Volume 28, Issue 1, January 1987
Index of content:

Commutants and bicommutants of algebras of unbounded operators
View Description Hide DescriptionThe first purpose of this paper is to show that for each O p*‐algebra (M,D) whose weak commutant M^{’} _{ w } is an algebra, there exists a closed O p*‐algebra (M̂,D̂), which is the smallest extension of (M,D) satisfying M̂_{ w } =M^{’} _{ w } and M̂_{ w } D̂ =D̂. The second purpose is to characterize an unbounded bicommutant M^{‘} _{ wσ} of an O p*‐algebra M. The third purpose is to generalize the well‐known Radon–Nikodym theorem for von Neumann algebras to O p*‐algebras M satisfying the von Neumann density type theoremM̄^{ t * s } =M^{‘} _{ wσ}.

On classical theory of moments: Finite‐set‐of‐moments approach. I. Non‐negative distribution: Its even moments and Hankel transform
View Description Hide DescriptionFor an unknown non‐negative distribution Ω(z), the corresponding Hankel transformF(k) is introduced. It is proposed to partition F(k) in such a way that each component satisfies a linear differential relation whose solution gives an approximate Hankel transform in terms of a given finite set of even moments. As a result, for a known finite set of even moments, the non‐negative distribution Ω(z) is obtained in the form of a finite sum of the definite differential and integral forms of the Gaussian distributions.

Approximate representations of SU(2) ordered exponentials in the adiabatic and stochastic limits
View Description Hide DescriptionApproximate representations for the SU(2) ordered exponential U(t‖E) =(exp[i∫^{ t } _{0} d t’ σ⋅E(t’)]) _{+}, written as a functional of its input field E(t), are derived in the adiabatic ( ρ≪1) and stochastic ( ρ≫1) limits, where ρ≡‖d Ê/d t‖/E, Ê=E/E, E=+(E ^{2})^{1} ^{/} ^{2}. An algorithm is set up for the adiabatic case, and fixed‐point equations are obtained for situations of possible convergence. In the stochastic regime, ‘‘averaged’’ functions describing U(t‖E) are derived which reproduce its slowly varying dependence of large magnitude while missing, or approximating, rapid oscillations of small magnitude. Several functional integrals, analytic and machine are carried out over these approximate forms, and their results compared with the same functional integrals over the exact U(t‖E).

Asymptotically flat space‐times have no conformal Killing fields
View Description Hide DescriptionIt is shown that asymptotically flat space‐times with certain properties do not admit conformal Killing fields which are not Killing fields. The space‐times must be vacuum, asymptotically Minkowskian, and have positive Bondi energy.

Quantum theory on a regular tetrahedron
View Description Hide DescriptionQuantum theory on the surface of a regular tetrahedron is discussed. The fact that the net of the tetrahedron tiles the plane allows the propagators to be constructed as image sums of standard ones. The effect of a constant magnetic field is discussed. The field theory Casimir energy is found.

Spectral function methods for nonlinear diffusion equations
View Description Hide DescriptionTwo spectral function methods are developed for linear and nonlinear diffusionequations in one dimension where the nonlinearity is in the inhomogeneous term and occurs as a power of the solution. In the single spectral function method polynomial spectral functions in the spatial variable are introduced. The spectral resolution of the diffusionequation in the Hilbert space spanned by these functions yields a system of ordinary differential equations which is then integrated in discrete steps of the time variable. The double spectral method introduces polynomial spectral functions in both space and time variables and thereby eliminates the need for time integration through application of an iterative algorithm. Both methods are compared against analytical solutions for the linear cases and against the numerical solutions for the nonlinear cases. The second spectral function method was found to be more efficient than the first by a factor of 6 in the case of nonlinear problems.

Islands of stability and complex universality relations
View Description Hide DescriptionFor complex mappings of the type z→λz(1−z), universality constants α and δ can be defined along islands of stability lying on filamentary sequences in the complex λ plane. As the end of the filament is approached, asymptotic values α_{ N }∼λ^{ N−1} _{∞}, δ_{ N }/α^{2} _{ N }∼1 are attained, where μ_{∞}=λ_{∞}(λ_{∞}−2)/4, is associated with the limiting form of the universal function for that sequence, g(z)=1−μ_{∞} z ^{2}. These results are complex generalizations of the real mapping case (applying to tangent bifurcations and windows of stability) where μ_{∞}=2 and δ/α^{2}→ (2)/(3) correspond to the filament running along the real axis.

Classical mechanics with respect to an observer’s past light cone
View Description Hide DescriptionThis work is motivated by the fact that it is impossible for an observer to know at time t _{0} all the initial data of a system, if that data is specified in the conventional manner on the spacelike surfacet=t _{0}. A Hamiltonian formulation for classical mechanics, first given by Dirac, is exploited, in which dynamical variables are specified by their values on an observer’s past light cones. Starting from initial data given on the past light cone of an observer at some initial space‐time point, the values of the variables on the observer’s current past light cone are given by a canonical transform of the initial data. The method is illustrated for a spinless particle of mass m, which is either free or interacts with an external electromagnetic field. The remarkable result is obtained that the dynamics of this classical one‐particle spin‐0 system can be formulated in terms of a Dirac‐like spinor, whose four components are formed from the generalized coordinates and momenta.

Maxwell’s equations for a transverse field
View Description Hide DescriptionHodge–deRham theory is applied to Maxwell’sequations for a transverse electromagnetic wave with a given wave‐front surface S. It is shown that the E and B fields, considered as tangent fields to S, are harmonic in the sense of Hodge theory. If S is a spheroid it is known that the space of harmonic fields on S has dimension zero, and hence transverse fields with spheroidal wave fronts do not exist. The same result holds, but for a different reason, if S is a noncircular cylinder or a surface of revolution, and it is conjectured that smooth, singularity‐free, transverse solutions to Maxwell’sequations exist only if S is a plane or a circular cylinder.

On hyper‐relativistic quantum systems
View Description Hide DescriptionA h y p e r‐r e l a t i v i s t i c s y s t e m is defined as one whose equation of motion is form invariant under coordinate transformations induced by a semisimple group whose algebra is contractible to the algebra of the Poincaré group. Such a system lies, categorically, in the domain between the special theory of relativity and the general theory, for whereas the former requires covariance under transformations between inertial systems, the latter imposes covariance with respect to arbitrary continuous transformations. In this paper, a new interpretation of a particular fiber‐bundle structure constructed on the timelike homogeneous space M=SO(4,2)/SO(4,1) is presented, and Minkowski space‐time is realized as a subspace of the standard fiber of the tangent bundle over this hyperquadric. Through the process of group contraction, coupled with the commutation of the momentum vector fields with the principal bundle of linear frames with which the tangent bundle is associated, a hierarchy of ‘‘Heisenberg commutation relations,’’ parametrized by the point spectrum of the center of the contracted algebra, is obtained. The classical Newtonian gravitational potential field enters as the fifth coordinate of an extended space‐time manifold.

Wiener measures for path integrals with affine kinematic variables
View Description Hide DescriptionThe results obtained earlier have been generalized to show that the path integral for the affine coherent state matrix element of a unitary evolution operator exp(−i T H) can be written as a well‐defined Wiener integral, involving Wiener measure on the Lobachevsky half‐plane, in the limit that the diffusion constant diverges. This approach works for a wide class of Hamiltonians, including, e.g., −d ^{2}/d x ^{2}+V(x) on L ^{2}(R_{+}), with V sufficiently singular at x=0.

Constructive representations of propagators for quantum systems with electromagnetic fields
View Description Hide DescriptionThe quantum evolution of an N‐body system of particles that mutually interact through scalar fields and couple to an arbitrary external electromagnetic field is rigorously described. Both operator and kernel valued solutions to the evolution problem are found. Based upon a particular realization of the Dyson expansion, a convergent series representation of the propagator (the kernel of the Schrödinger time evolution operator) is obtained. The basic approach is to embed the quantum evolution problem in the larger class of evolution problems that result if mass is allowed to be complex. Quantum evolution with real mass is considered to be the boundary value of the complex mass evolution problem. The constructive representation of the propagator is determined for the class of analytic scalar and vector fields that are given as Fourier transforms of time‐dependent scalar and vector‐valued measures.

Nontrivial zeros of the Wigner (3j ) and Racah (6j ) coefficients. II. Some nonlinear solutions
View Description Hide DescriptionAdditional formulas for nontrivial zeros in the 3j and 6j symbols have been found for some higher‐order cases, i.e., where k=2, 3, and 4 (numerical examples only).

The sum rule for spectroscopic factors in the seniority scheme of identical particles
View Description Hide DescriptionSpectroscopic factors partitioning an identical‐particle state into a couple of substates with definite seniorities are summed into a simple form. This sum rule is an extension of the reduction relation for the fixed‐seniority average of a many‐body operator. The seniority projection operator is utilized throughout the present formulation

The quasiperiodic solutions to the discrete nonlinear Schrödinger equation
View Description Hide DescriptionThe quasiperiodic solutions to the discrete nonlinear Schrödinger equation are obtained by a variant of a method due to Date and Tanaka [E. Date and S. Tanaka, Suppl. Prog. Theor. Phys. 5 9, 107 (1976)]. It is shown explicitly that the nonlinear field variable at the different lattice points can be determined in a recursive fashion in terms of combinations of Reimann’s functions depending on lattice position and time.

Towards the proof of the cosmic censorship hypothesis in cosmological space‐times
View Description Hide DescriptionA theorem supporting the view that the cosmic censorship hypothesis proved recently by Krȯlak [A. Krȯlak, Gen. Relativ. Gravit. 1 5, 99 (1983); J. Class. Quantum Grav. 3, 267 (1986)] for asymptotically flat space‐times, is true in general, is generalized so that it is applicable to cosmological situations.

On singularity theorems and curvature growth
View Description Hide DescriptionIt is shown that the proofs of a series of classical singularitytheorems of general relativity can be modified such that these theorems also state the maximality of the incomplete nonspacelike geodesics. Since along maximal incomplete nonspacelike geodesics with affine parameter u certain parts of the tidal curvature cannot blow up faster than (ū−u)^{−} ^{2}, where ū is the parameter value until which the geodesics cannot be extended, the classical singularitytheorems do restrict the behavior of the curvature.

Hamiltonian dynamics of higher‐order theories of gravity
View Description Hide DescriptionThe Hamiltonian dynamics of gravitational theories with general Lagrangians quadratic in components of the curvature tensor is investigated. It is shown that the Noether generator corresponding to the action of the diffeomorphism group of space‐time naturally defines the energy‐momentum function E. The analysis of the differential of E gives rise to a formula for the symplectic two‐form Ω and thereby defines symplectic (canonical) variables of the system. This construction is four‐covariant, that is, independent of a chosen slicing of space‐time. A more thorough analysis is performed in the (3+1) decomposition. In this scheme the canonical classification of quadratic Lagrangians is presented. It singles out the class of canonically regular Lagrangians as well as four classes of degenerate ones. The Cauchy–Kowalewska problem for all these classes is formulated.

On the physical properties of a nonquadratic solution for the McVittie metric
View Description Hide DescriptionA particular class of McVittie’s new nonquadratic solutions is examined with respect to its physical characteristics. It is found that center regularity is not compatible with negative pressure and density gradients. These solutions also have the strange geometric feature that the physical radius is a decreasing function of comoving radial coordinate. Models for nonstatic ‘‘gaseous’’ spheres (i.e., the density ρ vanishes at the outer surface of the perfect fluid sphere together with the pressure p) have been constructed. The global motion is studied, and it has been found that pulsations are not possible. It is shown, however, that the density and pressure gradients are both n e g a t i v e. The pressure and the density are thus p o s i t i v e within the boundary of these ‘‘gaseous’’ spheres, and it is also seen that the density is increasing for contracting models. For the layers close to the outer boundary, it is shown that the pressure is increasing when the sphere is contracting. The speed of sound is thus real, and it is further seen that this speed is less than the speed of light. It is found that there exist bouncing models where the rate of change of circumference as measured by an observer riding on the boundary of the ‘‘gaseous’’ sphere is zero just at the moment when the sphere starts its reexpansion.

Exact self‐gravitating disks and rings: A solitonic approach
View Description Hide DescriptionThe Belinsky–Zakharov version of the inverse scattering method is used to generate a large class of solutions to the vacuum Einstein equations representing uniformly accelerating and rotating disks and rings. The solutions studied are generated from a simple class of static disks and rings that can be expressed in a simple form using suitable complex functions of the usual cylindrical coordinates.