Volume 23, Issue 8, August 1982
Index of content:

Analytic SU(3) states in a finite subgroup basis
View Description Hide DescriptionA method is given for deriving branching rules, in the form of generating functions, for the decomposition of representations of SU(3) into representations of its finite subgroups. Interpreted in terms of an integrity basis, the generating functions define analytic polynomial basis states for SU(3), which are adapted to the finite subgroup.

Structure and matrix elements of the degenerate series representations of U(p+q) and U(p,q) in a U(p)×U(q) basis
View Description Hide DescriptionThe representations of the most degenerate series of the group U(p,q) which are induced by the representations of the maximal parabolic subgroup are considered in this article. By making use of the infinitesimal operators of these representations in the U(p)×U(q) basis the conditions are derived which are necessary and sufficient for i r r e d u c i b i l i t y. For the r e d u c i b l e representations we describe their structure (composition series). We select from among the irreducible representations which are obtained in this article all representations of U(p,q) which admit unitarization. As a result we obtain the principal degenerate series, the supplimentary degenerate series, the discrete degenerate series, and the exceptional degenerate series of unitary representations of U(p,q). The U(p)×U(q) spectrum of the representations of U(p+q) with highest weights (λ_{1}, 0,..., 0, λ_{2}) is defined. We obtain the integral representation for the matrix elements of the degenerate representations of U(p,q) in the U(p)×U(q) basis. The matrix elements of the irreducible representations of U(p+q) with highest weights (λ ,0,...,0), (0,...,0, λ) are evaluated in the U(p)×U(q) basis.

Congruence classes of finite representations of simple Lie superalgebra
View Description Hide DescriptionThe concept of congruence of representations of Lie algebras is generalized and applied to the finite‐dimensional representations of Lie superalgebras.

Supersymmetry and Lie algebras
View Description Hide DescriptionStarting from the standard supersymmetryalgebra, an infinite Lie algebra is constructed by introducing commutators of fermionic generators as members of the algebra. From this algebra a finite Lie algebra results for fixed momentum analogous to the Wigner analysis of the Poincaré algebra. It is shown that anticommutation of the fermionic charges plays the role of a constraint on the representation. Also, it is suggested that anticommuting parameters can be avoided by using this infinite Lie algebra with fermionic generators modified by a Klein transformation.

General superposition of solitons and various ripplons of a two‐dimensional nonlinear Schrödinger equation
View Description Hide DescriptionWe consider the two‐dimensional nonlinear Schrödinger equation of Benney–Roskes. It is shown that the equation admits superposition solutions of solitons and various ripplons.

Korteweg‐de Vries surfaces and Bäcklund curves
View Description Hide DescriptionIt is shown that every point w(ε) on the curve γ_{ r } (ε) representing a 1‐parameter family of integrable equations containing a given rth Korteweg–de Vries (KdV) equation w(0), also belongs to a different integrable curve Γ_{ r }(ε,ν) . Symmetries of the resulting surface make it possible to construct a curve of Bäcklund transformations, that is, infinitesimal automorphisms, of points on γ_{ r } (ε) starting with the usual infinitesimal automorphisms of w(0). In addition, we obtain four new Bäcklund transformations of the second order for all higher KdV equations.

Interior and exterior solutions for boundary value problems in composite elastic media. I. Two‐dimensional problems
View Description Hide DescriptionWe study two‐dimensional problems of elasticity when a homogeneous and isotropic solid of an arbitrary shape is embedded in an infinite homogeneous isotropic medium of different properties.Solutions are obtained both inside the guest and host media. These solutions are derived by first transforming the boundary value problems to the equivalent integral equations. The interior displacement field is obtained by a simple method of truncation. By this method the integral equations are recast into an infinite number of algebraic equations and a systematic scheme of solutions is constructed by an appropriate truncation. The exterior solutions are obtained by substituting the interior solutions in the integral equations valid for the entire medium. The boundaries considered are rectangular cylinder, equilateral triangular prism, and elliptic cylinder and its limiting configurations. It emerges that the solutions for the elliptic cylinder and its limiting configurations are exact.

Time‐dependent scattering by a bounded obstacle in three dimensions
View Description Hide DescriptionIn the prsent paper we introdue a new method of treating the scattering of transient fields by a bounded obstacle in three‐dimensional space. The method is a generalization to the time domain of the null field approach first given by Waterman. We define new sets of time‐dependent basis functions, and use these to expand the free space Green’s function and the incoming and scattered fields. The scattering problem is then reduced to the problem of solving a system of ordinary differential equations. One way of solving these equations is by means of Fourier transformation, and this leads to an efficient way of obtaining the natural frequencies of the obstacle. Finally, we have calculated the natural frequencies numerically for both a spheroid and a peanut‐shaped obstacle for various ratios of the axes.

Quantum evolution as a parallel transport
View Description Hide DescriptionWe point out that the evolution of a quantum system can be considered as a parallel transport of unitary operators in Hilbert spaces along the time with respect to a generalized connection. The different quantum representations of the system are shown to correspond to the choices of cross sections in the principal fiber bundle where the generalized connection is defined. This interpre‐ tation of time evolution allows us to solve the problem of the formulation of the evolution of a quantum particle in a four‐dimensional gauge field.

Feynman path integrals of operator‐valued maps
View Description Hide DescriptionFeynman integral is defined for operator‐valued maps which are Fourier transforms of an operator‐valued measure of bounded variation. Such an integral is then used to describe perturbation of certain unitary groups by certain cocycles.

Solutions of type D possessing a group with null orbits as contractions of the seven‐parameter solution
View Description Hide DescriptionIt is shown that several type D solutions with null group orbits of local isometries are limiting contractions of the seven‐parameter solution.

Explicit solutions of the conformal scalar equations in arbitrary dimensions
View Description Hide DescriptionSolutions of the equations of motion derived from the scalar conformal invariant Lagrangian in arbitrary dimensions are found. The solutions are invariant under the maximal compact subgroup of the corresponding conformal group. They have finite energy and action. In the case N = 2, we also find noticeable topological properties of the solutions.

Twistor bundles and gauge action principles of gravitation
View Description Hide DescriptionA gauge theory of gravitation is constructed with a twistor bundle as the starting point. Each fiber is a twistor space, acted upon by the Poincaré group, which forms an internal symmetry group. The formalism leads to a twistor action principle which overcomes difficulties encountered in previous attempts in the literature to formulate a true spinorial variational principle.

Properties of Mayer cluster expansion
View Description Hide DescriptionThe Umbral algebra developed by Rota and his co‐workers is used to show that the Mayer cluster expansion of the canonical partition function is related to the Bell polynomials. The algebra is also used to find a representation of the partition function and a rederivation of Mayer’s first theorem. Finally, it is shown that in the ’’tree approximation’’ for the cluster integrals, the summation of Mayer’s expression for the canonical‐ensemble partition function for a finite number of particles could be performed using Dénes’ and Rényi’s theorems in graph theory.

Continuity of sample paths and weak compactness of measures in Euclidean field theory
View Description Hide DescriptionUnder an assumption on the asymptotic u→∞ behavior of the generating functional F exp[uφ( g)] dμ(φ), an estimate is obtained on ‖χ(x)−χ(x′)‖, where χ(x) = φ_{ g }(x) ≡φ( g(⋅−x)), showing Hölder continuity of φ_{ g }(x) for a class of g. It is proved that the family of measures v _{γ}, with d v _{γ}(χ) = dμ_{γ}(φ_{ g }) and F expχ(x) d v _{γ}(χ) bounded in γ, is weakly conditionally compact. In application to the infinite volume cutoff μ_{κ} measures in P(φ)_{ d } , these results imply the continuity of μ_{κ}(C) when κ→∞ for a class of sets C. Such a property allows distinguishing the interacting measures from the free ones.

Comment on calculations of higher‐order tadpoles and triangle vertex integral
View Description Hide DescriptionIt is found that the calculations of higher‐order tadpoles and some integrals associated with the triangle diagram reported by Capper and Leibbrandt are incorrect within their dimensional regularization scheme.

Superfiber bundle structure of gauge theories with Faddeev–Popov fields
View Description Hide DescriptionIn this work the mathematical structure of principal superfiber bundle P _{s} is used to give a geometrical description of gauge theories. The base space of P _{s} is an S ^{4,2}‐supermanifold X _{s} (four commuting and two anticommuting variables), and the structure group an S ^{ n,0} supergroup G _{s}, where n is the dimension of the gauge group for the classical theory. The body of P _{s} is the usual principal fiber bundle P of gauge theories. Gauge and Faddeev–Popov fields arise as superfields, components of the connections in P _{s}, in a local coordinate system. BRS (Becchi, Rouet, and Stora) and anti‐BRS transformations are gauge transformations, in P _{s}, of parameters the ghost and antighost superfields, respectively. In the case of soul‐flat connections, which are connections in P _{s} coming from connections in P, the BRS and anti‐BRS transformations are finite translations along the anticommuting directions of X _{s}, and generate an S ^{0,2}‐supergroup.

Relativistic field theories in three dimensions
View Description Hide DescriptionWe provide a complete classification of the unitary irreducible representations of the (2+1)‐dimensional Poincaré group. We show, in particular, that only two types of ’’spin’’ are available for massless field theories. We also construct generalized Foldy–Wouthuysen transformations which connect the physical UIR’s with covariant field theories in three dimensions.

The nonlinear Schrödinger equation as a Galilean‐invariant dynamical system
View Description Hide DescriptionThe invariance of the nonlinear Schrödinger equation under the Galilei group is analyzed from the point of view of the inverse scattering transform. It is shown that this group induces an infinite‐dimensional nonlinear canonical realization which is locally equivalent to a direct product of the two well‐known Galilean actions describing classical particles and the free Schrödinger equation.

An analysis of Taylor’s theory of toroidal plasma relaxation
View Description Hide DescriptionThe Taylor theory of toroidal plasma relaxation is considered as a nonlinear eigenvalue problem. The analysis is rigorous and applies to quite general toroidal cross sections. Emphasis is placed on the symmetric state case, where the existence of field reversal and flux free states is demonstrated. Certain anomalies are revealed by the mathematical treatment and their significance is studied. Existence of a solution to the Taylor problem in the symmetric state is proved and the location of the eigenvalue of this solution state relative to other states is examined. The question of the existence of helical states is not resolved, but it is shown that in many respects any helical states behave like the anomalous cases in the symmetric problem and hence do not significantly affect the theory.