# 2 \(\times\) 2 Matrices: Manifolds, Realizations, Applications
## 3 Canonical Integral Transforms and the Universal \(2\times 2\) Matrix Covers
### Bargmann Parameters for the Metaplectic Group  
We now introduce appropriate coordinates to show that real \(2\times 2\) matrices are multiply covered in the same way that the real line covers circles. We first note that the upper-left \(2~{}\times~{}2\) blocks of the \(3~{}\times~{}3\) matrices in (3) and (5) represent rotations in the \(x\)-\(y\) plane in the three-dimensional vector spaces where they act. Their familiar form corresponds to the parameters \(\alpha=\exp(\mathrm{i}\frac{1}{2}\theta)\) and \(\beta=0\) in the \(2~{}\times~{}2\) complex matrices, and through (6), to the subgroup of rotation matrices \(\mathbf{R}(\theta):=\left( \begin{smallmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{smallmatrix} \right)\), with \(\theta\) modulo \(2\pi\). The polar decomposition of complex numbers \(z\in\mathbb{C}\) refers to their factorization into a phase and a positive radius as \(z=e^{\mathrm{i}\xi}|z|\), \(\zeta=\arg z\). For matrices, a corresponding decomposition is into a unitary and a positive definite symmetric matrix [9] [Sect. 15]. So let us write what we shall call Bargmann coordinates [20] for \(\mathbf{M}\in\mathbb{SL}(2,R)\),  
\[ \begin{pmatrix}a&b\\ c&d\end{pmatrix}  = \begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}\lambda+\operatorname{Re} \mu&\operatorname{Im}\mu\\ \operatorname{Im}\mu&\lambda-\operatorname{Re}\mu\end{pmatrix} \] \[= \begin{pmatrix}\lambda\cos\theta+\operatorname{Re}\left(e^{ \mathrm{i}\theta}\mu\right)&-\lambda\sin\theta+\operatorname{Im}\left(e^{ \mathrm{i}\theta}\mu\right)\\ \lambda\sin\theta+\operatorname{Im}\left(e^{\mathrm{i}\theta}\mu\right)& \lambda\cos\theta-\operatorname{Re}\left(e^{\mathrm{i}\theta}\mu\right) \end{pmatrix} ,\]  
\[\text{where}\quad\theta\in R\text{ modulo }2\pi,\qquad\lambda:=+\sqrt{|\mu|^{2}+1} \geq 1,\quad\mu\in C. \tag{35}\]  
When \(ad-bc=1\) and \(\theta\) is in the range (35), one can straightforwardly check that there is a 1:1 relation between the three independent coordinates \(\{a,\,b,\,c\}\) and \(\{\theta,\operatorname{Re}\mu,\operatorname{Im}\mu\}\), and conversely,  
\[\theta=\arg\left[a+d+\mathrm{i}(b-c)\right],\qquad\mu=\tfrac{1}{2}e^{- \mathrm{i}\theta}[a-d+\mathrm{i}(b+c)]. \tag{36}\]  
Upon writing out the product of two such matrices \(\mathbf{M}=\mathbf{M}_{2}\mathbf{M}_{1}\) with coordinates, after some algebra one arrives at:  
\[e^{\mathrm{i}\theta}\lambda = e^{\mathrm{i}(\theta_{2}+\theta_{1})}\lambda_{2}\lambda_{1}+e^{ \mathrm{i}(\theta_{2}-\theta_{1})}\mu_{2}\mu_{1}^{*}=:e^{\mathrm{i}(\theta_{2 }+\theta_{1})}\lambda_{2}\
u\lambda_{1}, \tag{37}\] \[e^{\mathrm{i}\theta} = e^{\mathrm{i}(\theta_{2}+\theta_{1})}\lambda_{2}\mu_{1}+e^{ \mathrm{i}(\theta_{2}-\theta_{1})}\mu_{2}\lambda_{1},\] (38) \[\text{where}\quad\
u := 1+e^{-2\mathrm{i}\theta_{1}}\mu_{2}\mu_{1}^{*}/\lambda_{2} \lambda_{1}. \tag{39}\]  
Now, since \(|\mu|/\lambda<1\) from (35), it follows that in the complex plane the auxiliary quantity \(\
u\) in (39) satisfies \(|\
u-1|<1\), so it is strictly inside a disk of center in \(\
u=1\) and unit radius that excludes the origin, and this means that the phase of \(\
u\) is uniquely defined. The phase of (37) and (38) thus lead to the composition law for two real \(2\,\times\,2\) in these Bargmann coordinates, as given by:  
\[\theta = \theta_{2}+\theta_{1}+\arg\
u,\qquad\qquad\quad\arg\
u\in(- \tfrac{1}{2}\pi,+\tfrac{1}{2}\pi), \tag{40}\] \[\mu = e^{-\mathrm{i}\arg\
u}(\lambda_{2}\mu_{1}+e^{-2\mathrm{i}\theta_ {1}}\mu_{2}\lambda_{1}),\quad\lambda=\lambda_{2}\,|\mu|\,\lambda_{1}\geq 1. \tag{41}\]  
We draw attention to Equation (40) because it allows \(\theta\) to evade the \(2\pi\) modularity condition by rendering a \(\theta\)-_line_, \(\theta\in R\). This coordinate will uniquely parametrize the universal cover group \(\overline{\mathbb{SL}}(2,R)\) of \(2\,\times\,2\) matrices. And depending on the modularity condition we may choose for \(\theta\), it serves to define and parametrize any multiple cover as well, delivering the sequence of covers:  
\[