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1.
It is shown that the extremal phase-space representations of quantum mechanics can be expressed in terms of wave-functions on L2-spaces which are embedded in L2(Γ). In L2(Γ) all these representations are restrictions of a globally defined representation of the canonical commutation relations. The master Liouville space 2(Γ) over L2(Γ) can accommodate representations of both classical and quantum statistical mechanics, and serves as a medium for their comparison. As a specific example, a Boltzmann-type equation on 2(Γ) is considered in the classical as well as quantum context. 相似文献
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S.N.M Ruijsenaars 《Annals of Physics》1980,126(2):399-449
Starting from the tensor product of N irreducible positive energy representations of the Poincaré group describing N free relativistic particles with arbitrary spins and positive masses, we construct an interacting positive energy representation by modifying the total 4-momentum operator. We first make a transformation to a Hilbert space on which the free total 4-momentum operator equals the product of a dimensionless center-of-mass 4-vector (, k) and a free “reduced Hamiltonian” Hr0, which is a positive operator acting only on internal variables, and then replace Hr0 by an interacting reduced Hamiltonian Hr = Hr0 + V, where V commutes with the Lorentz group and is such that Hr is a positive operator. The resulting product form is shown to imply that the wave operators interwine the free and interacting representations so that the S-operator is Lorentz invariant. From a physical point of view the scheme is related to the framework first introduced by Bakamjian and Thomas, in which the Hamiltonian and boost generators are modified, but the above procedure makes a mathematically rigorous discussion much simpler. In the spin-zero case we introduce a natural generalization of the pair potentials of nonrelativistic N-particle Schrödinger theory to the present relativistic setting, study its scattering theory, and point out some problems that do not have analogs at the nonrelativistic level. In the case we propose, inspired by the Dirac equation, explicit reduced Hamiltonians to describe atomic energy levels and present arguments making plausible that their eigenvalues are in closer agreement with the experimental data than their nonrelativistic counterparts. We also consider extensions to arbitrary spin and, in the case, coupling of a quantized radiation field. In view of eventual applications to “completely integrable” one-dimensional field theories the case of one space dimension is studied as well, both in quantum mechanics and in classical mechanics. 相似文献
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K. Drühl 《Reports on Mathematical Physics》1984,20(1):11-19
Using the general theory of classical limit developed by the author, we show the existence of classical limit for positive energy representations of the Poincaré group of arbitrary spin. The resulting classical phase space is an orbit of in the dual of its Lie algebra corresponding to given mass and spin. 相似文献
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Ralph F Guertin 《Annals of Physics》1975,91(2):386-412
We give the generalized Foldy-Wouthuysen transformation for any 2(2J + 1)-component Poincaré-invariant Hamiltonian theory that describes free massive spin-J particles and that is subject to the conditions: (a) every observable is either Hermitian ( = +) or pseudo-Hermitian ( = ?3+?3) and (b) the theory is invariant under the discrete symmetries. The requirement that the Hamiltonian be defined in the rest frame specifies one and only one boost generator that is also defined at p = 0. 相似文献
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Ralph F Guertin 《Annals of Physics》1974,88(2):504-553
We discuss 2(2J + 1)-component Poincaré-invariant Hamiltonian theories that describe free particles of definite mass and spin and that are subject to the conditions (a) every observable is either Hermitian or pseudo-Hermitian (i.e., O = ?3O+?3) and (b) the theory is invariant under the discrete symmetries. Our treatment is based on the Heisenberg equations of motion and on the Lie algebra of the Poincaré group. Explicit formulas are found for the generators of this algebra, including the Hamiltonian H, and all relations between the operators Γ and H that are both necessary and sufficient for to generate Lorentz boosts are found. To illustrate the utility of our results, we apply them to obtain explicit generalizations of the Dirac equation to any spin, by requiring that Γ = 0, and of the Sakata-Taketani spin-0 and spin-1 equations to any spin, by requiring that . 相似文献
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A. Jamiołkowski 《Reports on Mathematical Physics》1974,5(3):415-424
Let 1 be the algebra of linear operators on the n-dimensional Hilbert space 1, and let 2 be the algebra of linear operators of the m-dimensional Hilbert space 2. Let (1, 2) denote the complex space of linear maps from 1 to 2. By a positive map we mean an element of the space (1, 2) (superoperator with respect to 1) which maps positive definite operators in 1 into positive definite operators in 2. The aim of this paper is to present an effective method which allows to verify whether a given superoperator Λ∈(1, 2) is a positive map. Besides that necessary and sufficient conditions for the positive definiteness of even-degree forms in many variables are given. 相似文献
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Canonical and covariant representations of Lie groups of the semidirect product form G = N⊙K with N Abelian, are analyzed in a fibre bundle framework. We exhibit first the relationship between both kinds of representations in such framework. Two complementary methods of selecting irreducible representations from the covariant ones are developed. The first one proceeds by restriction to an invariant subspace and is exemplified in the case of massive integer spin representations of the Poincaré group. The second method takes quotients and is particularly useful when we deal with reducible but indecomposable representations. A family of stepped gauge transformations is generated when the method is used to obtain the covariant massless integer helicity representations of the Poincaré group; the electromagnetic and gravitational gauge transformations are just the first two cases of such a family. 相似文献
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W.E. Cleland A. Delfosse P.-A. Dorsaz J.-L. Gloor M.N. Kienzle-Focacci G. Mancarella A.D. Martin M. Martin P. Muhlemann C. Nef T. Pal J. Rutschmann H. Zeidler 《Nuclear Physics B》1982,208(2):228-261
The mass and momentum transfer spectra of the charged system produced in the reaction π±p→Ks0K±p are analyzed. The data have been collected at the CERN SPS with the Geneva-Lausanne two-arm, non-magnetic spectrometer at 30 and 50 GeV/c incident momenta. The general features of the reactions at these energies and the results of partial-wave analyses of the two kaon system are presented.The channel is dominated by the diffractive production of even spin resonances. The spin 4 recurrence of the A2(1320) is clearly observed at 2040 MeV (Γ=380 MeV. A new resonance is observed with a mass M=2450MeV and a width Γ=400 MeV; the quantum numbers of this state are found to be IG(JPC)=1?(6++). The analysis also shows the decay of the decay of the meson ?′(1600) through the channel at both energies.The production amplitudes are determined both as a function of the effective mass and of the momentum transfer. Isoscalar natural parity exchange is dominant. The energy dependence between 10 and 50 GeV/c is shown to be well described by a Regge pole model based on the f-dominated pomeron hypothesis. We compare the production mechanisms of the 2+ resonances A2(1320) and K1(1430). Finally, we estimate the branching ratios of the spin 4 A2(2040) and spin 6 A2(2450) resonances. 相似文献
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E. Angelopoulos 《Reports on Mathematical Physics》1977,11(2):239-258
All irreducible (unitary or not) ray representations of SL(3, R) obeying the Δj = 2 selection rule imposed by Regge trajectories are constructed. They provide irreducible ray representations of which restricted to the Poincaré subgroup yield unitary representation of real mass and of spin spectrum which statisfies the Δj = 2 selection rule. 相似文献
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Covariant fields: Poincaré group representations and metric structure in the space of quantum states
P M Mathews 《Pramana》1974,3(4):261-276
The representations of the Poincaré group realized over the space of covariant fields transforming according to any irreducible
representationD
(m,n) of the Lorentz group are constructed explicitly with reference to a helicity basis. The representation is indecomposable
in the massless case. The form of this representation together with the invariance of two-point Wightman functions of the
field (which follows from a weak set of axioms) determines the metric structure in the space of quantum states of the field.
This structure is explicitly determined for generalD
(m,n). Certain particular cases (especially the symmetric traceless tensor field) are discussed in detail. Finally we consider
the representation pertaining to massive fields, and examine the passage to the limit of vanishing mass. We present a limiting
procedure which leads from the unitary representation of the massive field to the indecomposable non-unitary representation
of the massless field. 相似文献
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A 2(2J + 1)-component relativistic Hamiltonian H that describes free particles of mass m and spin J is said to be linear if it has the form , where , h is a numerical factor, and g commutes with x and p. All such Hamiltonians are found, provided that the metric is either the unit matrix or ?3 and provided that the theory is invariant under the discrete symmetries. If the operator Γ in the generator of Lorentz boosts is required to be local, there are only two possibilities; either Γ = 0, which generalizes the Dirac spin- theory, or , which generalizes the Sakata-Taketani spin-0 and spin-1 theories. The relationship to linear manifestly covariant equations and its significance is discussed. 相似文献
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Bernhard F. Gächter 《Journal of Molecular Spectroscopy》1976,63(1):1-22
The irreducible components of the Raman scattering tensor operator under the symmetry of a general point group are calculated. The unitary transformations U(ΓγΓksΓk′s′, ρσ) from the Cartesian and spherical components, respectively, to the irreducible components for the 32 crystallographic point groups are collected in tables. As an example the unitary transformation U(ΓγΓksΓk′s′, ρσ) is used to discuss the behavior of the scattering tensor in a resonance Raman experiment. With the help of the general formalism the scattering tensor for electronic Raman transitions of transition metal ions is calculated. As an example the scattering tensors of electronic Raman transitions within the 5T2 state of the high-spin trigonal distorted octahedral Fe2+ are calculated and the refinement of the selection rules is discussed. 相似文献
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The proton spin-lattice relaxation time, T1, is measured as a function of temperature in α -(COOH)2·-2H2O, K2HgCl4· H2O and LiCHO·H2O. The relaxation is caused by 180° flips of the water molecules about their 2-fold axes and good agreement is obtained between calculated and observed values of T1. Empiricly the flip rate follows a classical Arrhenius equation: P· exp . A literature survey of values of P and ΔH obtained from similar investigations on other hydrates is given. The survey shows that the preexponential factor, P, is a function of the activation enthalpy, ΔH. P increases from 1012 to 1017 Hz when ΔH changes from 2 to 17 . Using a dynamical rate theory as formulated by Feit, we find the flip rate is given by: K2· √(ΔH)· exp (K1ΔH)· exp . This expression can be fitted to the observed data using K1 = 0.69 and K2 = 2 × 1011 Hz · . Thus both the frequency factor, K2√ (ΔH), and the entropic factor, exp (K1ΔH), have been obtained for flipping water molecules in hydrates. The values of K1 and K2 are shown to be physically reasonable. 相似文献
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J. Heintze G. Heinzelmann P. Igo-Kemenes R. Mundhenke H. Rieseberg B. Schürlein H.W. Siebert V. Soergel H. Stelzer K.-P. Streit A.H. Walenta 《Nuclear Physics B》1979,149(3):365-380
The decay K+ → e+υγ has been investigated. For the structure-dependent part with positive γ-helicity (SD+) the branching ratio is obtained from 51 ± 3 events observed in the kinematical region Ee ? 235 MeV, Eγ > 48 MeV and θeγ > 140°. For the corresponding part with negative γ-helicity we obtain an upper limit Γ(SD?)/Γ(SD+) < 11 (90% CL) from the sample of electrons with energies 220 MeV ? Ee < 230 MeV and with no γ in the backward direction. This upper limit implies that the ratio of structure-dependent axial vector amplitudes lies outside the region .For the decay the limit 90% confidence level) was found. 相似文献