Stone's theorem on one-parameter unitary groups

In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert spaceand one-parameter families of unitary operators that are strongly continuous, i.e., and are homomorphisms, i.e., Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups.The theorem was proved by Marshall Stone (1930, 1932), and John von Neumann (1932) showed that the requirement thatbe strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable.This is an impressive result, as it allows one to define the derivative of the mappingis defined by means of the functional calculus, which uses the spectral theorem for unbounded self-adjoint operators.will be a bounded operator if and only if the operator-valued mappingfor which the limit exists in the norm topology.Part of the statement of the theorem is that this derivative exists—i.e., thatis a densely defined self-adjoint operator.The result is not obvious even in the finite-dimensional case, sinceis only assumed (ahead of time) to be continuous, and not differentiable.The family of translation operators is a one-parameter unitary group of unitary operators; the infinitesimal generator of this family is an extension of the differential operator defined on the space of continuously differentiable complex-valued functions with compact support onThus In other words, motion on the line is generated by the momentum operator.Stone's theorem has numerous applications in quantum mechanics.For instance, given an isolated quantum mechanical system, with Hilbert space of states H, time evolution is a strongly continuous one-parameter unitary group onThe infinitesimal generator of this group is the system Hamiltonian.Stone's Theorem can be recast using the language of the Fourier transform.are in one-to-one correspondence with strongly continuous unitary representations ofi.e., strongly continuous one-parameter unitary groups.-algebra of continuous complex-valued functions on the real line that vanish at infinity.Hence, there is a one-to-one correspondence between strongly continuous one-parameter unitary groups and *-representations ofcorresponds uniquely to a self-adjoint operator, Stone's Theorem holds.Therefore, the procedure for obtaining the infinitesimal generator of a strongly continuous one-parameter unitary group is as follows: The precise definition ofwith compact support, where the multiplication is given by convolution.It is a non-trivial fact that, via the Fourier transform,A result in this direction is the Riemann-Lebesgue Lemma, which says that the Fourier transform maps, satisfying the canonical commutation relation, and shows that these are all unitarily equivalent to the position operator and momentum operator onThe Hille–Yosida theorem generalizes Stone's theorem to strongly continuous one-parameter semigroups of contractions on Banach spaces.
mathematicsone-parameterunitary groupsfunctional analysisself-adjoint operatorsHilbert spaceunitary operatorsstrongly continuousMarshall StoneJohn von Neumann weakly measurable separablederivative mapping continuousLie groupsLie algebrasfunctional calculusspectral theoremextensioncompact supportmomentum operatorquantum mechanicstime evolutionHamiltonianStone–von Neumann theoremHeisenberg groupFourier transformgroup C*-algebraRiesz-Markov Theoremprojection-valued measureself-adjoint operatorconvolutionRiemann-Lebesgue Lemmacanonical commutation relationposition operatorHille–Yosida theoremcontractionsBanach spacesBibcodeAnnals of MathematicsProceedings of the National Academy of Sciences of the United States of AmericatopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointBoundedCompactHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryBanach algebraC*-algebraSpectrum of a C*-algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceSpectral theory of ordinary differential equationsHeat kernelIndex theoremCalculus of variationsIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theory