Table of contents

The proper set of orthogonal polynomials to use in approximation work depends on the weight function of the problem. The author introduces a new set of polynomials orthogonal with respect to a simple closed-form approximate representation of the weight function of Case's half-range neutron transport theory (1967). The accuracy of this approximate weight function is shown to be very good, and the construction procedure of the new set of orthogonal polynomials, (C_i( mu ),c<or= mu <or=1), is demonstrated.

The set of L²(0, infinity ) functions (exp(-¹/₂ xi r^beta )r^{gamma (n+ alpha )}; n=0,1, . . . ), which is known to be complete for beta =1, gamma =2, is shown to be incomplete for all 0<2 beta < gamma and complete for all 0< gamma <or=2 beta .

The author considers certain nonlinear partial differential equations which are Korteweg-de Vries (KdV) like equations with higher order nonlinearity. It is shown that these have kink (domain wall) solutions for particular values of the coefficients of the nonlinear terms. The solutions are compared with the standard known solution of the lambda phi ²ⁿ field theories. Some conservation laws for these systems of equations are also given.

The authors present a method of calculating finite-size corrections for the isotropic (XXX) Heisenberg chain in the antiferromagnetic case. The leading corrections to the energy and the root density of the ground state are compared with the numerical data up to N=256 sites in the chain.

The magnetisation M in a three-dimensional Ising model, with only a fraction p of sites occupied randomly with spins, is found by Monte Carlo simulation to relax to zero logarithmically in time. The logarithm of the nonlinear relaxation time integral M(t) dt varies roughly as 1/(p_c-p)^1/2 in the paramagnetic region.

The authors consider indecomposable representations of the Poincare algebra iso(3,1) on the space Omega = Omega ₊ Omega _-H of its universal enveloping algebra. A master representation is obtained on Omega which induces representations on K, the invariant subalgebra of translations and on Omega _- and Omega ₊. These representations are discussed, in particular in view of finite dimensional indecomposable representations of iso(3,1). The approach taken is analogous to the approach chosen by the authors in their analysis of indecomposable representations of the Lorentz algebra so(3,1) (1983). Thus, under restriction of iso(3,1) to so(3,1) the earlier results are recovered. The interpretation of the finite dimensional indecomposable representations of iso(3,1) then follows easily as a coupling of a finite number of irreducible so(3,1) representations to an indecomposable iso(3,1) representation, with the dimension of the irreducible representations strictly increasing or strictly decreasing. The bases for the finite dimensional indecomposable iso(3,1) representations are explicitly determined, and thus also their matrix elements via the inducing representations. A formula for their dimensionalities is obtained. The methods employed are purely algebraic and follow the line of work of Jacobson and Dixmier. (1962, 1978).

Using an appropriate labelling operator constructed from representation generators, SU(3) Clebsch-Gordan coefficients are introduced whose symmetry properties are similar to those of their SU(2) counterparts. An algebraic algorithm for computing the coefficients is presented.

Starting from the s-classified SU(3) Clebsch-Gordan coefficients introduced previously (Pluhar et al. 1985) the s-classified SU(3) 3j-, 6j- and 9j-symbols are constructed and examined. They satisfy simple symmetry relations similar to those valid for the SU(2) case.

Each irreducible n-dimensional complex representation of an arbitrary group G is shown to be associated with a unique classical group. Such representations are thereby classified into 'ten classical types' associated with the respective classical groups GL(n;C), O(n;C), Sp(n;C), U(r,s), GL(n;R), GL(n/2;H), O(r,s), Sp(n;R), O(n/2;H), Sp(r/2,s/2), where r+s=n.

The stationary states of age-structured systems are analysed. The conditions of stationarity allows one to express the state probabilities in terms of age-averaged transition probabilities. A new scaling hypothesis for the age a= tau /V approximately V^-1 (V=the system extension and tau approximately _V⁰) is suggested which affords the applications of Van Kampen and Kubo-Matsuo-Kitahara approximations.

Half-order actions, i.e. actions where each term contains, at most, one spinorial derivative, of all the different known 3D supersymmetric massive vector systems are given. Since the two real Grassmann coordinates ( theta ¹, theta ²)= theta of the 3D superspace respectively constitute the two independent (fermionic) light-front projections of theta along the null bosonic directions x^+or-=2^1/2 (x⁰-or+x¹), one of them, theta ¹ is regarded as the spinorial time while the other one, theta ², is seen as a (spinorial) spacelike variable. In terms of the corresponding null spinorial timelike and spinorial spacelike derivatives the field equations contain algebraic as well as differential constraints. The authors solve them for the different known 3D vector systems and obtain their corresponding unconstrained actions in superspace. In each case their evolution is shown to be controlled by a quantity which, by analogy with the standard bosonic case, is called the superenergy of the system. The analysis of the scalar case illustrates the connection between null dynamics in superspace and the standard null dynamics in 3D bosonic spacetime. It also shows how much the superenergy contributes to the null bosonic generator of the 3D dynamics.

A detailed comparative study of the structure and workability of various rearranged versions of the Dirac method of variation of constants for perturbative treatment of quantum dynamics is presented. The variants include a modified strategy proposed recently and also a new rearrangement of this form; both of these contain lambda dependent phase factors associated with the amplitudes which are determined order-by-order by following a Brillouin-Wigner type perturbation procedure, lambda being the perturbation parameter. The following classes of perturbation problems are considered: V, (V exp(i omega t)+V Dagger exp(-i omega t)), f(t)V and f(t)(V exp(i omega t)+V Dagger exp(-i omega t)), where f(t) defines some suitable adiabatic 'switching' function. Special attention is paid to (i) the avoidance of divergent parts in the amplitude-correction terms from the scheme, (ii) applicability to cases involving a degenerate initial state and a 'resonant' harmonic perturbation, (iii) use of both exponential and non-exponential forms for f(t) and (iv) possible sources of convergence difficulties in such variants. The merits and demerits of the schemes concerned for the different problems under investigation are also listed, for convenience, in a tabular form.

Quantum mechanics is formulated on a quantum mechanical phase space. The algebra of observables and states is represented by an algebra of functions on a phase space that fulfils a certain coherence condition, expressing the quantum mechanical superposition principle. The trace operation is an integration over phase space. In the case where the canonical variables independently run from - infinity to + infinity formalism reduces to the representation of quantum mechanics by Wigner distributions. However, the notion of coherent algebra allows one to apply the formalism to spaces for which the Wigner mapping is not known. The quantum mechanics of a particle in a plane in polar coordinates is discussed as an example.

Based on previous work of Cardy (1984), the authors show in a systematic way how using conformal invariance one can determine the anomalous dimensions of various operators from finite quantum chains with different boundary conditions. The method is illustrated in the case of the three- and four-state Potts models where the anomalous dimensions of the para-fermionic operators are found.

The general solution for the Liouville equation with any given number N of singularities is considered. With the help of the inverse scattering transform method (IST) the set of regular continuous and discrete canonical variables is derived. The dynamical generators of Poincare and dilatation groups and N-soliton solutions are constructed in terms of these variables.

The authors show that an interpolated loop expansion produces a convex effective potential for Higgs fields in the vector representations of SU(N) and SO(N) and the adjoint representation of any simple Lie group, provided one considers the Higgs fields as a sector of a gauge theory and use the gauge fixing freedom to choose a 't Hooft-type gauge fixing term. The adjoint case is considered in some detail, exploiting the correspondence between the Dynkin diagrams used to classify the symmetry breaking and the Coexeter graphs that describe the symmetries of regular polytopes to determine the shape of the linearly interpolated regions that appear in the true, convex effective potential.

The action s and pseudocharge q of an electromagnetic wave are defined by 2s= integral (E.E-B.B) d⁴x and q=- integral (E.B) d⁴x. Both s and q are shown to vanish for an electromagnetic wave (in free space) which is spatially bounded at some time t=0.