Volume 31, Issue 12, December 1990
Index of content:

Hidden symmetry and potential group of the Maxwell fish‐eye
View Description Hide DescriptionThe Maxwell fish‐eye is an exceptional optical system that shares with the Kepler problem and the point rotor (mass point on a sphere) a hidden, higher rotation symmetry. The Hamiltonian is proportional to the Casimir invariant. The well‐known stereographic map is extended to c a n o n i c a l transformations between of the phase spaces of the constrained rotor and the fish‐eye. Their d y n a m i c a l group is a pseudoorthogonal one that permits a succint ‘‘4π’’ wavization of the constrained system. The fish‐eye exhibits, unavoidably, chromatic dispersion. Further, a larger conformal dynamical group contains the p o t e n t i a l group, that relates the closed, inhomogeneous fish‐eye system to similar, scaled ones. Asymptotically, it is related to free propagation in homogenous media.

On Clebsch–Gordan coefficients and matrix elements of representations of the quantum algebra U _{ q }(su_{2})
View Description Hide DescriptionClebsch–Gordan coefficients and matrix elements of irreducible representations of the quantum algebraU _{ q }(su_{2}) were considered in several papers. In particular, a few expressions for them were derived. An approach to Clebsch–Gordan coefficients and to matrix elements of representations of U _{ q }(su_{2}) on the base of the theory of basic hypergeometric functions is given. This approach allows one to obtain q‐analogs of all well‐known classical expressions for Clebsch–Gordan coefficients (most of them were absent). New symmetry relations, generating functions, and recurrence formulas for Clebsch–Gordan coefficients of U _{ q }(su_{2}) are obtained. Unlike other papers, Clebsch–Gordan coefficients and matrix elements are considered on the base of minimal theoretical constructions (in fact, without using the notion of a C* algebra and of a Hopf algebra).

Vector coherent state constructions of U(3) symmetric tensors and their SU(3)⊇SU(2)×U(1) Wigner coefficients
View Description Hide DescriptionGeneralized vector coherent state constructions of totally symmetric U(3) tensors are used to gain new expressions for the SU(3)⊇SU(2)×U(1) Wigner coefficients for the coupling (λ_{1}μ_{1})×(λ_{2}0)→(λ_{3}μ_{3}). These expressions show how the extremely simple formulas of Le Blanc and Biedenharn, involving a single 9‐j coefficient, arise as special cases of a general result that involves 12‐j coefficients. A simpler general result involving only 9‐j coefficients and K‐normalization factors is derived in a way that can, in principle, be generalized to the generic coupling with multiplicity.

New inhomogeneous boson realizations and inhomogeneous differential realizations of Lie algebras
View Description Hide DescriptionThe inhomogeneous boson realizations (IHBR) and the corresponding inhomogeneous differential realizations (IHDR) of Lie algebras, which play an important role in the search of quasi‐exactly solvable problems (QESP) of quantum mechanics, are studied. All possible IHDR of semisimple Lie algebras can be obtained in this way. As examples, the IHBR and the corresponding IHDR of Lie algebras SU(2) and SU(3) are studied in detail.

Branching rules for a class of typical and atypical representations of gl(m‖n)
View Description Hide DescriptionThe irreducible representations of the Lie superalgebra gl(m‖n) with highest weights of the form Λ=(λ_{1},λ_{2},...λ_{ m }‖ω̇) are investigated using a recently introduced induced module construction for atypical modules. The gl(m‖n)↓gl(m‖n−1) branching rules are obtained and a suitable Gel’fand–Tsetlin basis is introduced. The class of representations considered includes some multiply atypical irreducible representations of gl(m‖n) and all irreducible representations of gl(m‖1).

Local realizations of kinematical groups with a constant electromagnetic field. II. The nonrelativistic case
View Description Hide DescriptionIn this paper, nonrelativistic elementary physical systems interacting with constant external electromagnetic fields are studied. The method is to construct a special kind of realizations of the Galilei group, which depend on the electromagnetic field. The linearization of this problem, which consists in obtaining these local realizations via the linear representations of another group, leads to a new representation group: the nonrelativistic Maxwell group. The study of the representations of this group and the related invariant equations completes this work.

Cohomology theory and deformations of Z _{2}‐graded lie algebras
View Description Hide DescriptionThe algebraic cohomology and the spectral sequences for a Z _{2}‐graded Lie algebra are briefly reviewed. The reducibility property of a strongly semisimple Lie superalgebra is established. The role of second and third cohomologies in the deformation of a Lie superalgebra is discussed. Using spectral sequences, the second cohomology of the full BRS algebra is shown to be the ground field and the third cohomology being trivial implies that osp(1,2) is the only graded Lie algebra obtained by deformation of the full BRS algebra. A similar analysis yields the superconformal algebra as a deformation of the super Poincaré algebra. The superconformal algebra so derived contains so(4,1) as the even part, ruling out the existence of negative curvature of a de Sitter universe!

Note on asymptotic series expansions for the derivative of the Hurwitz zeta function and related functions
View Description Hide DescriptionAsymptotic series for the Hurwitz zeta function, its derivative, and related functions (including the Riemann zeta function of odd integer argument) are derived as an illustration of a simple, direct method of broad applicability, inspired by the calculus of finite differences.

Integrable Hamiltonian systems related to the polynomial eigenvalue problem
View Description Hide DescriptionThe independent integrals of motion in involution for the Hamiltonian system related to the second‐order polynomialeigenvalue problem are constructed by using relevant recursion formula. The hierarchy of Hamiltonian systems obtained from the above problem and the time part of the Lax pair are shown to be completely integrable and they are shown to commute with each other. Furthermore, their solution solves the evolution equation associated with the Lax pair.

Introduction to a covariant theory of special functions of mathematical physics
View Description Hide DescriptionUsing harmonic analysis techniquies, the covariant expression of Jacobi and Hermite polynomials on an n‐dimensional space endowed with a metric g of signature (p _{+}, q _{−}) is given. The properties of these polynomials are studied and their relations with the hypergeometric function are given.

Complete integrability and analytic solutions of a KdV‐type equation
View Description Hide DescriptionThe complete integrability of the variable coefficient version of a KdV equation via the Painlevé approach is analyzed. Through the Painlevé–Bäcklund equations, its auto‐Bäcklund transformation, Lax pairs, symmetry, strong symmetry, bilinear form, and analytic solutions are obtained.

Lie algebraic methods and solutions of linear partial differential equations
View Description Hide DescriptionIn this paper, an algebraic method to obtain the solution of linear partial differential equations of the evolution type is discussed. The proposed method exploits the Lie differential operators and their matrix realization, to reduce the equation to an easily solvable generalized matrix form. Some applications to problems of specific interest are also discussed.

A geometric approach to the path integral formalism of p‐branes
View Description Hide DescriptionThe configuration space for a path integral description of a p‐brane is seen as a vector bundle over moduli spaces. The Einstein condition, applied to such vector bundles over compact Kähler manifolds, provides the required stability conditions. Consequently moduli spaces for such extended objects of higher dimensionality are constructed. Finally a Hermitian metric can be introduced in these moduli spaces.

Bäcklund transformations for surfaces in Minkowski space
View Description Hide DescriptionA Bäcklund transformation is constructed between spacelike surfaces of constant negative curvature and timelike surfaces of constant negative curvature in three‐dimensional Minkowksi space. The transformation gives a differential geometric interpretation to a Bäcklund transformation between the elliptic sine‐Gordon equation and the elliptic sinh‐Gordon equation studied by Leibbrandt [J. Math. Phys. 1 9, (1978)].

Extremal properties of Synge’s world function and discrete geometry
View Description Hide DescriptionProperties of σ space [a set Ω of points P with a real function σ(P,P’) given on Ω] are investigated. A continuity of the set Ω is not necessary and, generally, geometry is discrete. The properties of the world function σ are investigated. At certain (extremal) world function properties the σ space is shown to be a subset of points of Euclidean space or Riemannian space. The presented approach has the peculiarity that no operation other than the function σ is given on σ space. In particular, all such operations as linear operation over vectors, constructing lines and planes, and dimension of the space are expressed through the world function σ and only through it (if it is extremal). A violation of the σ‐space extremality leads to going out beyond the frames of Riemannian geometry (lines are substituted by tubes of lines, etc.). The presented approach can be useful in quantum gravitation, string models, and other problems, where the properties of the event space at small distances are important.

Quantum mechanics as an infinite‐dimensional Hamiltonian system with uncertainty structure: Part I
View Description Hide DescriptionSchrödinger quantum mechanics is formulated as an infinite‐dimensional Hamiltonian system whose phase space carries an additional structure(uncertaintystructure) to account for the probabilistic character of the theory. The algebra of observables is described as an algebra of smooth functions on the quantal phase space, with a product naturally induced by the geometrical structures living on that manifold. The possibility of generalizing Schrödinger mechanics along these lines is discussed.

Quantum mechanics as an infinite‐dimensional Hamiltonian system with uncertainty structure: Part II
View Description Hide DescriptionMaking reference to the formalism developed in Part I to formulate Schrödinger quantum mechanics, the properties of Kählerian functions in general, almost Kählerian manifolds, are studied.

Bäcklund transformation, conservation laws, and inverse scattering transform of a model integrodifferential equation for water waves
View Description Hide DescriptionThe Bäcklund transformation (BT), an infinite number of conservation laws, and the inverse scattering transform (IST) of a model integrodifferential equation for water waves in fluids of finite depth [Y. Matsuno, J. Math. Phys. 2 9, 49(1989)] are constructed by employing the bilinear transformation method. The model equation is also shown to pass the Painlevé test. These facts prove the complete integrability of the equation. Both the deep‐ and shallow‐water limits of various results thus obtained are then investigated in detail. In addition, a new method to evaluate conserved quantities for pure N‐soliton is developed by utilizing actively the time part of the BT. It is found that the structure of conservation laws exhibits peculiar characteristics in comparison with those of usual water wave equations such as the Benjamin–Ono and the Korteweg–de Vries equations. The most important problem left open in this paper is to solve various IST equations.

Exact plane‐wave solutions of the coupled Maxwell–Klein–Gordon equations
View Description Hide DescriptionSolutions of the classical Maxwell–Klein–Gordonequations are investigated for which the Klein–Gordon field is assumed to be ψ(x)=αe ^{ i p μ x μ }. It is shown that for this class the exponential factor can be ‘‘gauged away’’ and the resulting system of equations can be reduced to a s i n g l e (complicated) nonlinear equation. Furthermore, the electromagnetic four‐potential field becomes m a s s i v e ‘‘absorbing scalar particles.’’ The steady‐state (or stationary) subclass of the resulting system of equations is examined. It is proved that in absence of any magnetic field, the steady‐state system does not have a solution. In the simple case for which four‐potential components A ^{μ} depend on o n espatial coordinate, the equations are completely solved and explicitly analyzed.

The functional Ito formula in quantum stochastic calculus
View Description Hide DescriptionUsing an Op‐*‐algebraic approach, noncommutative analogs of the Ito formula of classical stochastic calculus within the framework of the Hudson–Parthasarathy formulation of Boson quantum stochastic calculus are proven.