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        "quote": "We thus see that, at least in the case of finite dimensional $\\mathcal{H}$, the unitary representation $\\pi$ of $G$ on $\\mathcal{H}$ coming from an action of $G$ on our physical system gives us not just unitary matrices $\\pi ( g )$ , but also corresponding <u id=\"concept-1-self-adjoint-operator\" class=\"concept-term\" data-concept-id=\"concept-1-self-adjoint-operator\" data-concept-role=\"discussion\" data-concept-identity=\"urn:uuid:9f2dec8c-99ca-42d9-be8f-4f6c039a058a\" data-concept-route=\"/concept-identities#concept-9f2dec8c-99ca-42d9-be8f-4f6c039a058a\">self-adjoint operators</u> $B$ on $\\mathcal{H}$. Lie group actions thus provide us with a class of quantum mechanical observables, with the self-adjointness property of these operators corresponding to the unitarity of the representation on state space. It is a remarkable fact that for many physical systems the class of observables that arise in this way include the ones of most physical interest.",
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