Segal–Bargmann space

It is a Hilbert space with respect to the associated inner product: The space was introduced in the mathematical physics literature separately by Bargmann and Segal in the early 1960s; see Bargmann (1961) and Segal (1963).Basic information about the material in this section may be found in Folland (1989) and Hall (2000) .Segal worked from the beginning in the infinite-dimensional setting; see Baez, Segal & Zhou (1992) and Section 10 of Hall (2000) for more information on this aspect of the subject.A basic property of this space is that pointwise evaluation is continuous, meaning that for eachthere is a constant C such that It then follows from the Riesz representation theorem that there exists a unique Fa in the Segal–Bargmann space such that The function Fa may be computed explicitly as where, explicitly, The function Fa is called the coherent state (applied in mathematical physics) with parameter a, and the function is known as the reproducing kernel for the Segal–Bargmann space.Note that meaning that integration against the reproducing kernel simply gives back (i.e., reproduces) the function F, provided, of course that F is an element of the space (and in particular is holomorphic).Note that It follows from the Cauchy–Schwarz inequality that elements of the Segal–Bargmann space satisfy the pointwise bounds One may interpret a unit vector in the Segal–Bargmann space as the wave function for a quantum particle moving inThe restriction that F be holomorphic is essential to this interpretation; if F were an arbitrary square-integrable function, it could be localized into an arbitrarily small region of the phase space, which would go against the uncertainty principle.Since, however, F is required to be holomorphic, it satisfies the pointwise bounds described above, which provides a limit on how concentrated F can be in any region of phase space.Since the above quantity is manifestly non-negative, it cannot coincide with the Wigner function of the particle, which usually has some negative values.Indeed, Bargmann was led to introduce the particular form of the inner product on the Segal–Bargmann space precisely so that the creation and annihilation operators would be adjoints of each other.We may now construct self-adjoint "position" and "momentum" operators Aj and Bj by the formulas: These operators satisfy the ordinary canonical commutation relations, and it can be shown that they act irreducibly on the Segal–Bargmann space; see Section 14.4 of Hall (2013).Since the operators Aj and Bj from the previous section satisfy the Weyl relations and act irreducibly on the Segal–Bargmann space, the Stone–von Neumann theorem applies.The map B may be computed explicitly as a modified double Weierstrass transform, where dx is the n-dimensional Lebesgue measure onOne can also describe (Bf)(z) as the inner product of f with an appropriately normalized coherent state with parameter z, where, now, we express the coherent states in the position representation instead of in the Segal–Bargmann space.We may now be more precise about the connection between the Segal–Bargmann space and the Husimi function of a particle.Another useful inversion formula is[1] where This inversion formula may be understood as saying that the position "wave function" f may be obtained from the phase-space "wave function" Bf by integrating out the momentum variables.This is to be contrasted to the Wigner function, where the position probability density is obtained from the phase space (quasi-)probability density by integrating out the momentum variables.The various Gaussians appearing in the ordinary Segal–Bargmann space and transform are replaced by heat kernels.This generalized Segal–Bargmann transform could be applied, for example, to the rotational degrees of freedom of a rigid body, where the configuration space is the compact Lie groups SO(3).
mathematicsIrving SegalValentine Bargmannholomorphic functionsHilbert spaceRiesz representation theoremcoherent statein mathematical physicsreproducing kernelCauchy–Schwarz inequalityWigner functionHusimi functionStone–von Neumann theoremWeierstrass transformLie groupheat kernelscoherent statesloop quantum gravityTheta representationHardy spaceLynne H. WallingBargmann, V.Communications on Pure and Applied MathematicsSegal, I. E.Folland, G.Princeton University PressBaez, J.Springer VerlagFunctional analysistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological 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 conjectureSpectral theory of ordinary differential equationsHeat kernelIndex theoremCalculus of variationsFunctional calculusIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theoryHilbert spacesInner productL-semi-inner productPrehilbert spaceOrthogonal complementOrthonormal basisBessel's inequalityHilbert projection theoremParseval's identityPolarization identityCompact operator on Hilbert spaceDensely definedSelf-adjointSesquilinear formCn(K) with K compact & n<∞