# 2 \(\times\) 2 Matrices: Manifolds, Realizations, Applications
## 3 Canonical Integral Transforms and the Universal \(2\times 2\) Matrix Covers
### [PERSON]'s Geometric Triangle and Wave Hexagon  
A memorable line of work by [PERSON] concerns those quantum systems which, after a given cycle of transformations, return the input states of the system to their original values [22]. In particular, he found that six oscillator-potential jolts spaced by six appropriate periods of free drift, will comply with this requirement. Here we have the same requirement for a wave optical signal. In (17)-(19) we introduced blocks composed of a lens and a free flight, \(\mathbf{B}_{G,z}=\left(\begin{smallmatrix}1&z\\ -G&1-Gz\end{smallmatrix}\right)\), constructing a cycle of up to three such elements to build the identity geometric transformation, \(\mathbf{B}_{3/z,z}^{3}=\mathbf{1}\).  
We now build the corresponding integral canonical transforms of those matrices and their first few powers with the Bargmann parameters, (34) and (36) for the parameter values \(Gz\in\{1,2,3\}\) with \(z=1\), whose powers yield the \(\pm\mathbf{1}\) unity matrices [23]. We can proceed using numerical computation and recognize the \(\pm\pi\) values and its multiples, to write:  
\[\mathbf{B}_{1,1}=\left(\begin{smallmatrix}1&1\\ -1&0\end{smallmatrix}\right), \phi =-1.10715, \mu =0.223607+0.447214\,\mathrm{i}, \tag{43}\] \[\mathbf{B}_{1,1}^{3}=-\mathbf{1}, \phi =-\pi, \mu =0,\] \[\mathbf{B}_{1,1}^{6}=+\mathbf{1}, \phi =-2\pi, \mu =0;\] \[\mathbf{B}_{2,1}=\left(\begin{smallmatrix}1&1\\ -2&-1\end{smallmatrix}\right), \phi =-\tfrac{1}{2}\pi, \mu =\tfrac{1}{2}+\mathrm{i}\] (44) \[\mathbf{B}_{2,1}^{2}=-\mathbf{1}, \phi =-\pi, \mu =0,\] \[\mathbf{B}_{2,1}^{4}=+\mathbf{1}, \phi =-2\pi, \mu =0,\] \[\mathbf{B}_{2,1}^{6}=-\mathbf{1}, \phi =-3\pi, \mu =0;\] \[\mathbf{B}_{3,1}=\left(\begin{smallmatrix}1&1\\ -3&-2\end{smallmatrix}\right), \phi =-1.81577, \mu =0.606339+1.697750\,\mathrm{i},\] (45) \[\mathbf{B}_{3,1}^{3}=+\mathbf{1}, \phi =-2\pi, \mu =0,\] \[\mathbf{B}_{3,1}^{6}=+\mathbf{1}, \phi =-4\pi, \mu =0.\]  
For \(\phi=\pm\pi\), the \(\phi\)-factor in (34) is the first inversion matrix \(-\mathbf{1}\) while the \(\mu\)-factor, for \(\mu=0\), is the unit matrix; when \(\phi=\pm 2\pi\) the first factor is the unit matrix which is the symplectic unit that is the second metaplectic unit. Then, \(\phi=\pm 3\pi\) is again an inversion (metaplectically distinct from the previous one), while only \(\phi=\pm 4\pi\) returns the 'true' symplectic and metaplectic unit. Hence, while \(\mathbf{B}_{1,1}^{6}\), \(\mathbf{B}_{2,1}^{4}\), and \(\mathbf{B}_{3,1}^{3}\) yield the identity geometric system, in wave optics it will change the signal values by a minus sign; only the six-block setup \(\mathbf{B}_{3,1}^{6}\) truly reproduces the initial state as the symplectic and metaplectic unit. This is the optical realization of [PERSON]'s hexagon.  
One topic that we must skip is that of complex symplectic \(2\times 2\) matrices [24] which are used to describe linear systems that are diffusive [18] [Ch. 9]. Two cases of interest are diffusion itself, \(\mathbf{D}_{\tau}^{c}:=\left(\begin{smallmatrix}1&-\mathrm{i}\tau\\ 0&1\end{smallmatrix}\right)\), \(\tau\geq 0\), which has a widening Gaussian integral kernel that corresponds to the Gauss-Weierstrass transform, and the Bargmann transform \(\mathbf{B}:=\frac{1}{\sqrt{2}}\left(\begin{smallmatrix}1&-\mathrm{i}\\ -\mathrm{i}&1\end{smallmatrix}\right)\) that intertwines the Schrodinger oscillator energy states and their corresponding Schwinger power-function states [25]. The integral kernel (26) and its coefficient (31) present no problem as they are extended into the complex \((q,q^{\prime})\)-plane, but the unitarity--implying reversibility--of the transform does require a specific measure over the now-complex plane. This extension of the \(2\times 2\) group of matrices does not seem to apply directly to the geometric model of optics, so we mention it only for general interest.