Volume 20, Issue 1, January 1979
Index of content:

Geometry of homothetic Killing trajectories and stationary limit surfaces
View Description Hide DescriptionWe investigate the intrinsic geometry of timelike and null homothetic Killing trajectories. We do this through the use of two different Frenet–Serret formalisms developed by Synge and Bonnor. The curvature of such a nongeodesic timelike curve passing through a spacetime point is a direct measure of the expansion of the congruence at that point. Moreover, the rotation of this congruence is directly related to the torsions of its individual curves. For both classes of curves, timelike and null, the intrinsic scalars can be expressed as exponential functions of suitable parameters. We prove the following theorem twice, first by using the timelike formalism, and then through the null one. A homothetic stationary limit surface (where the homothetic Killing vector becomes null) is a null geodesic hypersurface if and only if the rotation of the homothetic Killing congruence vanishes on that hypersurface. The second, ’’null’’ approach allows us to derive this theorem directly on the hypersurface. Finally, we note that by setting a parameter equal to zero we recover all of the corresponding results for ordinary Killing vectors.

The ionization T matrix
View Description Hide DescriptionThe existence of the energy‐shell limit of the ’’post’’ and ’’prior’’ forms of the half‐off‐shell T matrix is considered for N‐particle scattering involving at most one charged fragment in the incoming channel. An approximate expression for the T matrix corresponding to the ionization of an uncharged fragment, consisting of two charged particles, by a charged particle is proposed.

An angle‐dependent lower bound on the solution of the elastic unitarity integral and a new uniqueness condition resulting from it
View Description Hide DescriptionThrough an iteration process we construct, from the angle‐dependent upper bound to the unitarity integral, an angle‐dependent lower bound and use this result to derive a uniqueness condition on the solution of the integral equation.

Topological solitons and graded Lie algebras
View Description Hide DescriptionThis paper is an extension of our previous investigation on the crossing of defects in ordered media. We provide a general mathematical framework, in which the obstructions for crossing p‐dimensional defects with q‐dimensional defects in a (p+q+1) ‐dimensional sample are the brackets of a certain graded Lie algebra (connected with homotopy theory). This ’’confinement mechanism’’ brings together the mathematical structures of two not yet related themes in present‐day physics: solitons and supersymmetry.

Application of Altman’s contractor techniques to nonlinear integral equations for Green’s functions of augmented quantum field theory
View Description Hide DescriptionAltman’s theory of contractors and contractor directions is applied to equations for connected irreducible Green’s functions of Klauder’s augmented quantum field theory in the φ^{4}model, and sufficient conditions for existence of solutions are discussed.

Spherically symmetric static conformally flat solutions in Brans–Dicke and Sen–Dunn theories of gravitation
View Description Hide DescriptionVacuum field equations for the static spherically symmetric conformally flat metric are obtained in Brans–Dicke and Sen–Dunn scalar‐tensor theories of gravitation. Closed form exact solutions to the field equations are presented and studied in both the theories.

Slow motion approximation in predictive relativistic mechanics. II. A noninteraction theorem for interactions derived from the classical field theory
View Description Hide DescriptionBy adopting an Aristotle invariant Lagrangian formalism (equivalent to canonically representing only this subgroup of the Poincaré group) and imposing a certain separability condition and a Newtonian limit on the Lagrangian, we obtain the most general Lagrangian up to c ^{−3} order that verifies these properties and leads to a relativistic invariant dynamic (i.e., it satisfies the Currie–Hill equations). It contains up to c ^{−2} order, all the Lagrangians known up to the present time. It is shown that the interactions derived from the classical field theory (CFT) do not admit approximated Lagrangians up to c ^{−4} order, and thus this constitutes a noninteraction theorem for said interactions and somehow justifies some authors’ attitudes of abandoning the Lagrangian formalism (dropping the canonical character of position coordinates) when they construct a Hamiltonian formalism for these systems.

Group theory of the interacting Boson model of the nucleus
View Description Hide DescriptionRecently Arima and Iachello proposed an interacting boson model of the nucleus involving six bosons, five in a d and one in an s state. The most general interaction in this model can then be expressed in terms of Casimir operators of the following chains of subgroups of the fundamental group U(6): U(6) ⊆U(5) ⊆O(5) ⊆O(3) ⊆O(2), U(6) ⊆O(6) ⊆O(5) ⊆O(3) ⊆O(2), U(6) ⊆SU(3) ⊆O(3) ⊆O(2). To determine the matrix elements of this interaction in, for example, a basis characterized by the irreducible representations of the first chain of groups, then we only need to evaluate the matrix elements of the Casimir operators of O(6) and SU(3) in this basis as the others are already diagonal in it. Using results of a previous publication for the basis associated with U(5) ⊆O(5) ⊆O(3), we obtain the matrix elements of the Casimir operators of O(6) and SU(3). Furthermore, we obtain explicitly the transformation brackets between states characterized by irreducible representations of the first two chains of groups. Numerical programs are being developed for these matrix elements from the relevant reduced 3j symbols for the O(5) ⊆O(3) chain of groups that were programmed previously.

Exact solutions of some multiplicative stochastic processes
View Description Hide DescriptionThe theory of multiplicative stochastic processes with completely and quasicompletely random Gaussian statistics is discussed. Operator valued equations with stochastic coefficients are solved exactly for various types of statistics using the path integral technique. Generalizations of previous results for such stochastic processes are obtained.

Variational formalism for spinning particles
View Description Hide DescriptionWe relate here the geometric formulation of Hamilton’s principle presented in our previous paper to the usual one in terms of a Lagrangian function. The exact conditions for their equivalence are obtained and a method is given for the construction of a Lagrangian function. The formalism is extended to spinning particles and a local Lagrangian is constructed in this case also. However, this function cannot be extended to a global one.

Nonlinear approach to inverse scattering
View Description Hide DescriptionThe inverse scattering problem for the scalar wave equation associated with propagation through a medium whose index of refraction differs from that of free space in a region of compact support is treated when the scattered data is given for diverse directions of (plane wave) incidence, scattered directions, and frequencies. The problem is formulated in terms of the minimization of a nonlinear functional which is bounded below, subject to constraints. It is shown that the conditional‐gradient method may be employed, the iteration process converging to stationary points. The linearized version (corresponding to the perturbed wave equation with only the linear perturbed terms retained) of the nonlinear functional is considered as a special case. In particular the linearized version related to the Born approximation leads to some additional new results.

Completely integrable systems and symplectic actions
View Description Hide DescriptionWe study results on a class of completely integrable systems, for instance, with Hamiltonian H (x,y) = (1/2) J^{ n } _{ i=1} y ^{2} _{ i } +J_{ i<j }(x _{ i }−y _{ j })^{−2} +αJ^{ n } _{ i=1} x ^{2} _{ i }, using quotient manifolds induced by symplectic group actions, which enables us to integrate the systems and understand their complete integrability. In addition, we give a natural interpretation for the scattering maps associated with these systems.

Formal quark binding and geometric strings
View Description Hide DescriptionGeometric quantization is applied to a line bundle over spacetime with structure group GL(2,C)/SL(2,C) and curvature form determined by the Ricci tensor. The Ricci form is interpreted as the Hamiltonian form for a particle moving on a string associated with a harmonic oscillator spectrum. Strings are characterized as minimal surfaces, and quarks as the gradients of minimal immersions of surfaces in a three‐dimensional spatial hypersurface, leading to a model of a baryon as the intersection of three such surfaces, and their associated quarks.

Polynomial tensors for double point groups
View Description Hide DescriptionGenerating functions for (Γ_{ r },Γ_{ m }) tensors for each pair of irreducible representations Γ_{ r } and Γ_{ m } are calculated for each double point group. A (Γ_{ r },Γ_{ m }) tensor transforms according to Γ_{ r } and its components are polynomials in another tensor transforming by Γ_{ m }. The actual tensors are given for the groups ^{(d)} C _{ n }, ^{(d)} D _{ n }, ^{(d)} T and for some representations Γ_{ m } of ^{(d)}0. Certain of the polynomialtensors provide polynomial SU(2) bases reduced according to the double point group in question.

Higher order modified potentials for the effective phase integral approximation
View Description Hide DescriptionHigher order modified potentials for the effective phase integral approximation are derived from first principles. Application is made to computing eigenvalues for a nearly free wavefunction in a lattice potential.

On nonlinear transformations for time‐dependent polynomial Hamiltonians
View Description Hide DescriptionA formal method is demonstrated for the transformation of polynomial Hamiltonians with time‐dependent coefficients to time‐independent polynomial Hamiltonians. The transformation functions are themselves polynomials in the canonical variables with time‐dependent coefficients. These coefficients are determined by sets of first order linear differential equations. The results apply equally well to multidimensional problems. The work is a generalization of the author’s earlier work on time‐dependent linear transformations for quadratic Hamiltonians.

Towards an invariant for the time‐dependent anharmonic oscillator
View Description Hide DescriptionThe Hamiltonians H= (1/2)(p ^{2}+q ^{2})+λ (t) q ^{3} and = (1/2)(P ^{2}+Q ^{2})+Q ^{3}, where (t) =1, are related by time‐dependent polynomial canonical transformations. Formulas are constructed for the generating function F _{2}(q,P,t) as well as for the direct relations between (Q,P) and (q,p). These formulas are expressed fairly concisely in terms of time integrals. The form is seen to be applicable to all polynomial transformations of time‐dependent anharmonic oscillator systems.

Percolation theory on multirooted directed graphs
View Description Hide DescriptionThe multiroot connectedness P _{ uv } of a directed graph G between the vertex u and the collection of vertices v ={v _{1},...,v _{ n }} is the probability that there are (directed) paths from u to each of the vertices v _{ i }, i=1, ...,n, when each edge and vertex has a given probability of being independently deleted. The properties of the coefficients in the expansion P _{ uv }(G) =Σ_{ A′ }⊆Ad⃗_{ uv }(G′) Π_{ a }∈A′ p _{ a } Π_{ w }∈V′ p _{ w }, where A and V are the arc and vertex sets of G respectively, p _{ a }(p _{ w }) is the probability that an arc a (vertex w) is not deleted, and G′ is the arc set A′ together with its incident vertices V′, are considered. The values of d⃗_{ uv }(G) are characterized as follows: d⃗_{ uv }(G) is shown to be nonzero if and only if G is coverable by paths and has no directed circuit. For this case d⃗_{ uv}(G′) = (−1)^{ t } _{ uv } ^{+n }=(−1)^{ν} ^{(G)}, where t _{ uv} is the maximal number of independent directed paths between u and the set v, and t _{ uv} is shown to be equal to ν (G)+n, where ν (G) =‖E‖−‖V‖+1 is the cyclomatic number of G.

Predictive relativistic mechanics
View Description Hide DescriptionWe integrate the partial differential equations on the accelerations in predictive relativistic mechanics by means of integro‐functional equations. We investigate some problems concerning the Hamiltonian formulation of this mechanics: Poisson brackets of position variables of each particle, Hamilton–Jacobi equations, and Hamiltonian formulation of the separation of the external and internal motions.

Canonical Fourier transforms
View Description Hide DescriptionAn analog of the Fourier–Plancherel transformation is developed which maps to themselves the ’’square‐traceable’’ operators in the von Neumann algebra generated by a quantum‐mechanical canonical pair (p,q). The role of translation‐invariant integral is played by the trace. The Fourier transform is given formally by (p′,q′) =sinαtr_{ p,q } e ^{−iα} ^{(p p′+q q′)} X (p,q), where α is a fixed parameter in the interval of (0,π). Comparisons are made with other Fourier‐type transformations associated with a canonical pair.