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  "body": "\n## Statement\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]].\nLet $(U(t))_{t \\in \\R}$ be a family of [[unitary operators|MM0EJ2GDE4]] on $H$ such that:\n\n1. $U(0) = I$ and $U(s + t) = U(s)U(t)$ for all $s, t \\in \\R$;\n2. $t \\mapsto U(t)x$ is continuous from $\\R$ to $H$ for every $x \\in H$.\n\nThen there is a unique [[self-adjoint operator|0NEGN5CRSC]] $A$ on $H$ with $U(t) = e^{itA}$ for all $t \\in \\R$.\nIts domain is the set of $x$ for which $\\lim_{t \\to 0} (U(t)x - x)/t$ exists, and $Ax = \\lim_{t \\to 0} (U(t)x - x)/(it)$.\nConversely, for every self-adjoint operator $A$ on $H$, the operators $e^{itA}$, defined by the [[spectral theorem|18KPYRRPZY]], satisfy clauses 1 and 2.\n\n## Remarks\n\nThe converse is a direct consequence of the spectral theorem and dominated convergence.\nFor the main direction, $A$ is defined by the limit in the statement.\nVectors $\\int \\varphi(t) U(t)x \\, dt$ with smooth compactly supported $\\varphi$ lie in its domain, so $A$ is densely defined.\nThe group law shows that $A$ is symmetric, and the vectors $\\int_0^\\infty e^{-t} U(\\pm t)x \\, dt$ show that $A \\pm iI$ are onto, which makes $A$ self-adjoint.\n\nExample: on $L^2(\\R)$, the translations $(U(t)f)(s) = f(s + t)$ form such a group.\nIts generator is $A = -i \\, d/ds$ with domain the Sobolev space $H^1(\\R)$, the momentum operator of quantum mechanics.\n\nStrong continuity in clause 2 is needed only in a weak form: on a separable Hilbert space, weak measurability of $t \\mapsto \\ip{U(t)x}{y}$ already implies it, by a theorem of von Neumann.\n\n## Explanations\n\n### L1 human\n\nA one-parameter unitary group is a way of moving the vectors of a Hilbert space continuously in time without changing lengths.\nStone's theorem says that every such motion has the form $e^{itA}$ for one self-adjoint operator $A$.\nIn quantum mechanics, the motion is the time evolution of a state and $A$ is the energy, the Hamiltonian.\nSo the theorem explains why observables and symmetries correspond to self-adjoint operators.\n\n### L2 human\n\nThe theorem is the unbounded counterpart of the fact that a continuous group of unitary matrices is $e^{itA}$ for a Hermitian matrix $A$.\nIt gives a bijection between self-adjoint operators and strongly continuous unitary groups.\nIt is the special case of semigroup theory where the semigroup extends to negative times and each operator is unitary; compare the [[Lumer–Phillips theorem|Z2N8MPD8DH]].\nIt is the starting point of the representation theory of $\\R$, and with the SNAG theorem, of locally compact abelian groups.\n",
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