orthogonal group]] $SO(3)$. # Elements The most general form of an element in this group is: $ \left(\begin{array}{rr} \alpha & \beta \\ -\beta^* & \alpha^* \end{array}\right) \quad \alpha, \beta \in \mathbb{C}, \quad|\alpha|^{2}+|\beta|^{2}=1 $ which means it is 3 [[dimension|dimensional]] (i.e. 3 free parameters). We could also use three real parameters with no constrains, $\phi, \theta, \psi$: $ \left(\begin{array}{c} e^{i(\psi+\phi) / 2} \cos \frac{\theta}{2} \quad i e^{i(\psi-\phi) / 2} \sin \frac{\theta}{2} \\ i e^{-i(\psi-\phi) / 2} \sin \frac{\theta}{2} e^{-i(\psi+\phi) / 2} \cos \frac{\theta}{2} \end{array}\right) $ These have a 1-1 correspondence to the Euler angles used to parameterize [[Special Orthogonal Group in 3 Dimensions#Definition|SO(3)]]. Alternatively, rotating along an axis described by a unit vector $\pmb{n} = (n_x, n_y, n_z)$, we get: $ \left(\begin{array}{cc} \cos (\theta / 2)-i n_{z} \sin (\theta / 2) & \left(-i n_{x}-n_{y}\right) \sin (\theta / 2) \\ \left(-i n_{x}+n_{y}\right) \sin (\theta / 2) & \cos (\theta / 2)+i n_{z} \sin (\theta / 2) \end{array}\right) \label{eq:rot_33} \tag{1} $ # Infinitesimal Generators The [[Lie Algebra]] of $SU(2)$, denoted $\mathfrak{su}(2)$, is the vector space of all $2\times2$ traceless anti-[[Hermitian Operator|Hermitian matrices]]. An arbitrary element $X \in \mathfrak{su}(2)$ can be written as: $ X = \frac{1}{2} \begin{pmatrix} -i z & -y - i x \\ y - i x & i z \end{pmatrix} \quad x,y,z \in \mathbb{R} $ which can be written as: $ X = x S_x + y S_y + z S_z $ where $ S_i \equiv -\frac{i}{2} \sigma_i $ where $\sigma_\alpha$ are the 3 [[Pauli matrices]]. Note that the commutation relation is: $ [S_i, S_j] = \sum_{k=1}^3 \epsilon_{ijk} S_k $ which is identical to the $\mathfrak{so}(3)$ commutation relation. Moreover, note that, given $X \in \mathfrak{so}(3)$ such that $[X]_{S_i} = (n_x, n_y, n_z)$, we can show that, by direct computation: $ \begin{align} e^{\theta n^iS_i} &= \left(\begin{array}{cc} \cos (\theta / 2)-i n_{z} \sin (\theta / 2) & \left(-i n_{x}-n_{y}\right) \sin (\theta / 2) \\ \left(-i n_{x}+n_{y}\right) \sin (\theta / 2) & \cos (\theta / 2)+i n_{z} \sin (\theta / 2) \end{array}\right)\\ &= \cos(\theta/2) I + 2 \sin(\theta/2) n^iS_i \end{align} $ which is just Equation $\eqref{eq:rot_33}$ above. # $SU(2)$ as a manifold To figure out which [[manifold]] describes $SU(2)$ (recall that [[Lie group|Lie groups]] are also [[Differentiable Manifold|differentiable manifolds]]), we rewrite the condition $ |\alpha|^2 + |\beta|^2 = 1 $ as $ u^2 + v^2 + a^2 + b^2 = 1 $ where now $u, v, a, b \in \mathbb{R}$. This is just the 3-sphere, $S^3$. A schematic is shown below ![[Pasted image 20210703133839.png]]