Stone–von Neumann theorem

In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators.[1][2][3][4] In quantum mechanics, physical observables are represented mathematically by linear operators on Hilbert spaces.Already in his classic book,[5] Hermann Weyl observed that this commutation law was impossible to satisfy for linear operators p, x acting on finite-dimensional spaces unless ħ vanishes.For notational convenience, the nonvanishing square root of ℏ may be absorbed into the normalization of p and x, so that, effectively, it is replaced by 1.The idea of the Stone–von Neumann theorem is that any two irreducible representations of the canonical commutation relations are unitarily equivalent.Since, however, the operators involved are necessarily unbounded (as noted above), there are tricky domain issues that allow for counter-examples.(There is also a discrete analog of the Weyl relations, which can hold in a finite-dimensional space,[6]: Chapter 14, Exercise 5  namely Sylvester's clock and shift matrices in the finite Heisenberg group, discussed below.)One would like to classify representations of the canonical commutation relation by two self-adjoint operators acting on separable Hilbert spaces, up to unitary equivalence.By Stone's theorem, there is a one-to-one correspondence between self-adjoint operators and (strongly continuous) one-parameter unitary groups.(For the explicit operators x and p defined above, these are multiplication by eitx and pullback by translation x → x + s.) A formal computation[6]: Section 14.2  (using a special case of the Baker–Campbell–Hausdorff formula) readily yieldsformally differentiating at 0 shows that the two infinitesimal generators satisfy the above canonical commutation relation.Since the operators involved are unbounded, technical issues prevent application of the Baker–Campbell–Hausdorff formula without additional domain assumptions.The problem thus becomes classifying two jointly irreducible one-parameter unitary groups U(t) and V(s) which satisfy the Weyl relation on separable Hilbert spaces.The answer is the content of the Stone–von Neumann theorem: all such pairs of one-parameter unitary groups are unitarily equivalent.[6]: Theorem 14.8  In other words, for any two such U(t) and V(s) acting jointly irreducibly on a Hilbert space H, there is a unitary operator W : L2(R) → H so thatWhen W is U in this equation, so, then, in the x-representation, it is evident that P is unitarily equivalent to e−itQ P eitQ = P + t, and the spectrum of P must range along the entire real line.More formally, there is a unique (up to scale) non-trivial central strongly continuous unitary representation.This was later generalized by Mackey theory – and was the motivation for the introduction of the Heisenberg group in quantum physics.[clarification needed] If the center does not map to zero, one has a more interesting theory, particularly if one restricts oneself to central representations.Thus the representation of the center of the Heisenberg group is determined by a scale value, called the quantization value (in physics terms, the Planck constant), and if this goes to zero, one gets a representation of the abelian group (in physics terms, this is the classical limit).Let G be a locally compact abelian group and G^ be the Pontryagin dual of G. The Fourier–Plancherel transform defined byIn fact, using the Heisenberg group, one can reformulate the Stone von Neumann theorem in the language of representation theory.Theorem —  For each non-zero real number h there is an irreducible representation Uh acting on the Hilbert space L2(Rn) byOne representation of the Heisenberg group which is important in number theory and the theory of modular forms is the theta representation, so named because the Jacobi theta function is invariant under the action of the discrete subgroup of the Heisenberg group.acting on holomorphic functions, satisfy the same commutation relations as the usual annihilation and creation operators, namely,In 1961, Bargmann showed that a∗j is actually the adjoint of aj with respect to the inner product coming from the Gaussian measure.By taking appropriate linear combinations of aj and a∗j, one can then obtain "position" and "momentum" operators satisfying the canonical commutation relations.The Heisenberg group Hn(K) is defined for any commutative ring K. In this section let us specialize to the field K = Z/pZ for p a prime.By the orthogonality relations for characters of representations of finite groups this fact implies the corresponding Stone–von Neumann theorem for Heisenberg groups Hn(Z/pZ), particularly: Actually, all irreducible representations of Hn(K) on which the center acts nontrivially arise in this way.Much of the early work of George Mackey was directed at obtaining a formulation[7] of the theory of induced representations developed originally by Frobenius for finite groups to the context of unitary representations of locally compact topological groups.
mathematicstheoretical physicsuniquenesscanonical commutation relationspositionmomentumoperatorsMarshall StoneJohn von Neumannquantum mechanicsobservableslinear operatorsHilbert spacesreal lineposition operatormomentum operatorreduced Planck constantcanonical commutation relationHermann Weylfinite-dimensionalbounded Wielandtnormed algebraSylvesterStone's theoremfunctional calculusBaker–Campbell–Hausdorff formulairreducibleHeisenbergmatrix mechanicsSchrödingerSchrödinger pictureHeisenberg groupWeyl algebraCCR algebraMackey theorycentral extensiontrivial bracketcenterFourier theorycenter of the algebrascalar matricesgroup algebrascalarslocally compact abelian groupPontryagin dualFourier–Plancherel transformgroup C*-algebraspectrumcrossed productirreducible representationscompact operatorsLie algebraLie groupidentity operatorirreducible representationunitarily inequivalentabsolute valuefinitenumber theorymodular formstheta representationJacobi theta functionautomorphismup to a scalarSchur's lemmaFourier transformunitaryPlancherel theoremFourier inversion formulaSegal–Bargmann spaceadditive groupcardinalitycharacter functionsorthogonality relationsinner product spaceGeorge Mackeyinduced representationsFrobeniusOscillator representationWigner–Weyl transformCCR and CAR algebrasMoyal productStone's theorem on one-parameter unitary groupsHille–Yosida theoremC0-semigroupvon Neumann, J.Mathematische AnnalenAnnals of MathematicsStone, M. H.Proceedings of the National Academy of Sciences of the United States of AmericaBibcodeWeyl, H.Springer-VerlagFunctional analysistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointCompactHilbert–SchmidtNormalTrace classTransposeUnboundedBanach 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