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Dec 1983

Volume 24, Issue 12, pp. 2695-2882


Transformation coefficients of permutation groups

Jin‐Quan Chen, David F. Collinson, and Mei‐Juan Gao

J. Math. Phys. 24, 2695 (1983); http://dx.doi.org/10.1063/1.525668 (11 pages) | Cited 16 times

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The eigenfunction method is used to calculate the transformation coefficients 〈[ν]m‖[ν],τ[ν1][ν2]m1 m2〉 from the Yamanouchi basis of the permutation group Sf1+f2 to the Sf1+f2⊃Sf1⊗Sf2 irreducible basis. A program in fortran is written, and tables of the transformation coefficients for the permutation group Sf up to f=6 are given. Possible applications of the transformation coefficients are sketched.
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02.20.Qs General properties, structure, and representation of Lie groups
03.65.Fd Algebraic methods

Unitary representations of the (4+1)‐de Sitter group on irreducible representation spaces of the Poincaré group

P. Moylan

J. Math. Phys. 24, 2706 (1983); http://dx.doi.org/10.1063/1.525669 (16 pages) | Cited 14 times

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The construction of the principal continuous series of unitary representations of the simply‐connected covering group of the (4+1)‐de Sitter group on unitary irreducible representation spaces of the Poincaré group is presented. A unitary irreducible representation space of this covering group of the de Sitter group is realized as the direct sum of two irreducible representation spaces of the Poincaré group. Possible physical implications are indicated. In particular, an interpretation of the instantaneous velocity operator in the Dirac theory as the spin part of the de Sitter boosts is given. We obtain a simple method of computing the matrix elements of the generators of the de Sitter group in an SO(4) basis using the matrix elements of the generators of the four‐dimensional Euclidean group. Also we obtain explicit expressions for certain matrix elements between the spinor and SO(4) basis of the representation space as functions on the coset space SO(4)/SO(3).
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02.20.Qs General properties, structure, and representation of Lie groups
03.65.Pm Relativistic wave equations
11.30.Cp Lorentz and Poincaré invariance

General indices of simple Lie algebras and symmetrized product representations

S. Okubo and J. Patera

J. Math. Phys. 24, 2722 (1983); http://dx.doi.org/10.1063/1.525670 (12 pages) | Cited 21 times

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In many branches of physics, it is important to know the decomposition of a product representation ρ⊗ρ⊗⋅⋅⋅⊗ ρ (n times) of identical representations ρ of a simple Lie algebra into irreducible components with a given Young tableau symmetry. We show that the notion of representation indices introduced elsewhere is a very useful tool in dealing with this problem. We calculate explicit formula for general pth order indices D( p) ( ρ) for all classical simple Lie algebras. Sixth‐order indices for exceptional Lie algebras are also discussed.
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02.20.Sv Lie algebras of Lie groups

Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods

Alex J. Dragt and Etienne Forest

J. Math. Phys. 24, 2734 (1983); http://dx.doi.org/10.1063/1.525671 (11 pages) | Cited 43 times

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Lie algebraic methods are developed to describe the behavior of trajectories near a given trajectory for general Hamiltonian systems. A procedure is presented for the computation of nonlinear effects of arbitrarily high degree, and explicit formulas are given through effects of degree 5. Expected applications include accelerator design, charged particle beam and light optics, other problems in the general area of nonlinear dynamics, and, perhaps, with suitable modification, the area of S‐matrix expansions in quantum field theory.
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45.05.+x General theory of classical mechanics of discrete systems
02.20.Sv Lie algebras of Lie groups
02.30.Hq Ordinary differential equations
11.55.-m S-matrix theory; analytic structure of amplitudes

Generalized canonical transformations for time‐dependent systems

Manuel Asorey, José F. Cariñena, and Luis A. Ibort

J. Math. Phys. 24, 2745 (1983); http://dx.doi.org/10.1063/1.525672 (6 pages) | Cited 13 times

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We introduce the concept of generalized canonical transformations as symplectomorphisms of the extended phase space. We prove that any such transformation factorizes in a standard canonical transformation times another one that changes only the time variable. The theory of generating functions as well as that of Hamilton–Jacobi is developed. Some further applications are developed.
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45.05.+x General theory of classical mechanics of discrete systems
02.40.-k Geometry, differential geometry, and topology

Stochastic electrodynamics for the free particle

L. de la Peña and A. Jáuregui

J. Math. Phys. 24, 2751 (1983); http://dx.doi.org/10.1063/1.525673 (11 pages) | Cited 4 times

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The theory of stochastic electrodynamics is applied to the free particle and to the particle moving in a homogeneous field, leading to a complete temperature‐ and time‐dependent description in phase space. After a transient time, the marginal description in configuration space coincides entirely with quantum mechanics, while the phase‐space description is only mathematically related to the Wigner distribution. The Schrödinger equation appears as a natural—though incomplete—means of describing the statistical behavior of the electron under these conditions.
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03.65.Ta Foundations of quantum mechanics; measurement theory
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion

Some properties of the eigenfunctions of the dilated model Hamiltonians with complex potentials

Piotr Froelich

J. Math. Phys. 24, 2762 (1983); http://dx.doi.org/10.1063/1.525674 (2 pages) | Cited 1 time

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A pair of operators H(θ) and H(θ∗) obtained by dilation into opposite directions of a model Hamiltonian with nonreal potential is considered. Relations between the resonant eigenfunctions of H(θ) and H(θ∗) are studied.
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03.65.Ge Solutions of wave equations: bound states

Evolution theorem for a class of perturbed envelope soliton solutions

E. W. Laedke, K. H. Spatschek, and L. Stenflo

J. Math. Phys. 24, 2764 (1983); http://dx.doi.org/10.1063/1.525675 (6 pages) | Cited 40 times

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Envelope soliton solutions of a class of generalized nonlinear Schrödinger equations are investigated. If the quasiparticle number N is conserved, the evolution of solitons in the presence of perturbations can be discussed in terms of the functional behavior of N(η2), where η2 is the nonlinear frequency shift. For ∂η2N >0, the system is stable in the sense of Liapunov, whereas, in the opposite region, instability occurs. The theorem is applied to various types of envelope solitons such as spikons, relatons, and others.
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03.65.Ge Solutions of wave equations: bound states
52.35.-g Waves, oscillations, and instabilities in plasmas and intense beams

Extension of Fuda’s off‐shell analysis to screened Coulomb potentials for arbitrary l and limiting relations

Ranabir Dutt and Y. P. Varshni

J. Math. Phys. 24, 2770 (1983); http://dx.doi.org/10.1063/1.525676 (6 pages) | Cited 5 times

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The Ecker–Weizel approximation technique is applied to the Schrödinger equation for a class of screened Coulomb potentials (Yukawa, Exponential cosine screened Coulomb and Hulthén) for any arbitrary angular momentum l. We find that the centrifugal term can be combined with the central screening potential to generate an effective Eckart potential with energy dependent strength parameters for which the s‐wave Schrödinger equation is exactly solvable. Using this effective s‐wave potential in the formalism of Fuda and Whiting for off‐shell analysis, we obtain a closed expression for the off‐shell Jost solution fS,l (k,q,r) in which k is the on‐shell momentum, q is the off‐shell momentum and the subscript S means screening. It turns out that for nonzero angular momentum, usual Jost function  fS,l (k,q) can not be defined for finite screening parameter λ. However, we find that the Jost solution, as well as the Jost function defined in the limit λ → 0, show discontinuities at the on‐shell point q=k, similar to the observation made by van Haeringen [Phys. Rev. A 18, 56 (1978)] for the s‐wave Hulthén potential. For the l=0 case, we obtain explicit expressions for the off‐shell and on‐shell Jost solutions and Jost functions which possess the limiting behaviors discussed by van Haeringen for the Hulthén potential only. Our results not only extend previous works to higher partial waves, but at the same time indicate that certain limiting properties of the Jost solutions and the Jost functions are generally true for a class of screened Coulomb potentials.
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03.65.Nk Scattering theory

A new semiclassical interpretation of the Lamb shift

John T. F. Barwick

J. Math. Phys. 24, 2776 (1983); http://dx.doi.org/10.1063/1.525677 (4 pages) | Cited 1 time

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A modification of a previous semiclassical explanation of the Lamb shift is shown to be applicable to all levels in hydrogenic ions. The phenomenon responsible for the level shifts has not been considered explicitly in other quantum electrodynamic or semiclassical theories, but it is shown that it should be a source of observable energy changes. An approximate calculation for hydrogen s states gives ΔE1s=0.25748 cm−1 (experimental ΔE1s=0.2722 cm−1), ΔE2s=0.03755 cm−1 =1125.7 MHz (experimental ΔE2s=0.03528 cm−1), and ΔE3s=0.01166 cm−1 (experimental ΔE3s =0.0088 cm−1), but more important is the demonstration of an effect which should apparently be involved in any theory of the Lamb shift.
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03.65.Sq Semiclassical theories and applications

B∗‐algebra representations in a quaternionic Hilbert module

A. Soffer and L. P. Horwitz

J. Math. Phys. 24, 2780 (1983); http://dx.doi.org/10.1063/1.525656 (3 pages) | Cited 2 times

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It is shown that the Gel’fand–Naimark–Segal (GNS) construction can be generalized to real B∗‐algebras containing an algebra ∗‐isomorphic to the quaternion algebra by the use of quaternion linear functionals and Hilbert Q‐modules. An extension of the Hahn–Banach theorem to such functionals is proved.
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03.70.+k Theory of quantized fields
11.15.-q Gauge field theories

A one‐fixed‐point Killing parameter transform

J. P. Krisch

J. Math. Phys. 24, 2783 (1983); http://dx.doi.org/10.1063/1.525657 (3 pages) | Cited 1 time

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A single fixed‐point transformation which generates solutions to the field equations is discussed. The method is applied to several examples.
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04.20.Jb Exact solutions

Solutions of Einstein’s equations involving arbitrary functions

Peter Szekeres

J. Math. Phys. 24, 2786 (1983); http://dx.doi.org/10.1063/1.525658 (7 pages) | Cited 1 time

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Necessary and sufficient conditions are derived for a solution of Einstein’s vacuum equations to depend on an arbitrary function of some scalar function ϕ. Unlike the case of the scalar wave equation the constant surfaces of the function ϕ need not be null. This apparent anomaly is discussed.
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04.20.Jb Exact solutions
02.30.-f Function theory, analysis

The Cauchy problem for the R+R2 theories of gravity without torsion

P. Teyssandier and Ph. Tourrenc

J. Math. Phys. 24, 2793 (1983); http://dx.doi.org/10.1063/1.525659 (7 pages) | Cited 110 times

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The exterior Cauchy problem is discussed for the fourth‐order theories of gravity derived from the Lagrangian densities L=(−g)1/2 (R+ (1/2)aR2+bRμν Rμν) −κLm. When b≠0, the Cauchy problem can be solved by the standard method already used in general relativity. When b=0, the problem cannot be formulated as in the case where b≠0, since the corresponding fourth‐order theory is shown to be equivalent to a second‐order scalar–tensor theory. This scalar–tensor theory is proved to coincide with one of the models of gravity proposed by O’Hanlon in order to present a covariant version of the massive dilaton theory suggested by Fujii. This result is generalized: The models of O’Hanlon are shown to be indistinguishable from the fourth‐order theories derived from the Lagrangian densities L=(−g)1/2 F(R)−κLm, where F is any real function such that F″(R) does not identically vanish.
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04.50.-h Higher-dimensional gravity and other theories of gravity

A probabilistic rejection test for multivariable sensitivity analysis

Y. Ronen and A. Dubi

J. Math. Phys. 24, 2800 (1983); http://dx.doi.org/10.1063/1.525660 (9 pages)

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A probabilistic rejection test for multivariable sensitivity analysis is presented. The test is applied by randomly changing all the assumed unimportant (those having low sensitivity values) input parameters simultaneously and calculating the appropriate response. It is shown that by repeating this procedure N times, where N is much smaller than the number of input parameters, it is possible to assign a probability limit to the assumption that a high sensitivity parameter exists. The application of the test is demonstrated in a nuclear waste disposal problem.
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06.20.Dk Measurement and error theory
02.50.Cw Probability theory

A dense set of cyclic vectors for quantum field polynomial algebras

Wulf Driessler and Stephen J. Summers

J. Math. Phys. 24, 2809 (1983); http://dx.doi.org/10.1063/1.525661 (11 pages)

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It is shown that in the Hilbert space of a quantum field theory with a nonzero mass gap there exists a dense set of vectors, each entire analytic for the energy–momentum operators, that are cyclic for the polynomial algebra P(Rd) [and for the local polynomial algebras P(O), for any nonempty O ⊆ Rd]. It is proven that for every vector Φ from this dense set there exists an element Q ∊ P(Rd) such that QΦ=Ω, where Ω is the vacuum, and QΩ=0. A similar, stronger result is proven for free field theories (including mass zero).
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11.10.Cd Axiomatic approach

Galilean field theories and nonlocal S‐matrix symmetries

Wolf‐Dieter Garber

J. Math. Phys. 24, 2820 (1983); http://dx.doi.org/10.1063/1.525662 (3 pages)

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Any operator that commutes with the S matrix and is additive, i.e., transforms an asymptotic incoming n‐particle state as a sum of its constituent one‐particle states, is called a symmetry of the S matrix. The structure of local S‐matrix symmetries in Galilean field theories is known. In this paper, S‐matrix symmetries that are nonlocal, i.e., may transform asymptotic fields into nonlocal operators, are investigated under the assumption that there is a finite number of such symmetries in the theory.
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11.10.Lm Nonlinear or nonlocal theories and models
11.10.Jj Asymptotic problems and properties
11.55.Fv Dispersion relations

A Lagrangian of Bargmann–Wigner equations for massive particles of spin 2

M. Lorente and M. A. Rodriguez

J. Math. Phys. 24, 2823 (1983); http://dx.doi.org/10.1063/1.525663 (5 pages)

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Following the ideas of Guralnik and Kibble and those of Larsen and Repko, we introduce a general method to calculate the first‐order Lagrangian of Bargmann–Wigner equations (BWE) of arbitrary spin, and make explicit calculations in case of spin 2. Finally, some considerations on the motivation of this method and on the invariance of the Lagrangian under the symmetric group and the general Lorentz group are discussed.
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03.65.Pm Relativistic wave equations

Conformal QED

B. Binegar, C. Fronsdal, and W. Heidenreich

J. Math. Phys. 24, 2828 (1983); http://dx.doi.org/10.1063/1.525664 (19 pages) | Cited 49 times

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A conformally invariant quantum electrodynamics is constructed. The setting is realistic space‐time (rather than Euclidean), and a complete Gupta–Bleuler quantization scheme is carried out. Conformal invariance of the quantum field theory (as opposed to either classical field theory or to a theory defined by its Feynman rules) requires a richer Gupta–Bleuler structure than has been considered previously. Yet the essential features of this structure are preserved. The requirement that the wave equation be of second order fixes a unique action that already contains the gauge‐fixing terms that are required in any complete quantum field theory. The ‘‘Lorentz condition’’ turns out to be the transversality condition yαaα(y)=0 (in the manifestly covariant six‐dimensional notation); this condition has to be treated in the same way as the Lorentz condition ∂μAμ(x)=0 (four‐dimensional notation), as a boundary condition on the physical states.
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12.20.Ds Specific calculations

Cross sections with polarized spin‐1/2 particles in terms of helicity amplitudes

Till B. Anders and Wolfgang Jachmann

J. Math. Phys. 24, 2847 (1983); http://dx.doi.org/10.1063/1.525665 (8 pages) | Cited 7 times

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The derivation of cross sections for collisions with polarized particles of spin 1/2 can be simplified considerably if the scattering amplitude is calculated explicitly before the transition probability is obtained by a simple squaring. The states of the polarized particles are represented as superpositions of states with definite helicity. The coefficients of the superposition relate directly to the strength of the transversal and longitudinal polarization. The helicity amplitudes are products of helicity currents for which detailed formulas have been elaborated. Two‐component spinors have been used. All the contractions of vector indices are done with a new set of general formulas. The resulting cross sections show terms with separate factors due to the polarization, the energy, and the directions of the particles. Therefore, the high energy approximation can be achieved very conveniently. Applications of the described method have been performed to the scattering of electrons by electrons or positrons including the exchange of Z0 and Higgs particles.
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13.85.Lg Total cross sections
11.80.-m Relativistic scattering theory
12.20.-m Quantum electrodynamics

Quasiclassical trajectory‐coherent states of a particle in an arbitrary electromagnetic field

V. G. Bagrov, V. V. Belov, and I. M. Ternov

J. Math. Phys. 24, 2855 (1983); http://dx.doi.org/10.1063/1.525666 (5 pages) | Cited 28 times

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In this paper we show that for a nonrelativistic charged particle moving in an arbitrary external electromagnetic field there exist approximate solutions of the Schrödinger equation, such that the quantum‐mechanical averages of the coordinates and the momenta with respect to these states are general exact solutions of the classical Hamiltonian equations. Such states are called trajectory‐coherent states. The wave functions of the trajectory‐coherent states are obtained by the complex germ method by V. P. Maslov. The simplest properties of these states are studied.
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41.60.-m Radiation by moving charges
03.65.Ge Solutions of wave equations: bound states

On the Ohm–Navier–Stokes system in magnetohydrodynamics

Zensho Yoshida and Yoshikazu Giga

J. Math. Phys. 24, 2860 (1983); http://dx.doi.org/10.1063/1.525667 (5 pages) | Cited 10 times

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We study a mathematical structure of the Ohm–Navier–Stokes system that describes the incompressible dissipative evolution of a plasma. We apply the nonlinear semigroup theory and construct a unique regular solution which satisfies the system at least locally‐in‐time. We show that, for small initial data, this solution solves the system globally‐in‐time. We also introduce another scheme to construct solutions for less regular initial data.
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52.30.-q Plasma dynamics and flow
52.55.Jd Magnetic mirrors, gas dynamic traps
02.30.-f Function theory, analysis

Structural phase transitions in crystals: Broken‐symmetry (isotropy) groups

Marko V. Jarić

J. Math. Phys. 24, 2865 (1983); http://dx.doi.org/10.1063/1.525678 (18 pages) | Cited 11 times

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A systematic procedure ensuring the determination of all isotropy groups of a given representation of a space group is presented for the first time. Isotropy groups play an important role in various areas of theoretical solid state physics. For example, only isotropy groups may occur in a structural phase transition driven by an order parameter belonging to a given representation. The method uses the chain criterion directly on the image of the representation, employing a labeling of the matrices by space group elements. A notion of a substar of a wave vector associated with the representation is central to the method. Finally, as a detailed illustration the method is applied to a structural phase transition in A15 systems driven by an X‐point order parameter. The result agrees with previously reported ones.
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64.60.-i General studies of phase transitions
61.50.Ah Theory of crystal structure, crystal symmetry; calculations and modeling
02.20.-a Group theory
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