# 2 \(\times\) 2 Matrices: Manifolds, Realizations, Applications
## 3 Canonical Integral Transforms and the Universal \(2\times 2\) Matrix Covers
### The Canonical Transform Kernel  
The canonical transform integral kernel, \(C_{\mathbf{M}}(q,q^{\prime})\) in (22), can be found straightforwardly applying the integral action of \(\mathcal{C}_{\mathbf{M}}\) to \(\widehat{q}f(q)\) and \(\widehat{p}f(q)\), the latter subject to integration by parts, that yield the following two simultaneous linear differential equations of first degree for the two arguments of the kernel [7; 8],  
\[-\mathrm{i}\partial_{q^{\prime}}C_{\mathbf{M}}(q,q^{\prime}) = (cq+\mathrm{i}a\partial_{q})C_{\mathbf{M}}(q,q^{\prime}), \tag{24}\] \[q^{\prime}C_{\mathbf{M}}(q,q^{\prime}) = (dq+\mathrm{i}b\partial_{q})C_{\mathbf{M}}(q,q^{\prime}),\] (25) \[\Rightarrow\quad C_{\mathbf{M}}(q,q^{\prime}) = K_{\mathbf{M}}\exp\mathrm{i}\big{[}(dq^{2}-2 qq^{\prime}+aq^{ \prime 2})/2b\big{]}. \tag{26}\]  
Here \(K_{\mathbf{M}}\) is an \(\mathbf{M}\)-dependent normalization factor that makes \(C_{\mathbf{M}}(q,q^{\prime})\) the kernel of unitary transformations, whose inverse is the transpose conjugate: \(C_{\mathbf{M}^{-1}}(q,q^{\prime})=C_{\mathbf{M}}(q^{\prime},q)^{*}\) under the \(\mathcal{L}^{2}(\mathsf{R})\) inner product, provided that its scale constant obeys \(K_{\mathbf{M}^{-1}}=K_{\mathbf{M}}^{*}\). In the limit to the identity transformation \(\mathcal{I}\), the unit element under the integral transform group, the kernel becomes a Dirac \(\delta\), as:  
\[\lim_{\mathbf{M}\to\mathbf{1}}C_{\mathbf{M}}(q,q^{\prime})=\delta(q-q^{\prime }). \tag{27}\]  
This manifold of integral kernels \(C_{\mathbf{M}}(q,q^{\prime})\), \(\mathbf{M}\in\text{SL}(2,\mathsf{R})\) form another, distinct representation of the group of \(2\times 2\) matrices. Due to (23) this representation inherits the group product law as:  
\[\mathbf{M}_{2}\mathbf{M}_{1}=\mathbf{M}_{3}\quad\Longrightarrow\quad\int_{ \mathsf{R}}\!\mathrm{d}q^{\prime}\,C_{\mathbf{M}_{2}}(q,q^{\prime})\,C_{ \mathbf{M}_{1}}(q^{\prime},q^{\prime\prime})=\pm C_{\mathbf{M}_{3}}(q,q^{ \prime\prime}), \tag{28}\]  
with the inevitable sign ambiguity. Performing the integration of the \(q^{\prime}\)-dependent factor in (28), we come to shift contours in the complex plane, resulting in:  
\[I(r,s) := \int_{\mathsf{R}}\mathrm{d}q^{\prime}\,\exp\mathrm{i}(r^{2}q^{ \prime 2}+sq^{\prime})=\varphi(r)\frac{e^{\mathrm{i}\pi/4}\sqrt{\pi}}{r}\exp \left(\frac{-\mathrm{i}s^{2}}{4r^{2}}\right), \tag{29}\] \[\text{where }\varphi(r) := \left\{ \begin{array}{cc}1&\text{when }\arg r\in[0,\tfrac{1}{2}\pi] \mod 2\pi,\\ -1&\text{when }\arg r\in[-\pi,-\tfrac{1}{2}\pi]\mod 2\pi,\end{array} \right. \tag{30}\]  
for \(r=a_{2}/2b_{2}+d_{1}/2b_{1}=b_{21}/2b_{2}b_{1}\) with specified phases \(\arg r=0\) for \(r>0\) or \(\arg r=-\pi\) for \(r<0\), and \(s=-q/b_{2}-q^{\prime\prime}/b_{1}\). The factors in \(q,q^{\prime\prime}\) then reconstitute the kernel of the product (28), and the sign ambiguity is resolved by \(\varphi(b_{21}/b_{2}b_{1})\). Approtioning the factors in (29) to (28) determines the normalization and phase of the coefficient function in (26) that should be written with a carefully determined phase [7,18] [S9.1] as,  
\[K_{\mathbf{M}}=\frac{1}{\sqrt{2\pi\mathrm{i}b}}:=\frac{e^{-\mathrm{i}\frac{1}{2 }\pi}\exp(-\mathrm{i}\frac{1}{2}\arg b)}{\sqrt{2\pi\left|b\right|}}. \tag{31}\]  
The singular limit \(b\to 0\) of the kernel of \(\mathbf{M}:=\left( \begin{smallmatrix}a&b\\ c&d\end{smallmatrix} \right)\) is an oscillating Gaussian with a Dirac-delta limit that has a stationary point at \(q=aq^{\prime}\), \(a=1/d\). Under the integral this is:  
\[\lim_{\left|b\right|\to 0}C_{\mathbf{M}}(q,q^{\prime})=\frac{1}{\sqrt{a}}\exp \mathrm{i}\left(\frac{cq^{2}}{2a}\right)\delta\left(q^{\prime}-\frac{q}{a} \right). \tag{32}\]  
The paraxial lens transformation \(\mathbf{L}_{\mathrm{G}}=\left( \begin{smallmatrix}1&0\\ -C&1\end{smallmatrix} \right)\) in (12) thus multiplies the signal by a quadratic phase \(\sim\exp\mathrm{i}(-\frac{1}{2}Gq^{2})\). The free flight transformation \(\mathbf{D}_{z}:=\left( \begin{smallmatrix}1&z\\ 0&1\end{smallmatrix}