Volume 33, Issue 7, July 1992
Index of content:

Weyl orbits and their expansions in irreducible representations for affine Kac–Moody algebras
View Description Hide DescriptionWeyl orbits of affine rank 2 and 3 algebras are obtained analytically, along with the depths of their weights, in terms of an integrity basis consisting of sets of compatible (or adjacent) weights, one from each fundamental orbit. The fundamental orbits and others of the same level and congruence class are then decomposed into irreducible representations (IR). Since the orbit →IR matrix is triangular, it will be easy to invert it to get orbit multiplicities and subalgebra branching rules.

A supergroup based supersigma model
View Description Hide DescriptionThe supergroup UOSp(1,2) is used to construct geometrically a twice graded supersymmetric supersigma model. In an appropriate contraction limit, the model reduces to the conventional supersymmetric sigma model.

Models of q‐algebra representations: Tensor products of special unitary and oscillator algebras
View Description Hide DescriptionThis paper begins a study of one‐ and two‐variable function space models of irreducible representations of q analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q‐hypergeometric functions. The algebras considered are the quantum algebra U_{ q }(su_{2}) and a q analog of the oscillatoralgebra (not a quantum algebra). In each case a simple one‐variable model of the positive discrete series of finite‐ and infinite‐dimensional irreducible representations is used to compute the Clebsch–Gordan coefficients. It is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and the corresponding matrix elements of the ‘‘group operators’’ on these representation spaces are computed. It is shown that the matrix elements are polynomials satisfying orthogonality relations analogous to those holding for true irreducible group representations. It is also demonstrated that general q‐hypergeometric functions can occur as basis functions in two‐variable models, in contrast with the very restricted parameter values for the q‐hypergeometric functions arising as matrix elements in the theory of quantum groups.

Symmetry scattering on the hyperboloid SO(2,1)/SO(2) in different coordinate systems
View Description Hide DescriptionThe symmetry scattering theory based on the Harish‐Chandra–Helgason theory of spherical functions on noncompact Riemannian symmetric spaces is extended to treat all spherical harmonics on the hyperboloid SO(2,1)/SO(2). The required conditions for an extension of the symmetry scattering theory to treat all spherical harmonics on arbitrary noncompact Riemannian symmetric spaces are elaborated.

The Euclidean root of Snell’s law I. Geometric polarization optics
View Description Hide DescriptionCoset spaces of the Euclidean group of rigid motions in three‐space are used as a model of the rays in geometric polarizationoptics. The Haar measure invariance leads to canonicity of the phase space transformations. The phase space of optical rays undergoes a canonical map also under the effect of smooth refracting or reflecting surfaces between two optical media that is governed by Snell’s law. This is a conservation law between two irreducible representations of the Euclidean group labeled by the refractive indicesn, n’ of the media. These surface transformations are canonical and, furthermore, factorize into two root transformations that are also canonical. This factorization applies equally to the transformation of the polarization vector. The root transformation permits the computation of an aberration expansion of the polarization field over the object screen. The generators of infinitesimalsurface transformations are found, i.e., those that transform between media n and n+dn. It is shown that, together with displacements, they yield the Hamilton equations of optics for inhomogeneous media.

Quantum spectra of zero modes, topological vacua and dynamical supersymmetry breaking in non‐abelian gauge theories
View Description Hide DescriptionSome aspects of supersymmetric gauge theories are discussed intimately connected to the Lie algebras of the gauge groups. From the discussions of zero modes in non‐abelian gauge theories, a formula for the quantum spectra of the classical zero modes in supersymmetric gauge theories is obtained. Some relevant aspect of topological vacua in higher dimensions are also discussed. As a physics application, the spectrum formula and the discussions of the topological vacua are applied to the determinations of dynamical supersymmetry breaking. Especially, for gauge theories without gauge symmetry breaking in quantum dynamics, dynamical supersymmetry breaking does not occur in supersymmetric non‐abelian gauge theories in D=4 and higher dimensions for any value of the gauge coupling constant and independent of the vacuum angle in the theories if the theory is asymptotically free.

Tensor product of group representations and structural zeros of Racah coefficients
View Description Hide DescriptionIt is shown that structural (or nontrivial) zeros of Racah coefficients (6‐j symbols) can be related to decompositions of tensor products of group representations reduced to SO(3). This extends a previous relation between structural zeros of 6‐j symbols and boson realizations of exceptional Lie groups. In the present case, both classical and exceptional groups may appear, and examples are given for SU(3), SO(7), SO(9), and F _{4}.

Some simple representations of the affine Lie superalgebra A ^{(1)}(1,0)
View Description Hide DescriptionIn this paper, some simple modules of the affine Lie superalgebraA ^{(1)}(1,0) are considered which, although they are neither highest nor lowest weight modules, are constructed in a manner closely analogous to the construction of highest weight modules of the finite‐dimensional Lie superalgebrasA(m,n).

Diffusion in asymptotically logarithmic potentials
View Description Hide DescriptionBehavior of solutions to the one‐dimensional Fokker–Planck equation with potentials U(x) which diverge logarithmically as x→±∞ is investigated through the analysis of the frequency component of the Green’s function. It is shown that solutions of the equation decrease for large time t either logarithmically or as t ^{−α}, where α may take arbitrary positive value depending on a parameter.

Renormalized stress tensors on ellipses
View Description Hide DescriptionThe stress‐energy tensor of a conformally invariant field is calculated in a two‐dimensional Euclidean space‐time with elliptical boundary. The results are expressed in terms of elliptic functions and are displayed.

Families of invariants of the motion for the Lotka–Volterra equations: The linear polynomials family
View Description Hide DescriptionThe modified Carleman embedding method already introduced by the authors to find first integrals (invariants of the motion) of polynomial form to the Lotka–Volterra system is described in detail, and its efficiency to treat the N‐dimensional system proved. Using this method, an extensive investigation is performed for polynomials of the first degree, which allow a classification of the integrals in three families. For some systems possessing one invariant it is possible to find a second invariant using rescaling methods. They represent very restrictive solutions, implying that there exists a great number of conditions among the equation’s coefficients to satisfy. A proof is given that the Volterra invariants can be deduced as a limit. Finally, the interesting properties of the solutions of these systems are studied in detail.

Painlevé test and exact similarity solutions of a class of nonlinear diffusion equations
View Description Hide DescriptionFor (1+1)‐dimensional diffusionequations where the diffusion coefficient is an arbitrary power of the dependent variable, a Painlevé test is performed on both the partial differential equation and the similarity reductions. The invariance of the Painlevé property under a large class of similarity reductions is shown. For the reductions with the Painlevé property, the explicit similarity solutions are given. It is demonstrated how results of a Painlevé analysis can be used to obtain further exact similarity solutions, even in cases without the Painlevé property.

Lax representations and Lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations
View Description Hide DescriptionA method of constructing Lax representations for isospectral and nonisospectral hierarchies of evolution equations is proposed. It is shown that under some simple but essential conditions, the corresponding Lax operators may constitute infinite‐dimensional Lie algebras with respect to the binary operation ■⋅,⋅■ given in this paper. Furthermore, the detailed analyses for the KdV couple, the AKNS couple, and a new couple of isospectral and nonisospectral hierarchies of integrable equations are presented as examples.

A new family of integrable models in (2+1) dimensions associated with Hermitian symmetric spaces
View Description Hide DescriptionIn a series of papers Fordy and his collaborators have studied families of integrable models in (1+1) dimensions associated with Hermitian symmetric spaces. These models were also generalized to (2+1) dimensions. However, the generalization used is not unique, and in this paper a different generalization is considered, resulting in new families of integrable models in (2+1) dimensions.

Two‐point quasifractional approximant in physics: Method improvement and application to J _{ν}(x)
View Description Hide DescriptionAn improved two‐point quasifractional approximant method is presented that includes a free parameter in the denominator. This parameter is determined by minimizing the theoretical truncation error bound, thus ensuring high accuracy. An added advantage of introducing a free parameter is that defects that are usual in Padé approximants are always avoided. The method is applied to the fractional Bessel functions J _{ν}(x), obtaining a unique approximant, valid for all positive x, that is more accurate than previously published.

The Hamiltonian structures associated with a generalized Lax operator
View Description Hide DescriptionIt is shown that with every Lax operator, which is a pseudodifferential operator of nonzero leading order, is associated a KP hierarchy. For each such operator, we construct the second Gelfand–Dikii bracket associated with the Lax equation and show that it defines a Hamiltonian structure. When the leading order is positive the corresponding compatible first Hamiltonian structure, which turns out, in general, to be different from the naive first Gelfand–Dikii bracket is derived. The corresponding Hamiltonian structures for the constrained Lax operator, where the next to leading‐order term vanishes or has a constant coefficient, is discussed.

A family of nonlinear Klein–Gordon equations and their solutions
View Description Hide DescriptionVarious forms of the nonlinear Klein–Gordon equation are seen to have exact, solitonlike solutions when separation of variables is postulated. The family for which these exact solutions are found includes the sine‐Gordon equation as a special case. An interesting conclusion obtained is that both the soliton–antisoliton and breather solutions, hitherto known for the sine‐Gordon equations result from a broader class of Klein–Gordon equations. The method can be extended to other equations.

Hamiltonian theory over noncommutative rings and integrability in multidimensions
View Description Hide DescriptionA generalization of the Adler–Gel’fand–Dikii scheme is used to generate bi‐Hamiltonian structures in two spatial dimensions. In order to implement this scheme, a Hamiltonian theory is built over a noncommutative ring, namely the ring of formal pseudodifferential operators. Bi‐Hamiltonian structures generated in this way can be used for the Kadomtsev–Petviashvili equation as well as other integrable equations in 2+1.

A class of affine Wigner functions with extended covariance properties
View Description Hide DescriptionAffine Wigner functions are phase space representations based on the affine group in place of the usual Weyl–Heisenberg group of quantum mechanics. Such representations are relevant to the time–frequency analysis of real signals. An interesting family is singled out by the requirement of covariance with respect to each solvable three‐parameter group containing the affine group. Explicit forms are given in each case and properties such as unitarity and localization are discussed. Some particular distributions are recovered.

On the uniqueness of the Berry connection
View Description Hide DescriptionArguments are presented that make the choice of the connection, which gives rise to the Berry phase not only natural, but unique, both in the Abelian and the non‐Abelian cases. Invariance is invoked under unitary transformations of the probability amplitudes in quantum systems to force the connection to be invariant under the unitary group. Because the action is not free, the horizontal subspace chosen by the connection has to be invariant under the little group, and that makes it orthogonal to the fiber’s direction, yielding the conventional Berry connection. The argument works just as well for the non‐Abelian case, where the fibers are orthonormal frames (Stiefel manifolds), and the state space a Grassmanian, with a transitive but not free unitary action.