Volume 28, Issue 9, September 1987
Index of content:

Spectral theory for unitary operators on a quaternionic Hilbert space
View Description Hide DescriptionThe spectral theorem for a unitary operator on a quaternionic Hilbert space is established and a number of related results are proved.

Generating relations for reducing matrices. III. Kronecker products
View Description Hide DescriptionIn the first two of this series of papers [R. Dirl e t a l., J. Math. Phys. 2 7, 37 (1986); M. I. Aroyo e t a l., i b i d. 2 7, 2236 (1986)] a systematic method to calculate the elements of reducing matrices has been developed. In this paper this ‘‘auxiliary group approach’’ is adapted to multiple Kronecker products. Three examples are worked out to illustrate the efficiency of generating relations and the proposed reduction of the multiplicity problem.

Nonlinear representations of Poincaré group in three dimensions
View Description Hide DescriptionThe method of formal series is applied to the construction of nonlinear representations of Poincaré’s group in three dimensions. The first term of the series must be a linear massless representation. In the special case of discrete helicities, the cocycles of extension of these representations by their tensor product with a nonzero quadratic term are determined. It turns out that a nonlinear representation must have a helicity zero representation as leading term. It is proved by induction how to avoid the successive obstructions to the computation of each term in the series.

Path integration over compact and noncompact rotation groups
View Description Hide DescriptionApplications of group theoretical methods in the path integral formalism of nonrelativistic quantum theory are considered. Analysis of the symmetry of the Lagrangian leads to the expansion of the short time propagator in matrix elements of unitary irreducible representations of the symmetry group. Identification of the coordinates with the group parameters transforms the path integral to integrals over the group manifold. The integration is performed using the orthogonality of the representations. Compact and noncompact rotation groups are considered, where the corresponding path integral is embedded in Euclidean and pseudo‐Euclidean spaces, respectively. The unit sphere and unit hyperboloid may either be viewed as the group manifold itself or at least as a group quotient. In the first case Fourier analysis leads to an expansion in group characters. In the second case an expansion in zonal spherical functions is obtained. As examples the groups SO(n), SU(2), SO(n−1,1), and SU(1,1) are explicitly discussed. The path integral on SO(n+m) and SO(n,m) in bispherical coordinates is also treated.

The polynomial‐type analysis of SU(3) group‐theoretical quantities
View Description Hide DescriptionFormulas for the transformation matrix between the canonical and noncanonical SU(3) bases in both λ≥μ and λ<μ cases are derived. An expression for the isoscalar factor of the SU(3) coupling coefficient, in the noncanonical basis, with one symmetric representation is also derived. Both expressions are put in a polynomial‐type form suitable for coding computer programs to obtain explicit and exact algebraic expressions.

Classification and interpretation of the supertableaux of the orthosymplectic groups OSP(m‖4)
View Description Hide DescriptionThe supertableaux of the orthosymplectic group OSP (m‖4) are analyzed and interpreted as representations of the corresponding superalgebras. An extension of the results to the general case OSP (m‖2p) is proposed.

Integrable equations in (2+1) dimensions associated with symmetric and homogeneous spaces
View Description Hide DescriptionGeneralizations of the N‐wave, Davey–Stewartson, and Kadomtsev–Petviashvili equations associated with homogeneous and symmetric spaces are presented. These equations are (2+1)‐dimensional generalizations of those presented by Fordy and Kulish [Commun. Math. Phys. 8 9, 427 (1983)] and Athorne and Fordy [J. Phys. A, 2 0, 1377 (1987)]. Examples are explicitly presented that are associated with the simplest spaces. In particular, a new single component, (2+1)‐dimensional generalization of the KdV equation is presented.

Periodic fixed points of Bäcklund transformations
View Description Hide DescriptionThe discussion of the periodic fixed points of Bäcklund transformations for the Korteweg–de Vries equation is completed. It will be shown that the systems of equations defined by the KdV periodic fixed points are e q u i v a l e n t to the periodic Kac–Van Moerbeke systems. As a consequence, for even order fixed points, the KdV systems are equivalent to the periodic Toda lattice. The periodic fixed points of the Bäcklund transformation for the Boussinesq equation are found to have a Hamiltonian structure. The integrals of these systems are found.

Remarks on some hypergeometric orthogonal polynomials of mathematical physics
View Description Hide DescriptionThe main result is a simple evaluation of an integral related to the orthogonality property of hypergeometric polynomials occurring in mathematical physics. Some related formulas for generalized hypergeometric functions are also briefly discussed.

On some classes of unbounded commutants of unbounded operator families
View Description Hide DescriptionSome classes of unbounded commutants and bicommutants and their behavior with respect to the quasiweak*‐topology which seems to play here the role of the weak topology for bounded operators, are investigated. In particular, some sufficient conditions are given in order that the bicommutants be the quasiweak*‐closure of the original set of operators.

Affine collineations in Robertson–Walker space‐time
View Description Hide DescriptionIn a recent paper [M. L. Bedran and B. Lesche, J. Math. Phys. 2 7, 2360 (1986)], an attempt was made to find an affine collineation in Robertson–Walker space‐time. However, only a homothetic affine collineation was found, in the case of a linear scale factor, R(t)∼t. It is pointed out that another homothetic affine collineation exists when R(t)∼t ^{ b }, and a proper (nonhomothetic) affine collineation in the Einstein static space‐times has been found.

Graded spinors as an underlying geometry for extended supersymmetries
View Description Hide DescriptionSpinors associated with an eight‐dimensional Euclidean space are used for constructing a graded vector representation space for OSp(4/N,R). The underlying geometry fixes both the nature and highest dimensionality of the family of internal symmetry groups of the bosonic subspace, as well as those of the fermionic subspace through an appropriately defined inner product. The basic ingredients of the theory are fermionic variables while the bosonic fields occur as composite quantities made up from an even number of products of the fermionic entities.

The Schücking problem
View Description Hide DescriptionThe embedding problem for a three‐parametric family of homogeneous three‐spaces into a higher‐dimensional Euclidean space is considered. These three‐spaces occur as space sections in cosmological models. After general consideration a certain two‐parametric family is embedded into a five‐dimensional Euclidean space, deferring the solution of the general case to later papers.

The global Utiyama theorem in Einstein–Cartan theory
View Description Hide DescriptionA global formulation of Utiyama’s theorem for Einstein–Cartan‐type gravitational theories regarded as gauge theories of the group of space‐time diffeomorphisms is given. The local conditions for the Lagrangian to be gauge invariant coincide with those found by other authors [A. Pérez‐Rendón Collantes, ‘‘Utiyama type theorems,’’ in P o i n c a r é G a u g e A p p r o a c h t o G r a v i t y. I, Proceedings Journées Relativistes 1984; A. Pérez‐Rendón and J. J. Seisdedos, ‘‘Utiyama type theorems in Poincaré gauge approach to gravity. II, ’’ P r e p r i n t s d e M a t h e m a t i c a s, Universidad de Salamanca, 1986] in Kibble’s and Hehl’s approaches.

Rigorous estimates for a computer‐assisted KAM theory
View Description Hide DescriptionNonautonomous Hamiltonian systems of one degree of freedom close to integrable ones are considered. Let ε be a positive parameter measuring the strength of the perturbation and denote by ε_{ c } the critical value at which a given KAM (Kolmogorov–Arnold–Moser) torus breaks down. A computer‐assisted method that allows one to give rigorous lower bounds for ε_{ c } is presented. This method has been applied in Celletti–Falcolini–Porzio (to be published in Ann. Inst. H. Poincaré) to the Escande and Doveil pendulum yielding a bound which is within a factor 40.2 of the value indicated by numerical experiments.

Solitary wave solutions of a system of coupled nonlinear equations
View Description Hide DescriptionA class of coupled nonlinear waveequations is presented. It is shown that the coupled equation possesses solitary wave solutions. Some comments are made on the previously obtained solutions of a similar class of equations.

Newman–Penrose constants for zero‐rest‐mass fields on Minkowskian space‐time
View Description Hide DescriptionIt is proved that the Newman–Penrose constants associated with zero‐rest‐mass spin‐1 fields on Minkowskian space‐time are inherited from the Newman–Penrose constants associated with the complex wave function from which these fields are constructed. This latter construction is also given and a possible extension of these results to spin‐s (=0,1,2,...) fields is indicated.

A search for bilinear equations passing Hirota’s three‐soliton condition. II. mKdV‐type bilinear equations
View Description Hide DescriptionIn this paper (second in a series) [for part I, see J. Math. Phys. 3 0, 1732 (1987)] the search for bilinear equations having three‐soliton solutions continues. This time pairs of bilinear equations of the type P _{1}(D _{ x },D _{ t })F⋅G=0, P _{2}(D _{ x },D _{ t })F⋅G=0, where P _{1} is an odd polynomial and P _{2} is quadratic, are considered. The main results are the following new bilinear systems: P _{1}=a D _{ x } ^{7}+b D _{ x } ^{5} +D _{ x } ^{2} D _{ t }+D _{ y }, P _{2}=D _{ x } ^{2}; P _{1}=a D _{ x } ^{3}+b D _{ t } ^{3} +D _{ y }, P _{2}=D _{ x } D _{ t }; and P _{1}=D _{ x } D _{ t } D _{ y } +a D _{ x }+b D _{ t }, P _{2}=D _{ x } D _{ t }. In addition to these, several models with linear dispersion manifolds were obtained, as before.

Non‐Abelian Berry’s phase, accidental degeneracy, and angular momentum
View Description Hide DescriptionThe non‐Abelian Berry’s phase effect for a family of operators H _{0}+k⋅V is considered, where H _{0} is rotationally invariant, V is a vector operator, and k varies over the unit vectors in R^{3}. The parameter space is the two‐sphere. The time evolution in the adiabatic limit is given by a connnection on a fiber bundle over the two‐sphere. All connections consistent with the rotational symmetry are classified, and the time evolution is explicitly calculated for a nondegenerate Hamiltonian, as well as for a Hamiltonian with a double degeneracy. In the nondegenerate case, the connection is uniquely determined by the symmetry. In the doubly degenerate case, the connection is in some instances not determined by the symmetry. The case of approximate degeneracy is also discussed. A possible experimental test of this effect using optical pumping is described elsewhere.

Physical relations and the Weyl group
View Description Hide DescriptionLet a given particle symmetry be described by a reductive Lie groupG. It is proved that the corresponding Weyl group W(R) acts canonically in all zero‐weight spaces of G and hence, in particular, on observables. Moreover, it is shown how this W(R) action provides many physical relations, including those believed to be implied by G‐transformation properties of observables. The results simplify a testing of symmetries based on various Lie groups(algebras). Their use extends beyond particle physics, e.g., to nuclear physics.