I have noticed how similar the structure preserving property of Lorentz transformations and canonical transformations look, for them we have:
$$\Lambda^Tg\Lambda=g, \quad D^TJD=J,$$
with $\Lambda$ being a Lorentz transformation, $g$ is the metric, $D=\frac{\partial y}{\partial x}$ as the canonical transformation and $J$ being a symplectic form. My question, is if there is a similar structure preserving transformation in quantum mechanics? The reason I was suspecting this, is because of canonical relations (using the Poisson bracket):
$$\{x^\mu, p^\nu\}=g^{\mu\nu}, \quad \{q^i, p^j \}=\delta^{ij},$$
while in quantum mechanics, we have the commutator:
$$[\hat{X}^i, \hat{P}^j ] = i\hbar \delta^{ij}.$$
I "feel", like there must be a structure preserving transformation in quantum mechanics as well, similar to the Lorentz or the canonical transformation. But unfortunately I didn't find anything online, so thanks for any help!
