Volume 20, Issue 12, December 1979
Index of content:

Topological solution for systems of simultaneous linear equations
View Description Hide DescriptionUsing the discrete path approach, we derive an explicit topological expression for the solution of a system of simultaneous linear equations. This is done by establishing a homomorphism between the solution x _{ n }, and the set of paths, defined on the corresponding signal flow graph, from all sources to vertex n.

Ladder operators of group matrix elements
View Description Hide DescriptionAll ladder operators and some recurrence relations of the matrix elements of certain group elements of SO(3), SO(2,1), E(2), SO(4), SO(3,1), and E(3) have been explicitly determined and the underlying factorizations of the second‐ and the fourth‐order linear ordinary differential equations in terms of first‐ and second‐order ladder operators have been transparently demonstrated as an extension to the Schrödinger–Infeld–Miller factorization. These ladder operators are very useful in physical applications where the corresponding matrix elements represent certain physical transitions.

On the multiplicity‐free Wigner and Racah coefficients of U(n)
View Description Hide DescriptionA necessary and sufficient condition is found for the tensor operators of U(n) to be multiplicity‐free. At present all matrix elements for the multiplicity‐free tensor operators of U(n) are known explicitly in closed form. The phase convention given by Wong in a previous paper is modified slightly. Some inconsistencies in the equations found in the two previous papers of Wong regarding phase factors are corrected. The present phase convention for the multiplicity‐free Wigner and Racah coefficients of U(n) is the most general extension of the Condon–Shortley phase convention from U(2) to U(n).

On the definition and properties of generalized 6‐j symbols
View Description Hide DescriptionWhipple’s work on the symmetries of well‐poised _{7} F _{6}(1) and Saalschürtzian _{4} F _{3}(1) series with unit argument is applied to study the properties of 6‐j symbols generalized to any arguments. For SU2, we obtain eleven different looking _{4} F _{3}(1) series which can be used. Whipple’s parameter x s provide a good description of symmetries. We obtain quite simple recurrence relations, valid for any arguments, in terms of these parameters.

Meromorphic solutions of nonlinear partial differential equations and many‐particle completely integrable systems
View Description Hide DescriptionComplete description of meromorphic solutions of several two‐dimensional equations with algebraic laws of conservation is obtained. Among them are Zakharov–Shabat systems and, e.g., the Kadomtsev–Petiashvili equation.

A particular N‐soliton solution and scalar wave equations
View Description Hide DescriptionA special class of N‐soliton solutions of the Korteweg–deVries equation is examined. It is shown that the nonscattering wave equation based on the corresponding special reflectionless potentials can be obtained by a coordinate transformation of the ordinary wave equation and that these reflectionless potentials and others occur in the scalar wave equation on black hole geometries of general relativity.

Restoration of a functional from its functional derivatives
View Description Hide DescriptionInverse operation to a functional differentiation is considered.

Visual geometry and the algebraic properties of spinors
View Description Hide DescriptionKey words of the present paper are visualization and concrete geometry. We develop ordinary two‐component spinor algebra from a concrete geometrical model of spinor space. The null flag picture may be said to constitute such a model as far as topological and differential properties of spinors go. In our model it is also possible to visualize the a l g e b r a i c properties of spinors by straightforward geometrical constructions. Notably, an interpretation of spinor addition is given in terms of a geometrical procedure, analogous to the addition of real 3‐vectors via the parallelogram rule. By this procedure the relation between the projection of spinors on the 2‐sphere and the projection of their sum can directly be read off. The connection between null flags and our presentation of spinors is touched upon. It is planned to discuss the connection to Minkowski space more closely in a forthcoming paper.

Disequilibrium theory applied to two‐spin Glauber model
View Description Hide DescriptionThe macroscopic disequilibrium theory for Markovian relaxation processes is applied to the two‐spin Glauber model. The theory is tested rigorously by evaluation of all quantities explicitly. The probability distribution function is shown to be reproduced exactly by the minimal information procedure using the knowledge of certain macroscopic variables. The probability distribution functions obtained using a reduced number of macroscopic observables are examined and the deviations from the exact distribution function are discussed. It is concluded that the disequilibrium situation might be described satisfactorily if we choose a suitable set of macroscopic variables, but the result might crucially depend on the choice of these variables. The entropy deficiency, entropy production, and the first and the second time derivatives of the entropy production are evaluated explicitly for this model. The signs of such quantities are found to be in accordance with the predictions of more qualitative theories.

A guiding center Hamiltonian: A new approach
View Description Hide DescriptionA Hamiltonian treatment of the guiding center problem is given which employs noncanonical coordinates in phase space. Separation of the unperturbed system from the perturbation is achieved by using a coordinate transformation suggested by a theorem of Darboux. As a model to illustrate the method, motion in the magnetic field B=B (x,y) ? is studied. Lie transforms are used to carry out the perturbation expansion.

World‐line invariance in predictive mechanics
View Description Hide DescriptionThe Currie–Hill conditions for the relativistic world‐line invariance of a Newtonian‐like dynamical system of interacting particles are generalized to cover the case of invariance under any given finite‐dimensional continuous group of transformations of space–time. Necessary and locally sufficient conditions are obtained both in the general case and in the important particular case when the group of transformations includes time translations as a subgroup.

Time ordered operator cumulants: Statistical independence and noncommutativity
View Description Hide DescriptionA cumulant identity for noncommuting operator random processes is discussed. Its validity is negated by exhibiting a counterexample in which statistical independence and noncommutativity are clearly separated.

Solution of the wave equation for the logarithmic potential with application to particle spectroscopy
View Description Hide DescriptionWe present an almost complete solution of the Schrödinger equation for a logarithmic potential. In particular we obtain two pairs of high‐energy asymptotic expansions of the boundstate eigenfunctions together with a corresponding expansion of the eigenvalue determined by the secular equation. We also obtain a pair of uniformly convergent solutions and a pair of uniform asymptotic expansions. Various properties of the solutions and eigenvalues are examined, including the scattering problem of the cut‐off potential and the behavior of Regge trajectories. Finally the relevance of these investigations to the spectroscopy of heavy quark composites is discussed. In particular we point out that the relevance of the logarithmic potential can be tested only if more than two consecutive energy levels are known. In a separate paper the methods outlined here are applied to quark‐confining potentials of the generalized power type.

The complete exact solution to the translation‐invariant N‐body harmonic oscillator problem
View Description Hide DescriptionIt is shown that Schrödinger’s nonrelativistic equation for a translation‐invariant system of N particles with arbitrary masses interacting by harmonic pair potentials with arbitrary coupling constants is exactly soluble. An explicit matrix K V is given whose eigenvalues and eigenvectors determine all the exact energies and corresponding eigenfunctions of the N‐particle problem. The result is extended to include systems composed of an arbitrary number of groups of identical particles.

The radial reduced Coulomb Green’s function
View Description Hide DescriptionThe reduced Green’s function,g _{ n l }(r,r′), of the radial hydrogenic Schrödinger equation is simplified for all values of n and l, l⩽n−1, to a closed form appropriate for analytic treatments of Rayleigh–Schrödinger perturbation theory. Integral moments of the form ∫_{0} ^{∞} d r′ g _{ n l }(r,r′) (r′)^{ k+2}exp(−Z r′/n) are given. It is also shown how g _{ n l } is connected to g _{ n,l±1} by the ladder operators of the factorization method. Recursion relations are derived between integrals that arise in perturbation theory. The above results are generalized to the case l⩾n, which occurs in the Green’s functions required for the Rayleigh–Schrödinger perturbation treatment even though it does not arise for the eigenfunctions. As an example of the use of the reduced Green’s functions, the first‐order wave function and second‐order energy corresponding to the spin‐orbit interaction is evaluated for any bound state.

Does there exist a scattering theory for time automorphism groups of C*‐algebras corresponding to two‐body interactions?
View Description Hide DescriptionIt is shown that at least on the level of perturbation theory there does not exist a scattering automorphism between free time evolution and time evolution corresponding to two‐body interaction for the C*‐algebra of fermions on a lattice system.

Positive and negative frequency decompositions in curved spacetime
View Description Hide DescriptionIn this note we derive a formula for the positive and negative frequency parts of a solution in terms of the Feynman propagator. Our arguments are valid in the presence of particle creation. We also derive a formula for an operator J, that gives the particle creation rate. The formalism uses complex structures to capture the notion of positive and negative frequencies and thus avoids using analyticity arguments. The results obtained clarify the relation between approaches to quantum field theory based on the complex structure and approaches in which the propagator is the basic object. We will consider only scalar fields for simplicity.

Crossing properties of scattering operators
View Description Hide DescriptionFor scattering operators that have the cluster decomposition property, it is shown that crossing relations for the scattering operator are consequences of the assumed crossing relations of the kernels of the connected parts of the scattering operator.

A representation of multiparticle scattering operators satisfying unitarity and crossing properties
View Description Hide DescriptionA representation for multiparticle scattering operators satisfying unitarity and crossing properties is presented. The representation is given in terms of a set of functions that satisfy orthogonality, completeness and analyticity conditions. It is shown that integrals over these functions yield inclusive cross sections.

Off‐shell scattering by Coulomb‐like potentials
View Description Hide DescriptionWe derive closed expressions for and interrelationships between off‐shell and on‐shell scattering quantities for Coulomb plus short‐range potentials. In particular we introduce off‐shell Jost states and show how the transition matrices are obtained from these states. We discuss some formulas connecting the coordinate and momentum representatives of certain quantities. For the pure Coulomb case we derive analytic expressions for the Jost state and the off‐shell Jost state for l=0 in the momentum representation.