Volume 25, Issue 4, April 1984
Index of content:

The eigenvalue problem for ‘‘arrow’’ matrices
View Description Hide DescriptionThe eigenvalue problem for a particular class of ‘‘arrow’’ matrices where A is a Hermitian N×N matrix, C a real diagonal M×M matrix, B an arbitrary complex N×M matrix, and γ a real number, is investigated by means of a partitioning technique. Both Hermitian (γ=1) and non‐Hermitian (γ≠1) arrow matrices Z are studied. The one‐dimensional case (dimension N of A equal to 1) is briefly reviewed and a detailed treatment of the multidimensional case (N>1) is presented. For Hermitian arrow matrices, the analysis leads to a new algorithm for computing the eigenvalues and eigenvectors of Z which is particularly efficient if M≫N.

The relationship between finite groups and Clifford algebras
View Description Hide DescriptionClifford algebras are traditionally realized in terms of a specific set of representation matrices. This paper provides a more effective alternative by giving the finite group associated with each Clifford algebra. All the representation‐independent algebraic results, which are really direct consequences of the underlying group structure, can thus be derived in an easier and more general manner. There are five related but distinct classes of finite groups associated with the Clifford algebras. These groups are constructed from the complex, cyclic, quaternion, and dihedral groups in a way which is discussed here in detail. Of particular utility is a table which lists the order structure of each group: this permits the immediate identification of any Clifford algebra in any dimension.

Infinitesimal operators and structure of the most degenerate representations of the groups Sp( p+q) and Sp( p, q) in an Sp( p)×Sp(q) basis
View Description Hide DescriptionExplicit expressions are derived for the infinitesimal operators of the representations π_{λ} of the most degenerate series of the group Sp( p, q) and for the irreducible unitary representations of the group Sp( p+q) with highest weight (m, m, 0, ...,0) with respect to an Sp( p)×Sp(q) basis. The Sp( p)×Sp(q) basis can be chosen arbitrarily. The irreducibility conditions for the representations π_{λ} and the structure of the reducible representations π_{λ} are discussed. All the unitary irreducible representations of the most degenerate series of Sp( p, q) are obtained. It is shown that there are four series of unitary representations (principal, supplementary, discrete, and ladder).

Realizations of F _{4} in SO(3)⊗SO(3) bases and structural zeros of the 6 j‐symbol
View Description Hide DescriptionThe decompositon of F _{4} irreps in the group chains F _{4}⊇SO(3)⊗G _{2}⊇SO(3)⊗SO(3) and F _{4}⊇SO(3)⊗Sp(6)⊇SO(3)⊗SO(3) is considered. In both cases a realization of the F _{4}Lie algebra in terms of SO(3)⊗SO(3) tensor operators is established. From these, the nontrivial vanishing of certain 6 j‐coefficients is explained.

Matrix elements for indecomposable representations of complex su(2)
View Description Hide DescriptionIndecomposable representations of the simple complex Lie algebraA _{1} are investigated in this article from a general point of view. First a ‘‘master representation’’ is obtained which is defined on the space of the universal enveloping algebra Ω of A _{1}. Then, from this master representation other indecomposable representations are derived which are induced on quotient spaces or subduced on invariant subspaces. Finally, it is shown that the familiar finite‐dimensional and infinite‐dimensional irreducible representations of su(2) and su(1,1) are closely related to certain of the indecomposable representations. Indecomposable representations of A _{1} [su(2), su(1,1)] have found increased applications in physical problems, including the unusual ‘‘finite multiplicity’’ indecomposable representations. Emphasis is placed in this article on an analysis of the more unfamiliar indecomposable representations. The matrix elements are obtained in e x p l i c i t f o r m for all representations which are discussed in this article. The methods used are purely algebraic.

Formal linearization of nonlinear massive representations of the connected Poincaré group
View Description Hide DescriptionLet U ^{1} be an arbitrary finite direct sum of unitary irreducible representations, each of positive (mass)^{2}, of the connected Poincaré group P _{0}=R^{4}⊗SL(2,C). It is proved that each nonlinear representation of P _{0} with linear part U ^{1}, on the space of differentiable vectors of U ^{1}, is formally linearizable. Further, it is remarked that, commonly used, nonlinear massless representations of R^{2}⊗SO_{0}(1,1) are nonlinearizable and that the corresponding evolution equations do not have covariant wave operators.

On Lie algebras built from SO(3) tensor operators
View Description Hide DescriptionConcerning the Lie algebras generated by certain linear combinations of SO(3) tensor operators two theorems are given which disprove some widely accepted results in the standard literature.

The positivity of the Lyapunov exponent and the absence of the absolutely continuous spectrum for the almost‐Mathieu equation
View Description Hide DescriptionThis paper contains the rigorous proof of the formulated by Andre and Aubry following statement: the Cauchy solutions of the discrete Schrödinger equation with the potential q _{ n }=g cos(2πnθ+φ) grow exponentially for every irrational θ, g>1 and almost every φε[0,2π). According to known this fact implies the absence of the absolutely continuous component of the spectrum for the corresponding operator.

The geometric foundations of the integrability property of differential equations and physical systems. II. Mechanics on affinely‐connected manifolds and the work of Kowalewski and Painlevé
View Description Hide DescriptionIn the 1880’s Sophie Kowalewski proposed to study ‘‘integrability’’ of differential equations in terms of the analyticity properties of their solutions. The model she studied, the spinning top moving under gravity, has an interesting geometric and algebraic structure, which will be generalized. The differential geometricproperties of such models will be investigated in terms of the theory of affinely‐connected manifolds and Lie theory.

The Schrödinger–Langevin equation: Special solutions and nonexistence of solitary waves
View Description Hide DescriptionTwo types of solutions of the Schrödinger–Langevin equation are investigated. It is proved that a special type of Gaussian solutions exist globally in time for the harmonic oscillator Hamiltonian. Furthermore, it is shown that the Schrödinger–Langevin equation can have no solitary wave type solutions in the damped free‐particle case which lie in L ^{2}.

Symmetry reduction for nonlinear relativistically invariant equations
View Description Hide DescriptionSymmetry reduction is studied for the relativistically invariant scalar partial differential equationH(⧠u,(∇u)^{2},u)=0 in (n+1)‐dimensional Minkowski space M(n,1). The introduction of k symmetry variables ξ_{1}, ... ,ξ_{ k } as invariants of a subgroup G of the Poincaré group P(n,1), having generic orbits of codimension k≤n in M(n,1), reduces the equation to a PDE in k variables. All codimension‐1 symmetry variables in M(n,1) (n arbitrary), reducing the equation studied to an ODE are found, as well as all codimension‐2 and ‐3 variables for the low‐dimensional cases n=2,3. The type of equation studied includes many cases of physical interest, in particular nonlinear Klein–Gordon equations (such as the sine–Gordon equation) and Hamilton–Jacobi equations.

Representation of solutions to Helmholtz’s equation
View Description Hide DescriptionIt is proved that any potential of a single layer v is identically equal to a potential of a double layerw in the bounded domain, D, and a necessary and sufficient condition for v≡w in Ω=R^{3}/D, the exterior domain, is given.

On the solution of the two‐dimensional Helmholtz equation
View Description Hide DescriptionThe solution of the two‐dimensional Helmholtz equation has been obtained in the annular region having eccentric circular boundaries. It is shown that in the limit of eccentricity zero, the solution reduces to that corresponding to the case of concentric circular boundaries.

Convergence of a crossing‐symmetric perturbation series for the four‐point vertex
View Description Hide DescriptionA finite radius of convergence is established for a perturbation expansion of the crossing‐symmetric vertex function in a discrete model.

An integral transformation in Lobachevsky space
View Description Hide DescriptionThe integral formulas of the one‐to‐one correspondence between the one‐particle state F _{ sλ}( p) with nonzero mass p ^{2}=E ^{2}−p̄^{2}, helicity (−s≤λ≤s), and the state F _{ν}(k)(ν=±s,±(s−1),...,0 or ±( 1/2 ) with mass equal to zero (k ^{2}=ω^{2}−k̄^{2}=0) is obtained.

Planar coherent states
View Description Hide DescriptionPlanar coherent states with domains in the complex plane are defined by means of ladder operators acting in a separable Hilbert space. Some properties of the states are derived and examples provided to indicate areas of possible applications.

General transformation matrix for Dirac spinors and the calculation of spinorial amplitudes
View Description Hide DescriptionA general transformation matrix T( p̃,s̃;p,s) is constructed which transforms a Dirac spinor ψ( p,s) into another Dirac spinor ψ( p̃,s̃) with arbitrarily given momenta and polarization states by exploiting the so‐called Stech operator as one of generators for those transformations. This transformation matrix is then used in a calculation to yield the spinorial matrix element M=ψ̄( p̃,s̃)Γψ( p,s) for any spin polarization state. The final expressions of these matrix elements show the explicit structure of spin dependence of the process described by these spinorial amplitudes. The kinematical limiting cases such as very low energy or high energy of the various matrix elements can also be easily displayed. Our method is superior to the existing one in the following points. Since we have a well‐defined transformation operator between two Dirac spinor states, we can evaluate the necessary phase factor of the matrix elements in an unambiguous way without introducing the coordinate system. This enables us to write down the Feynman amplitudes of complicated processes in any spin basis very easily in terms of previously calculated matrix elements of ψ̄Γψ which are building blocks of those Feynman amplitudes. In contrast, in the existing method, one has to figure out the suitable multiplication factor for each individual case. Furthermore, in the previous method one needs a coordinate system to evaluate phase factors, thus making the calculation more cumbersome. Also, the expression of the matrix element in the previous method is in the fractional form whose denominator may have a singularity structure (and not always an isolated one at that), while the spinorial amplitudes in our method take a much simpler form, free of coordinate systems in any spin basis. The usefulness of the results is illustrated on Compton scattering and on the elasticscattering of two identical massive leptons where the phase factor is important. It is also shown that the Stech operator as a polarization operator is simply related to the operator K=β(Σ⋅L+1)/2, which is often used in bound state problems.

Orthogonal decompositions of certain finite‐dimensional vector spaces
View Description Hide DescriptionThe vector space of real functions, defined on the set of all mappings of a finite set P into another finite set L, splits into a sum of orthogonal subspaces, one for each subset of P. The orthogonal projections onto these subspaces merely involve averaging operations. Certain linear functional identities are equivalents of k‐representability, i.e., of location in the span of those subspaces that belong to subsets of cardinality k. Potential applications refer to complex systems where these results could be used to analyze empirically how their properties depend on properties of their components as well as on the interactions between them. Roughly this amounts to estimating the internal structure of a ‘‘black box’’ from measured properties.

Generalized supermanifolds. I. Superspaces and linear operators
View Description Hide DescriptionGeneralized superspaces are defined taking as a starting point the concept of Grassmann–Banach algebra. The category of B‐spaces with L _{ B } ‐maps as morphisms is introduced. This allows one to discuss linear and tensor algebra on generalized superspaces.

Generalized supermanifolds. II. Analysis on superspaces
View Description Hide DescriptionThe analysis on generalized superspaces is studied. G‐differentiability of functions and indetermination of partial G‐derivatives are treated in detail. A canonical expansion for supersmooth functions and a particular choice of partial G‐derivatives are given, and their properties studied. Existence of (nonanalytic) G ^{∞} ‐functions is proved.