Volume 24, Issue 12, December 1983
Index of content:

Transformation coefficients of permutation groups
View Description Hide DescriptionThe eigenfunction method is used to calculate the transformation coefficients from the Yamanouchi basis of the permutation group S _{ f 1+f 2 } to the S _{ f 1+f 2 }⊃S _{ f 1 }⊗S _{ f 2 } irreducible basis. A program in fortran is written, and tables of the transformation coefficients for the permutation group S _{ f } up to f=6 are given. Possible applications of the transformation coefficients are sketched.

Unitary representations of the (4+1)‐de Sitter group on irreducible representation spaces of the Poincaré group
View Description Hide DescriptionThe 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 Diractheory 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).

General indices of simple Lie algebras and symmetrized product representations
View Description Hide DescriptionIn 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.

Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods
View Description Hide DescriptionLie 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 particlebeam 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.

Generalized canonical transformations for time‐dependent systems
View Description Hide DescriptionWe 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 generatingfunctions as well as that of Hamilton–Jacobi is developed. Some further applications are developed.

Stochastic electrodynamics for the free particle
View Description Hide DescriptionThe 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.

Some properties of the eigenfunctions of the dilated model Hamiltonians with complex potentials
View Description Hide DescriptionA 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.

Evolution theorem for a class of perturbed envelope soliton solutions
View Description Hide DescriptionEnvelope 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(η^{2}), where η^{2} is the nonlinear frequency shift. For ∂_{η2 } N >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.

Extension of Fuda’s off‐shell analysis to screened Coulomb potentials for arbitrary l and limiting relations
View Description Hide DescriptionThe 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 momentuml. 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 f _{ S,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 f _{ S,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 1 8, 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.

A new semiclassical interpretation of the Lamb shift
View Description Hide DescriptionA 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 ΔE _{1s }=0.25748 cm^{−1} (experimental ΔE _{1s }=0.2722 cm^{−1}), ΔE _{2s }=0.03755 cm^{−1} =1125.7 MHz (experimental ΔE _{2s }=0.03528 cm^{−1}), and ΔE _{3s }=0.01166 cm^{−1} (experimental ΔE _{3s } =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.

B*‐algebra representations in a quaternionic Hilbert module
View Description Hide DescriptionIt 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.

A one‐fixed‐point Killing parameter transform
View Description Hide DescriptionA single fixed‐point transformation which generates solutions to the field equations is discussed. The method is applied to several examples.

Solutions of Einstein’s equations involving arbitrary functions
View Description Hide DescriptionNecessary 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.

The Cauchy problem for the R+R ^{2} theories of gravity without torsion
View Description Hide DescriptionThe exterior Cauchy problem is discussed for the fourth‐order theories of gravity derived from the Lagrangian densities L=(−g)^{1/2} (R+ (1/2)a R ^{2}+b R _{μν} R ^{μν}) −κL_{ m }. 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)−κL_{ m }, where F is any real function such that F″(R) does not identically vanish.

A probabilistic rejection test for multivariable sensitivity analysis
View Description Hide DescriptionA 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.

A dense set of cyclic vectors for quantum field polynomial algebras
View Description Hide DescriptionIt 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 polynomialalgebraP(R^{ d }) [and for the local polynomialalgebrasP(O), for any nonempty O ⊆ R^{ d }]. It is proven that for every vector Φ from this dense set there exists an element Q ∈ P(R^{ d }) 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).

Galilean field theories and nonlocal S‐matrix symmetries
View Description Hide DescriptionAny 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.

A Lagrangian of Bargmann–Wigner equations for massive particles of spin 2
View Description Hide DescriptionFollowing 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.

Conformal QED
View Description Hide DescriptionA 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 y _{α} a _{α}(y)=0 (in the manifestly covariant six‐dimensional notation); this condition has to be treated in the same way as the Lorentz condition ∂_{μ} A _{μ}(x)=0 (four‐dimensional notation), as a boundary condition on the physical states.

Cross sections with polarized spin‐1/2 particles in terms of helicity amplitudes
View Description Hide DescriptionThe 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 Z _{0} and Higgs particles.