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        "quote": "The group $\\mathrm{SL}(2,\\mathbf R)$ is noncompact, and its representation theory differs greatly from that of $\\mathrm{SU}(2)$. In particular, every nontrivial [<u class=\"concept-term\" data-concept-reference=\"concepts/representation-theory/irreducible-representation\" data-concept-identity=\"urn:uuid:4e9e2c89-0422-42cf-8762-6b5e59837ef0\" data-concept-route=\"/concept-identities#concept-4e9e2c89-0422-42cf-8762-6b5e59837ef0\">irreducible representation</u>](../../../../../concepts/representation-theory/irreducible-representation) that is [<u class=\"concept-term\" data-concept-reference=\"concepts/representation-theory/unitary-representation\" data-concept-identity=\"urn:uuid:d06a0f44-cd99-4902-bac1-36de0284c944\" data-concept-route=\"/concept-identities#concept-d06a0f44-cd99-4902-bac1-36de0284c944\">unitary</u>](../../../../../concepts/representation-theory/unitary-representation) is infinite-dimensional. This is an important topic in mathematics, but beyond our scope. We will study one such representation, arising from the quantum-mechanical state space; it is a representation of a double cover of $\\mathrm{SL}(2,\\mathbf R)$, not of $\\mathrm{SL}(2,\\mathbf R)$ itself.",
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