Mathematical Optics

applicable when $[\hat{\mathrm{A}},\hat{\mathrm{B}}]=c\hat{\mathrm{I}}$, which indeed, through the consequent commutation relation

$e^{i\text{α}\hat{X}}{e}^{i\text{β}\hat{\mathrm{P}}}=e^{i\text{β}\hat{\mathrm{P}}}{e}^{i\text{α}\hat{X}}{e}^{-i\propto \text{β}},$

determines the group structure. In fact, the composition law for operators in $\mathfrak{M}$ explicitly writes as

${\hat{\mathrm{W}}}_2({\text{α}}_2,{\text{β}}_2,{\text{γ}}_2){\hat{\mathrm{W}}}_1({\text{α}}_1,{\text{β}}_1,{\text{γ}}_1)=\hat{\mathrm{W}}({\text{α}}_1+{\text{α}}_2,{\text{β}}_1+{\text{β}}_2,{\text{γ}}_1+{\text{γ}}_2+\frac{1}{2}({\text{α}}_1{\text{β}}_2-{\text{α}}_2{\text{β}}_1)).$

In contrast, the exponential map of $sl(2,\mathbb{C})$ into $SL(2,\mathbb{C})$ is not surjective. This means that not all the elements of $SL(2,\mathbb{C})$ are of exponential type, that is, expressible as e^K for some matrix K in the relevant algebra $sl(2,\mathbb{C})$. However, it is a remarkable result of the Lie group theory that for every matrix $\mathbf{O}\in SL(2,\mathbb{C})$ either O or −O (= −IO) is in the image of the exponential map of $sl(2,\mathbb{C})$ (Gallier 2012). Therefore, being −I = e^i(2πi)K₃, we see that every operator $\hat{\mathrm{O}}\in$