Volume 16, Issue 1, January 1975
Index of content:

Prolongation structures of nonlinear evolution equations
View Description Hide DescriptionA technique is developed for systematically deriving a ’’prolongation structure’’—a set of interrelated potentials and pseudopotentials—for nonlinear partial differential equations in two independent variables. When this is applied to the Korteweg−de Vries equation, a new infinite set of conserved quantities is obtained. Known solution techniques are shown to result from the discovery of such a structure: related partial differential equations for the potential functions, linear ’’inverse scattering’’equations for auxiliary functions, Bäcklund transformations. Generalizations of these techniques will result from the use of irreducible matrix representations of the prolongation structure.

Singularities for fluids with p=ω equation of state
View Description Hide DescriptionThe structure of singularities is discussed for some exact solutions of Einstein equations for irrotational perfect fluids with equation of state pressure equal to energy density, p=ω. It is found that all singularities studied are velocity−dominated of the semi−Kasner class. It is also found that data on the singularity are not enough to generate space−time for all times.

Analytic vector harmonic expansions on S U (2) and S ^{2}
View Description Hide DescriptionIrreducible vector fields (vector harmonics) are introduced on S U (2). It is shown that an arbitrary vector field can be expanded in terms of these vector harmonics, and tight convergence conditions are derived for analytic vector fields. These expansions are related to well−known vector harmonic expansions on the two−sphere. The generalization to arbitrary tensor fields is discussed. A connection between the Lie algebra of vector fields on S U (2) and the Virasoro algebra is noted.

Dual trees and resummation theorems
View Description Hide DescriptionVarious resummation theorems known for graphical expansions in statistical physics and field theory are exhibited as special cases of a single symmetrical, easily remembered theorem on a generalized structure we call a dual tree.

A new derivation of some fluctuation theorems in statistical mechanics
View Description Hide DescriptionA simple derivation is given of some of the fluctuationtheorems of statistical mechanics which relate integrals of molecular distribution functions to thermodynamic properties. The derivation employs the generating function for the probability P _{ω}(n) that a domain ω contains n particles. Various forms of the generating function are derived, and each leads to a different form of the fluctuationtheorems.

Gravitational and electromagnetic radiation in Kerr−Maxwell spaces
View Description Hide DescriptionThe class of Kerr−Maxwell spaces is defined. This class consists of regular electrovac spacetimes in which a geodesic, diverging, shear−free principal null vector field of the Weyl tensor coincides with a principal null vector field of the Maxwelltensor. It is shown that this class admits no Petrov−Penrose type III or type N solutions. It is also shown that the most general nonradiative solution is the Kerr−Newman metric.

Stationary Kerr−Maxwell spaces
View Description Hide DescriptionWe consider algebraically special electrovac spacetimes in which a diverging, geodesic, shear−free (but twisting) degenerate principal null direction of the Weyl tensor coincides with a principal null direction of the Maxwelltensor. All stationary solutions of this class are solved exactly and found to be of Petrov−Penrose type D, the most general regular stationary solution being the Kerr−Newmann metric.

Propagation of transients in a random medium
View Description Hide DescriptionThe propagation of transient scalar waves in a three−dimensional random medium is considered. The analysis is based on the smoothing method. An integro−differential equation for the coherent (or average) wave is derived and solved for the case of a statistically homogeneous and isotropic medium and a delta−function source. This yields the coherent Green’s function of the medium. It is found that the waveform of the coherent wave depends generally on the distance from the source measured in terms of a certain dimensionless parameter. Based on the magnitude of this parameter, three propagation zones, called the near zone, the far zone, and the intermediate zone, are defined. In the near zone the evolution of the waveform is determined primarily by attenuation of the high−frequency components of the wave, whereas in the far zone it is determined mainly by dispersion of the low−frequency components. The intermediate zone is a region of transition between the near and far zones. The results show that, in general, the randomness of the medium causes a gradual smoothing and broadening of the waveform, as well as a decrease in amplitude of the wave, with propagation distance. In addition, the propagation speed of the wave is reduced. It is also found that an oscillating tail appears on the waveform as the propagation distance increases.

Discrete state perturbation theory via Green’s functions
View Description Hide DescriptionThe exposition of stationary state perturbation theory via the Green’s function method in Goldberger and Watson’s C o l l i s i o n T h e o r y is reworked in a way that makes explicit its mathematical basis. It is stressed that the theory consists of the construction of, and manipulations on, a mathematical identity. The perturbation series fall out of the identity almost immediately. The logical status of the method is commented on.

An exact determination of the gravitational potentials g _{ i j } in terms of the gravitational fields R ^{ l } _{ i j k }
View Description Hide DescriptionUnder a certain asymmetry assumption, the gravitational potentials g _{ i j } are determined up to a conformal factor from the field R ^{ l } _{ i j k }. This constitutes a partial solution to Einstein’s equations in the general case. Moreover, the solutions are simple in that the g _{ i j } are expressed as polynomial functions of the R ^{ l } _{ i j k }.

On resonance and stability of conservative systems
View Description Hide DescriptionResonance and stability of conservative systems are considered by means of a perturbation method similar to the averaging method of Bogoliubov. The accuracy of the method is tested by numerical simulations and by comparing the conditions for stability derived here with the well−known conditions given by Moser and Arnold.

Motion of a body in general relativity
View Description Hide DescriptionA simple theorem, whose physical interpretation is that an isolated, gravitating body in general relativity moves approximately along a geodesic, is obtained.

A dispersion series for nonlocal potentials
View Description Hide DescriptionFor a class of short−range nonlocal potentials, and for the energy variable E in a certain part of the complex plane, we obtain a generalized subtracted dispersion relation which relates the forward scattering amplitude to contributions from negative energy pole terms, a usual dispersion integral along the positive real axis of the complex energy plane, and a uniformly convergent infinite series, apart from subtraction terms, subject to the condition that no bound state exists with energy less than −γ^{2}, where γ is some parameter of

Casimir operators of complementary unitary groups
View Description Hide DescriptionA relationship between all the generalized Casimir operators of complementary unitary groups is derived, both in the fermion and boson realizations of the corresponding Lie algebras. It is shown that the number of independent Casimir operators of unitary groups reduces essentially to half the number for self−conjugate irreducible representations.

Ghost neutrinos in plane−symmetric spacetimes
View Description Hide DescriptionAn exact solution to the Einstein−Dirac equations is presented for a plane−symmetric spacetime generated by neutrinos. The neutrino field is nonzero and corresponds to a neutrino current along the symmetry axis of the space. The neutrinos yield a nonzero energy−momentum tensor, which we specialize to T _{ i j } = 0 for ’’ghost’’ neutrinos. We show that, since the energy−momentum tensor vanishes, the time−dependent ’’ghost’’ neutrino metric reduces to the static case. The time−dependent ’’ghost’’ current is then reduced to the static current through a Lorentz transformation and the ’’ghost’’ wavefunction reduced to the static wavefunction through a spinor transformation. The ’’ghost’’ neutrino current is geodesic and the spacetime is classified by the expansion, rotation, and shear of these geodesics. From previous results it follows that our plane−symmetric ’’ghost’’ solution is the most general solution to the Einstein−Dirac equations for a vanishing energy−momentum tensor and a neutrino current that is expanding. The solution is Petrov type D.

Neutrinos in cylindrically−symmetric spacetimes
View Description Hide DescriptionAn exact solution to the Einstein−Dirac equations is presented for a static, cylindrically−symmetric spacetime. The neutrino field is nonzero and corresponds to a neutrino current in the radial direction. The neutrinos yield a zero energy−momentum tensor and therefore the gravitational field is the same as for the vacuum case. The neutrinos in these static cylindrically−symmetric spacetimes can exist only if the spacetimes are locally equivalent to static plane−symmetric spacetimes. This type of ’’ghost neutrino’’ solution is already known to exist in plane−symmetric spacetimes.

Canonical transformation and accidental degeneracy. III. A unified approach to the problem
View Description Hide DescriptionWe continue the discussion of the groups of canonical transformations responsible for accidental degeneracy in quantum mechanical problems. A general unified treatment is provided for a wide class of two−dimensional physical systems, having an energy spectrum which is a linear combination of two quantum numbers. The general method involves the use of both nonorthonormal and orthonormal sets of states to construct groups of complex or real canonical transformations, mapping the problem under consideration onto the two−dimensional isotropic harmonic oscillator. The group responsible for the accidental degeneracy is then quite obviously S U (2). The problem of an isotropic oscillator in a sector π/q (q integer) was discussed previously using a nonorthonormal basis. In the present paper we carry the analysis in an orthonormal basis to establish the general procedure mentioned above. We also analyze in detail the Calogero problem for three particles which has a spectrum of the type given above, and obtain explicitly the canonical transformation that maps it on the anisotropicoscillator whose ratio of frequencies is 2/3 and subsequently on the isotropic one.

On the exact solutions of the Wick−rotated fermion−antifermion Bethe−Salpeter equation
View Description Hide DescriptionExact solutions of the fermion−antifermion Bethe−Salpeter equation with harmonic kernels are given for bound states with zero mass. The

Four examples of the inverse method as a canonical transformation
View Description Hide DescriptionThe Toda lattice, the nonlinear Schrödinger equation, the sine−Gordon equation, and the Korteweg−de Vries equation are four nonlinear equations of physical importance which have recently been solved by the inverse method. For these examples, this method of solution is interpreted as a canonical transformation from the initial Hamiltonian dynamics to an ’’action−angle’’ form. This canonical structure clarifies the independence of an infinite number of constants of the motion and indicates the special nature of the solution by the inverse method.

A stochastic Gaussian beam. II
View Description Hide DescriptionThe propagation of a Gaussian beam in a strongly focusing medium is considered. The medium is subject to random deformations of the beam axis. The average intensity and the intensity fluctuations on the beam axis and the mean population remaining in the fundamental mode are computed when the random inhomogeneities are weak and the distance between the source and observation points is large. All results for random axis deformations are compared to those obtained earlier for random width perturbations. The mean intensity off the beam axis and the mean population transfer into higher modes are also discussed.