Index of content:
Volume 26, Issue 5, May 1985

On reduction of two degrees of freedom Hamiltonian systems by an S ^{1} action, and SO_{0}(1,2) as a dynamical group
View Description Hide DescriptionReduction by an S ^{1} action is a method of finding periodic solutions in Hamiltonian systems, which is known rather as the method of averaging. Such periodic solutions can be reconstructed as S ^{1} orbits by pulling back the critical points of an associated ‘‘reduced Hamiltonian’’ on a ‘‘reduced phase space’’ along the reduction. For Hamiltonian systems of two degrees of freedom, a geometric setting of the reduction is already accomplished in the case where the reduced phase space is a two‐sphere in the Euclidean space R^{3}, and the reduced Hamilton’s equations of motion are Euler’s equations. This article deals with the case where the reduced phase space will be a two‐hyperboloid in the three‐Minkowski space, and the reduced Hamilton’s equations of motion will be Euler’s equations with respect to the Lorentz metric. This reduction is associated with SU(1,1) symplectic action on the phase space R^{4}. As a consequence of this association the reduced Hamiltonian system proves to admit a dynamical group SO_{0}(1,2). A well‐known reduction by an S ^{1} action occurs in the case of rotational‐invariant Hamiltonian systems, which will be associated with SL(2,R) symplectic action on R^{4}. It is shown that the reduction associated with SU(1,1) and with SL(2,R) are symplectically equivalent.

A pair of commuting scalars for G(2) ⊇ SU(2)×SU(2)
View Description Hide DescriptionG(2) ⊇ SU(2)×SU(2) is a two‐missing‐labels problem, and therefore in order to give a complete and orthogonal specification of states of irreducible representations of G(2) in an SU(2)×SU(2) basis, one needs to find a pair of commuting Hermitian operators which are scalar with respect to the SU(2)×SU(2) subalgebra. A theorem due to Peccia and Sharp states that there are, apart from the Lie algebra invariants, twice as many functionally independent scalars as missing labels. Here two commuting SU(2)×SU(2) scalars are obtained, both of sixth order in the G(2) basis elements. They are in fact combinations of five scalars of different tensorial types, indicating that the functionally independent ones are in general insufficient to provide the lowest‐order commuting scalars. An expression for the sixth‐order invariant of G(2) is also obtained.

Application of generating function techniques to Lie superalgebras
View Description Hide DescriptionThe construction of generating functions for multiplicities of irreducible representations of Lie superalgebras from generating functions for characters is examined, and applied in order to obtain polynomialtensors and branching rules. Problems arising because of the existence of typical and atypical representations are discussed in detail. The techniques are applied to osp(1,2), spl(1,2), osp(3,2), and osp(4,2).

Irreducible representations of the exceptional Lie superalgebras D(2,1;α)
View Description Hide DescriptionThe shift operator technique is used to give a complete analysis of all finite‐ and infinite‐dimensional irreducible representations of the exceptional Lie superalgebrasD(2,1;α). For all cases, the star or grade star conditions for the algebra are investigated. Among the finite‐dimensional representations there are no star and only a few grade star representations, but an infinite class of infinite‐dimensional star representations is found. Explicit expressions are given for the ‘‘doublet’’ representation of D(2,1;α). The one missing label problem D(2,1;α)→su(2)+su(2)+su(2) is discussed in detail and solved explicitly.

On U(1)‐gauge fields and their geometrical interpretation
View Description Hide DescriptionThe Harnad–Shnider–Vinet study of symmetry properties on gauge fields in terms of invariant connections on principal fiber bundles is reviewed in the simple U(1)‐gauge theory. It is extended to the case of invariant electromagnetic fields admitting nontrivial extensions of their symmetry groups. Some specific examples are discussed.

Linearization of nonlinear differential equations by means of Cauchy’s integral
View Description Hide DescriptionIt is suggested to transform a class of nonlinear differential equations with a holomorphic type of nonlinearity into linear integrodifferential equations. The method is presented in detail for first‐order ordinary differential equations. The transformedequation is studied and is found to have a unique solution with an analytical representation. In a numerical test calculation rapid convergence of an approximate solution of the linearized equation towards the reference solution is found. The method can be applied to higher‐order ordinary and partial differential equations. The transformation can be generalized also to operator‐valued differential equations.

Compact expression for Löwdin’s alpha function
View Description Hide DescriptionA simple analytical expression for Löwdin’s alpha function is derived. This expression is expected to be more convenient with the expansion of a general function about a displaced center than any other available in the literature.

The Hamiltonian structure of a complex version of the Burgers hierarchy
View Description Hide DescriptionIn this paper we construct a class of integrable Hamiltonian nonlinear evolution equations generated by a purely differential recursion operator. It turns out that this hierarchy is a complex version of the Burgers hierarchy and can be linearized through a generalization of the Cole–Hopf transformation.

Field equations and the tetrad connection
View Description Hide DescriptionA fundamental result of Geroch is that a space‐time admits a spinor structure if and only if it is parallelizable. A nonsymmetric, metric‐compatible curvature‐free connection is associated with a global orthonormal tetrad field on such a parallelizable space‐time. This connection is used to examine reported inconsistencies for S> 1/2 spinor field equations on general space‐times. It is shown that the assumed Levi–Civita transport of Clifford units causes the inconsistencies at the Klein–Gordon stage. The relation of the torsion tensor of the parallelization connection to the space‐time topology is indicated and the Lorentz covariance of the modified Klein–Gordon equations is demonstrated. A particularly simple plane‐wave solution form for free‐field equations is shown to result for locally flat space‐times for which the torsion tensor is necessarily zero.

Numerical integration in many dimensions. I
View Description Hide DescriptionIf a d‐dimensional integral involves an integrand of the functional form F ( f _{1}(x _{1})+f _{2}(x _{2})+⋅⋅⋅), then one can introduce an integral transform (Fourier or Laplace or variants on those) which allows all the integrals over the coordinates x _{ i } to factor. Thus a d‐dimensional integral is reduced to a one‐dimensional integral over the transform variable. This is shown to be a very powerful and practical numerical approach to a number of problems of interest. Among the examples studied is the computation of the volume of phase space for an arbitrary collection of relativistic particles. One important aspect of the approach involves numerical integration along various contours in the complex plane.

Numerical integration in many dimensions. II
View Description Hide DescriptionTwo new techniques are presented that appear to be useful in obtaining accurate numerical values for the numerical integration of fairly smooth functions in many dimensions. Both methods start with the idea of a mesh containing n points laid out in each of the d dimensions, then seek strategies that use far less than all n ^{ d } points in some systematically improved sequence of approximations.

On the intrinsic behavior of the internal variable in the Finslerian gravitational field
View Description Hide DescriptionA structural extension of the gravitational field is attempted in reference to the theory of Finsler spaces: The vector y is attached to each point x as the internal variable and the intrinsic behavior of y is reflected in the whole spatial structure.

A summation method for the Rayleigh–Schrödinger series for the anharmonic oscillator
View Description Hide DescriptionWe approximate the energy levels of the anharmonic oscillator with any coupling constant by eigenvalues λ_{ j }( g,T) of the operator −d ^{2}/d x ^{2}+x ^{2}+g V _{ T }(x) with V _{ T }(x) =x ^{4} when ‖x‖≤T and V _{ T }(x) =T ^{4} when ‖x‖>T. The functions λ_{ j }( g,T) are holomorphic with respect to g in a neighborhood of the non‐negative half‐axis. The conformal transformation maps this neighborhood onto the unit circle of the complex plane. It gives the summation method for the Rayleigh–Schrödinger series for every g≥0.

On the propagation of the operator average in truncated space
View Description Hide DescriptionThe propagation of the operator average is described in truncated space with some quantum number(s) being fixed. It is first shown that the propagation coefficient satisfies an analog of the Chapman–Kolmogorov equation. Next, particle‐hole symmetry is incorporated into the propagation of the operator average. It yields an expression that facilitates evaluation of many‐body trace. Fermion and boson systems are treated alike.

Strong approximation of time evolution operators for a finite system of oscillators with nonlinear coupling
View Description Hide DescriptionWe consider a system in three space dimensions consisting of a finite number of oscillators with a nonlinear interaction. Using projectors on N‐particle subspaces of the Fock space, we show that the time evolution operator is strongly approximatable by exponentials of self‐adjoint finite‐rank operators (finite‐dimensional Hermitian matrices), which can easily be calculated in the corresponding eigenrepresentation.

Lie algebras for systems with mixed spectra. I. The scattering Pöschl–Teller potential
View Description Hide DescriptionStarting from an N‐body quantum space, we consider the Lie‐algebraic framework where the Pöschl–Teller Hamiltonian, − 1/2 ∂^{2} _{χ} +c sech^{2} χ+s csch^{2} χ, is the single sp(2,R) Casimir operator. The spectrum of this system is m i x e d: it contains a finite number of negative‐energy bound states and a positive‐energy continuum of free states; it is identified with the Clebsch–Gordan series of the D^{+}×D^{−} representation coupling. The wave functions are the sp(2,R) Clebsch–Gordan coefficients of that coupling in the parabolic basis. Using only Lie‐algebraic techniques, we find the asymptotic behavior of these wave functions; for the special pure‐trough potential (s=0) we derive thus the transmission and reflection amplitudes of the scattering matrix.

The semi‐classical expansion for a charged particle on a curved space background
View Description Hide DescriptionWe give the semi‐classical expansion, with remainder to any order in ℏ, for the wave function of a nonrelativistic quantum particle in a classical external magnetic field on a curved space background. The basic assumption is of a ‘‘no caustics condition’’ on the underlying classical mechanics, at least up to the time in question. The gauge invariance of the result is emphasized together with a discussion of the geometric meaning of the classical mechanical quantities involved.

Hamiltonian representation for helically symmetric magnetic fields
View Description Hide DescriptionIt is proved that one of the two potentials used in a standard representation of helically symmetric magnetic fields is a Hamiltonian that generates the magnetic lines of force. It is proved also that if the other potential is restricted to be proportional to this Hamiltonian, then the radial components of current density and magnetic field are proportional. This restriction applies, for example, to constant‐λ, helically symmetric, force‐free magnetic fields [j(r)=λB(r)] relevant to both fusion physics and astrophysics.

Ladder and cross terms in second‐order distorted Born approximation
View Description Hide DescriptionIn the strong fluctuation theory for a bounded layer of random discrete scatterers, the second moments of the fields in the second‐order distorted Born approximation are obtained for copolarized and cross‐polarized fields. The backscattering cross sections per unit area are calculated by including the mutual coherence of the fields due to the coincidental ray paths, and that due to the opposite ray paths, corresponding to the ladder and cross terms in the Feynman diagramatic representation. It is proved that the contributions from ladder and cross terms for the copolarized backscattering cross sections are the same, while the contributions for the cross‐polarized backscattering cross sections are of the same order. The bistatic scattering coefficients in the second‐order approximation for both the ladder and cross terms are also obtained. The contributions from the cross terms explain the enhancement in the backscattering direction.

Calculating resonances (natural frequencies) and extracting them from transient fields
View Description Hide DescriptionMathematical formulation and analysis of numerical methods for calculating the natural frequencies (resonances) are given. Stability of these methods towards roundoff errors and small perturbations of the obstacles is established. Some formulas for the variations of the natural frequencies due to small perturbations of the surface of the obstacle are given. A simple new method for extraction of resonances from transient fields is given.