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I'm wondering something small about the link between representations and quantization. For quantization you start with some phase space (symplectic manifold) $M$, and you have a classical observable $f$. You want to construct a Hilbert space $H$ and some quantum observable $\hat{f}$. If you have some symmetry group $G$ acting on $M$ you want $G$ also to be a symmetry for $H$. On $M$ it works by symplectomorphisms, but on $H$ it needs to preserve linear and unitary structure, so it acts on $H$ by unitary representations. My question is, does there even pop up some representation by assigning $f \mapsto \hat{f}$? How is this linked to representation theory. Doesn't representation theory only pop up for quantization when we want to discuss conserved symmetries?

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