Index of content:
Volume 19, Issue 7, July 1978

Fourth degree Casimir operator of the semisimple graded Lie algebra (Sp(2N); 2N)
View Description Hide DescriptionThe Casimir operators of the graded Lie algebra (Sp(2N);2N) [denoted also by OSp(1/2N) in the literature] are discussed. A general method, according to which the higher degree Casimir operators of the graded Lie algebras, in our case of the (Sp(2N);2N), can be constructed is developed. It is shown that the third degree Casimir operator of this graded Lie algebra does not exist. The Casimir operator of the fourth degree is derived explicitly.

A note on representation of para‐Fermi algebra
View Description Hide DescriptionRepresentation of para‐Fermi algebras is obtained utilizing the operators of a s i n g l e Fermi field. In analogy with Kalnay’s para‐Grassman algebras, para‐Clifford algebras are defined in terms of Fermi operators.

Regge pole behavior in φ^{3} field theory
View Description Hide DescriptionWe consider the Feynman amplitudes for all the essentially and crossed planar graphs of the four‐point vertex function in φ^{3} field theory, and we evaluate their behavior at high energy (large s, fixed t). We compute the coefficients of all logarithms for the dominant amplitudes which behave in s ^{−1} (up to logarithms of s). This computation is performed by using the Bogoliubov–Parasiuk–Hepp R operation and the Mellin transform of the Feynman amplitudes. The geometrical structure of the coefficients is such that all logarithms of s of all dominant amplitudes can be summed to give the well‐known Regge behavior with signature +. The Regge trajectory verifies an equation which may be solved explicitly in the lowest order approximation; the residue is found to be the ratio of two functions of t, the upper one being factorized into two vertex functions expressed as infinite series and the lower one providing a ghost killing factor.

A note on recoupling coefficients for SU(3)
View Description Hide DescriptionA 6‐ (λμ) coefficient, denoted by Z and different from the usual U coefficient, associated with a specific recoupling of three irreducible representations of SU(3), is defined. A general 9‐ (λμ) coefficient, analogous to the unitary 9‐J coefficient of the angular momentum Racah algebra, is then expressed in terms of the Z coefficient and two U coefficients. In this way problems associated with the existence of outer multiplicities in the products of irreducible representations of SU(3) are circumvented.

Solution of the almost‐Killing equation and conformal almost‐Killing equation in the Kerr spacetime
View Description Hide DescriptionFour linearly independent classes of vector solutions to the generalized almost‐Killing equation in the Kerr spacetime are presented in terms of Teukolsky’s radial and angular functions. The vector solutions which are asymptotic to the ten Minkowski‐space Killing vectors are given by way of example.

Nonuniqueness in the inverse scattering problem
View Description Hide DescriptionThe inverse scattering problem consists of determining the functional form of a scattering potential given the scattering matrix A (k _{0}s, k _{0}s_{0}) for all scattering directions s and one or more values of the wave vector k _{0}s_{0}. In this paper it is shown that within the framework of the first Born approximation the inverse scattering problem as defined above does not possess a unique solution. It is also shown that within the framework of exact (potential) scattering theory the problem does not admit a unique solution given only the scattering matrix for a single fixed value of the wave vector k _{0}s_{0} as data. The final section in the paper considers scattering experiments using incident fields other than plane waves and where knowledge of the scattered field at all points exterior to the scattering volume is available as data. It is found that, within the framework of exact scattering theory, the data generated by any s i n g l e such experiment is not sufficient to uniquely specify the scattering potential while, within the framework of the first Born approximation, the data generated by any f i n i t e number of such experiments is not sufficient to uniquely specify the potential.

Classical and relativistic vorticity in a semi‐Riemannian manifold
View Description Hide DescriptionIt is shown that a form of the Cauchy–Lagrange formula for the evolution of vorticity in a barotropic flow generalizes to the case of ideal fluid motion on higher‐dimensional Riemannian or semi‐Riemannian manifolds.

Spatial conformal flatness in homogeneous and inhomogeneous cosmologies
View Description Hide DescriptionIn the past few years there has been a growing interest in cosmological models which are not spatially homogeneous. The assumption of spatial homogeneity simplifies the Einstein equations to ordinary differential equations. If the assumption of spatial homogeneity is relaxed, some other symmetries are needed to make the Einstein equations mathematically tractable. The recently discovered solutions of Szekeres have been found to possess an interesting type of symmetry: The three spaces orthogonal to the fluid flow are conformally flat. Herein, we prove a theorem restricting the possible inhomogeneous cosmologies with conformally flat 3‐surfaces. We determine which spatially homogeneous models admit conformally flat 3‐surfaces. This information, although interesting in its own right, will serve as a guide in determining those spatially homogeneous models that may be generalized by retaining spatial conformal flatness but relaxing the condition of spatial homogeneity.

A unified treatment of null and spatial infinity in general relativity. I. Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity
View Description Hide DescriptionA new definition of asymptotic flatness in both null and spacelike directions is introduced. Notions relevant to the null regime are borrowed directly from Penrose’s definition of weak asymptotic simplicity. In the spatial regime, however, a new approach is adopted. The key feature of this approach is that it uses only those notions which refer to space–time as a whole, thereby avoiding the use of a initial value formulation, and, consequently, of a splitting of space–time into space and time. It is shown that the resulting description of asymptotic flatness not only encompasses the essential physical ideas behind the more familiar approaches based on the initial value formulation, but also succeeds in avoiding the global problems that usually arise. A certain 4‐manifold—called Spi (spatial infinity) —is constructed using well‐behaved, asymptotically geodesic, spacelike curves in the physical space–time. The structure of Spi is discussed in detail; in many ways, Spi turns out to be the spatial analog of I. The group of asymptotic symmetries at spatial infinity is examined. In its structure, this group turns out to be very similar to the BMS group. It is further shown that for the class of asymptotically flat space–times satisfying an additional condition on the (asymptotic behavior of the ’’magnetic’’ part of the) Weyl tensor, a Poincaré (sub‐) group can be selected from the group of asymptotic symmetries in a canonical way. (This additional condition is rather weak: In essence, it requires only that the angular momentum contribution to the asymptotic curvature be of a higher order than the energy–momentum contribution.) Thus, for this (apparently large) class of space–times, the symmetry group at spatial infinity is just the Poincaré group. Scalar, electromagnetic and gravitational fields are then considered, and their limiting behavior at spatial infinity is examined. In each case, the asymptotic field satisfies a simple, linear differential equation. Finally, conserved quantities are constructed using these asymptotic fields. Total charge and 4‐momentum are defined for arbitrary asymptotically flat space–times. These definitions agree with those in the literature, but have a further advantage of being both intrinsic and free of ambiguities which usually arise from global problems. A definition of angular momentum is then proposed for the class of space–times satisfying the additional condition on the (asymptotic behavior of the) Weyl tensor. This definition is intimately intertwined with the fact that, for this class of space–times, the group of asymptotic symmetries at spatial infinity is just the Poincaré group; in particular, the definition is free of super‐translation ambiguities. It is shown that this angular momentum has the correct transformation properties. In the next paper, the formalism developed here will be seen to provide a platform for discussing in detail the relationship between the structure of the gravitational field at null infinity and that at spatial infinity.

A technique for analyzing the structure of isometries
View Description Hide DescriptionA new technique is introduced to investigate the structure of isometry Lie algebras. Some general results are first proved by applying this technique to n‐dimensional manifolds equipped with metrics of arbitrary signature. A restriction is then made to 3‐manifolds representing the space of orbits of the timelike Killing field in stationary space–times. Under the assumption of asymptotic flatness at s p a t i a l infinity, a complete description of isometry Lie algebras of these 3‐manifolds is obtained. As corollaries, several results about symmetries of stationary isolated systems in general relativity are proved.

On the separability of the sine‐Gordon equation and similar quasilinear partial differential equations
View Description Hide DescriptionThe separability of the sine‐Gordon equation (SGE) is defined and studied in detail. We find a general class of dependent‐variable transformations under which the SGE is separable. This class may be reduced to a two‐parameter generalization of the usual transformation adopted, by requiring the transformations to reduce to the identity in the linear limit of the SGE (i.e., the Klein–Gordon equation). The method developed for studying the separability of the SGE is then applied to more general quasilinear equations and a discussion of the limitations of the method, and of separable solutions in general, is also given.

Existence of instantaneous Cauchy surfaces
View Description Hide DescriptionSeveral properties of instantaneous Cauchy surfaces are obtained. It is shown that a strongly causal spacetime admits an instantaneous Cauchy surface through each of its points, that there is a close and reversible relationship between these surfaces and maximal open globally hyperbolic subsets, that every instantaneous Cauchy surface is contained in a maximal instantaneous Cauchy surface, and that the latter surface is a maximal achronal surface which separates spacetime into past, present, and future. Some other properties of instantaneous Cauchy surfaces are discussed along with a refinement of an earlier topology change property.

Discrete finite nilpotent Lie analogs: New models for unified gauge field theory
View Description Hide DescriptionTo each finite dimensional real Lie algebra with integer structure constants there corresponds a countable family of discrete finite nilpotent Lie analogs. Each finite Lie analog maps exponentially onto a finite unipotent group G, and is isomorphic to the Lie algebra of G. Reformulation of quantum field theory in discrete finite form, utilizing nilpotent Lie analogs, should elminate all divergence problems even though some non‐Abelian gauge symmetry may not be spontaneously broken. Preliminary results in the new finite representation theory indicate that a natural hierarchy of spontaneously broken symmetries can arise from a single unbroken non‐Abelian gauge symmetry, and suggest the possibility of a new unified group theoretic interpretation for hadron colors and flavors.

On a possible experiment to evaluate the validity of the one‐speed or constant cross section model of the neutron‐transport equation
View Description Hide DescriptionThe inverse problem for a half‐space is solved (for isotropic scattering) to yield results that suggest an idealized experiment that could be used to evaluate in a new way the validity of the one‐speed or constant cross section model of the neutron‐transport equation.

Lorentz transformations of observable and ghost particle states in quantum electrodynamics and in a massive gauge theory
View Description Hide DescriptionDerivations are given of the effect of Lorentz boosts on physical particle and ghost states in quantum electrodynamics. It is shown that the photon helicity is an invariant even though, in general, Lorentz boosts transform the transverse, longitudinal, and timelike components of the vector potential into each other. A similar calculation is made for an Abelian gauge theory in which the particles have dynamical mass.

Solution of a second‐order integro–differential equation which occurs in laser modelocking
View Description Hide DescriptionWe solve the following integro–differential equation for the eigenfunction u (x): s u (x) ={d ^{2}/d x ^{2}−[1−6sech^{2}(x)]}u (x) −εF^{∞} _{−∞} u (x′)sech(x′) d x′ (1+ν^{2} d ^{2}/d x ^{2})sech(x), where s is the eigenvalue and ε and ν are arbitrary parameters which need not be small. This equation occurs in laser modelocking theory in the analysis of pulse stability, and s is proportional to the rate of growth of perturbations. We expand the eigenfunction u (x) in terms of a convenient basis set Λ (x,k) satisfying −k ^{2}Λ (x,k) =[d ^{2}/d x ^{2}+6sech^{2}(x)]Λ (x,k). We find two discrete eigenfunctions u _{0}(x) and u _{1}(x) and a continuum u (x,s). We find that the lowest eigenvalues _{0}(ε) is −3 at ε=0 for finite or zero ν, and that the point where s _{0} =0 the parameters ε and ν obey ε=2/(1−ν^{2}). This is the zero growth point at which no eigenfunction u (x) has a positive eigenvalues.

A theory of classical limit for quantum theories which are defined by real Lie algebras
View Description Hide DescriptionA theory of classical limit is developed for quantum theories, the basic observables of which correspond to elements in some real Lie algebraL _{0}. For both quantum and classical systems based on L _{0} the basic observables are contained in a unique universal algebra. This is the universal enveloping algebraU for the quantum case, and a universal commutative Poisson algebraL for the classical case. U and L are connected by a system of contraction maps. For certain sequences of representations and of vector states defined by them renormalized expectation values of the quantum variables are shown to converge to values of the corresponding classical variables at some point in the classical phase space. The classical phase space is obtained as a limit of certain systems of coherent states. The general theory is illustrated by several examples and counterexamples.

Exact vacuum solutions of Einstein’s equation from linearized solutions
View Description Hide DescriptionIt is proved that if (M,g _{ a b }) is an exact vacuum solution of Einstein’s equation,l _{ a } a null vector field and if l _{ a } l _{ b } satisfies the linearized equation on background (M,g _{ a b }), then g _{ a b }+l _{ a } l _{ b } is an exact vacuum solution. Applications to the search for asymptotically flat spacetimes are discussed.

N‐body quantum scattering theory in two Hilbert spaces. II. Some asymptotic limits
View Description Hide DescriptionWithin the framework of two‐Hilbert space scattering theory the existence of the strong Abel limit of a certain operator is proved, leading to the following results. A generalized Lippmann identity is derived that is valid for all channels, rather than only two‐body channels. On shell equivalence of the prior, post and AGS transition operators is rigorously proved, thus closing a gap in previous proofs. Results concerning the existence of the scattering operator as a strong, rather than weak, Abel limit are presented, and their implications with respect to the problem of unitarity are discussed. Finally, the possibility of exploiting operator limits of the Obermann–Wollenberg type is studied, with negative results.

An investigation of some of the kinematical aspects of plane symmetric space–times
View Description Hide DescriptionA short review of the literature on plane symmetric space–times (PSSTS) is given in the Introduction. The rest of the paper concerns itself with an investigation of some of the kinematical aspects of PSSTS, i.e., properties of PSSTS which do not depend on the field equations. In particular, the existence of four special coordinate systems is considered. It is shown that the existence of these coordinate systems is not guaranteed for a general C ^{ k } (k⩽1) plane symmetric metric (PSM). For k=2, two of the coordinate systems exist in a weak sense whereas the existence of the other two is not guaranteed in any sense. A local intrinsic type classification is introduced in Sec. 3, and it is shown that the existence of an extra Killing vector is correlated to the classification. Finally, the local equivalence of two given PSSTS is considered in Sec. 4. It is shown that some algebraic equations arise from the analysis. These algebraic equations may lead directly to the solution of the problem of the local equivalence of two given PSSTS.