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In quantum optics, I am trying to explore the group generated by squeezing and rotation operators. These are closely related to area-preserving linear transforms, which they induce on the phase space, and indeed are generated by 3 quantum mechanical operators $$\begin{align} \hat K_+ &= \frac{(\hat a^\dagger)^2}2 \\ \hat K_- &= \frac{\hat a^2}2 \\ \hat K_3 &= \frac{\hat a \hat a^\dagger + \hat a^\dagger \hat a}4 \end{align}$$ which follow $sl(2,\mathbb{R})$ commutation relations, $$[\hat K_3, \hat K_\pm] = \pm \hat K_\pm, \quad [\hat K_-, \hat K_+] = 2\hat K_3.$$

It takes a lot of work to derive results like ordering theorems—expansion of an exponential of a linear combination of generators into a product of separate exponentials of the generators—but I have been successful at reaching the same formulas in $SL(2,\mathbb{R})$ with ease. An example of such formula is a rather well-known ordering theorem of the squeezing operator $$\exp(-re^{i\phi} \hat K_+ + re^{-i\phi} \hat K_-) = \exp(-e^{i\phi} \tanh r \hat K_+) \cdot \exp(-2\ln(\cosh r) \hat K_3) \cdot \exp(e^{-i\phi} \tanh r \hat K_-)$$

The problem is that the actual group is different; namely, $\exp(4 \pi i \hat K_3)$ is minus identity, an operator which leaves phase space (pseudo-)distribution intact but inverts the sign of the wave function, resemblant of the sign change in fermionic rotation by $2\pi$. This is an additional degree of freedom outside $SL(2,\mathbb{R})$, suggesting that the group generated by $\{\hat K_\pm,\hat K_3\}$ might be a double cover thereof ($Mp_2$, according to Wikipedia). I'm not sure how to prove it is not even a higher order cover, but this is out of the scope of this question.

The formulas I work with seem to have no problem with this difference but I am afraid the generalization might break up at some point. My question is, under what conditions is it justifiable to extend the validity of theorems valid for the matrix representation to other representations as well? What should one be cautious of in this step?

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