Table of contents

For particles with speed v and mass m emitted isotropically from a point source and falling under gravity g, the quantal probability density is dominated by a paraboloidal caustic decorated with paraboloidal interference fringes. Near the caustic, the fringes have a spacing independent of v and hence of wavelength, given by Delta R=3.53897*(h(cross)²/m²g)^1/3 for the first two (brightest) fringes at the level of the source. For neutrons in the Earth's field, Delta R=0.02617 mm. The effect might be difficult to detect.

Continuous symmetries of the Lorenz model and the Rikitake two-disc dynamo are given. From these symmetries the author derives constants of motion.

Manifestly conserved but time-non-local currents are exhibited for source-free Yang-Mills and gravity theories in arbitrary dimensions. They are just the familiar spinorial currents of the associated supersymmetric models, properly reinterpreted, and are infinite in number. The underlying hidden symmetry is super symmetrisability.

The problem of random graphs, which arises in the analysis of network reliability in communication theory, is considered here as a bond percolation. A closed form expression is obtained for the cluster-size generating function from which the mean cluster size as well as the percolation probability are derived. In a network of N to infinity stations in which the communication between any two stations is intact with a probability alpha /N, it is found that for alpha <or=1 the network breaks into clusters of average size of (1- alpha )^-1 stations and alpha /(1- alpha ) links, while for alpha >1 there is a non-zero percolation probability.

In the dilute s-state Potts formulation of the site-bond correlated percolation, two current treatments of the single-site interaction in the Migdal-Kadanoff renormalisation group are analysed. It is pointed out that they respect the Ising symmetry and would lead always to a critical site percolation probability p_Sc>or=¹/₂, giving a phase diagram qualitatively wrong for dimensions d>2. It is found that the origin of this defect is traced to the preservation of the Ising symmetry which is absent in the percolation problem. An alternative scheme is proposed which gives the correct behaviour.

A new method of estimating the critical percolation threshold is proposed, based on Stauffer's cluster number scaling hypothesis and the universality with respect to lattice structure of the corresponding Stauffer scaling function. This method is illustrated by obtaining estimates of the site percolation threshold for the honeycomb lattice, p_c=0.6962+or-0.0006, and for the square lattice, p_c=0.5923+or-0.0007. The error bars or 'confidence limits' of our estimates are substantially smaller than previous series estimates, and-for the honeycomb lattice-would have to be multiplied by the factor 18 to include the value, p_c=2^-12/, recently proposed by Kondor (1980) to be exact. An additional result is R=4.95+or-0.15, where R=lim_{s to infinity} R_s, and R_s is the ratio of the cluster number scaling function at its maximum to its value at p_c for clusters of size s.

The properties of the interface of a q-state Potts chain are calculated exactly. For q>2 and non-zero temperature, non-boundary states are created at the interface. The number of such states is calculated as a function of position, temperature and q.

Several extant series for the d=2, q=3 Potts model are analysed with a method that explicitly accounts for the effect of confluent corrections to scaling. Excellent agreement with the conjectured exponents for this model is found, as well as confirmation of renormalisation group estimates of the sub-leading eigenvalue.

An Ising model with competing nearest and more-than-nearest neighbour interactions on a hierarchical lattice is solved by decimation. In the context of nonlinear mappings the renormalisation group trajectories are analysed as the competition parameter is varied. Commensurate and incommensurate phases are identified.

A Hamiltonian approach to the monomer-dimer model is derived by a special truncation of the spin operators in the Yang-Lee model. At its critical point, the negative fugacity dimer problem is proved to belong to the universality class of the Yang-Lee singularity as previously conjectured.

A Heisenberg spin glass with local uniaxial anisotropy is studied at mean field level in terms of a Parisi symmetry breaking scheme. It is shown that replica symmetry breaking leads to important modifications to the replica symmetric expressions for the susceptibilities chi _T, chi _L which signal the transitions in this system.

The authors extend the work of a previous paper (1980) in which they showed that some of the phase choices in the 3jm factors of certain point group embeddings affect the orientation of the symmetry axes. The occurrence of such choices is unpredictable and their effect is sometimes subtle but once a set of 3jm factors has been calculated they can determine the orientations of the symmetry axes and mirror planes and investigate how these orientations vary with the phase choices. The properties of the reflection-rotation groups are obtained from the the isomorphic pure rotation groups but care is necessary in interpreting the effects of the corresponding operations of, for example, D₃ and C_3v.

Using the shift operators of Hughes for the group SU(3) in an O(3) basis, a simple method is developed to obtain the three-body hyperspherical functions of a definite symmetry for any angular momentum in a given SU(3) representation.

The field equation for a classical nonlinear O(2,1)-invariant sigma -model in two dimensions may be written as the single scalar equation z_uv=2z_uz_v(z+z)^-1 for a complex field z. Two infinite families of conserved currents for this equation are obtained by means of its representation as the integrability condition of a linear system of equations. One of the resulting families is of non-local type.

The differential equation for the harmonic oscillator is generalised to include an interaction potential containing a positive definite quadratic denominator. Conditions are developed under which certain eigenfunctions take the form of an exponential function multiplied by a polynomial. The problem reduces to that of finding the eigenvalues of a certain matrix that is tridiagonal in form. Properties of the eigenvalues of this matrix are investigated, since they are functions of a parameter occurring in the positive definite quadratic form. The asymptotic forms of these eigenvalues are developed together with computed results expressed as curves showing the variation of the eigenvalues with respect to this parameter.

The analyticity of the Schrodinger energy levels in the coupling constant beta for a class of confining potentials of the type V(r)=- zeta /r+ alpha r+ beta r² has been studied by using Kato-Rellich perturbation theory for linear operators.

The authors discuss a number of papers which claim that the EPR 'paradox' may be resolved using density-matrix methods. They first show that it is only inside the Copenhagen interpretation that the EPR experiment has any appearance of paradox; inside an ensemble interpretation, or hidden-variable theory, it may be analysed in a straightforward way. They then show that these 'resolutions' are implicitly using an ensemble interpretation.

The method of using square-integrable trial functions for representing resonant and scattering states is adapted to the generator-coordinate description of two-cluster systems. Numerical tests in an exactly soluble two-particle model indicate that the method is suitable for locating a resonance and is surprisingly accurate in predicting scattering phase shifts. The method is shown to be closely related to the Kohn variational scattering formalism used in the framework of the generator-coordinate method.

R-separable coordinate systems are introduced as a new class of systems in which the equations of general relativity may be solved. The static, axially symmetric vacuum and electrovac Einstein field equations are solved for two such systems, tangent spheres and bispherical. The bispherical is found to be more versatile than any previously used coordinate system in that its eigensolutions can represent the exteriors of single point, double point and line sources. The tangent sphere eigensolutions are found to be generalisations of the Curzon solution. The relatively simple nature of the individual bispherical eigensolutions allows explicit integration of the field equations for a completely general, static, two-body source. The bodies are then charged, according to the Weyl formalism, and the conditions for balance obtained. Finally, it is shown that all vacuum Weyl solutions are either type I or type D.

It is shown that if a space-time has an isometry group whose Killing vector is time-like in a region V and if there exists one closed time-like curve somewhere in the space-time, then under suitable conditions on V (including the absence of horizons) there is a closed time-like curve through every point of the space-time.

The author examines anisotropic (principal stresses unequal), conformally flat, spherically symmetric interior solutions. The properties of several solutions are compared with those of the conformally flat Schwarzschild interior.

A number of exact solutions of Einstein's equations are obtained, which describe the collisions between one scalar plane wave and one scalar, neutrino, electromagnetic or gravitational plane wave.

It is shown that a solution to this problem proposed by Komar is invalid. The author proves that the canonically quantised ADM constraints must be symmetrically ordered.

The author deals with ³/₂-spinor fields in the framework of the non-abelian Klein-Kaluza theory. He introduces a dimensional reduction procedure for a ³/₂-spinor field and a generalisation of the minimal coupling scheme. He gets dipole electric moments for a ³/₂-spin particle of value 10^-31 cm and PC breaking for a gauge group G with odd parameters. Reflections in higher (additional) dimensions were proposed as a conjugation of 'colour' charges connected with Yang-Mills fields. His approach avoids the Velo-Zwanziger paradox (acausal propagation in an external gauge field).

The geometry of four-dimensional Kahler manifolds is discussed, and it is shown that the existence of a certain constant spinor enables one to obtain relations between the spectra of wave operators of different spins in an Einstein-Kahler space. This result can be regarded as a generalisation of one discovered recently by Hawking and Pope (1978) in spaces with half-flat Riemann tensor.

The authors prove that the pressure of a system of fermion particles, interacting through a generalised mean field potential, is always greater than the pressure of an identical system of boson particles at the same density and temperature, even when phase transitions occur.

The location of the zeros of the partition function for Ising models having four-, six-, or eight-body interactions are studied. For quasi-one-dimensional models the zeros are found to lie on intersecting circles in the y plane where y=exp (2 beta h). These circles are part of a family of circles which includes the unit circle of Lee and Yang. Two, three, and four circles are needed for the four-body, six-body, and eight-body interaction systems, respectively. A counter example is found to show this behaviour does not hold for all such systems.

A large q expansion of the partition function of the anisotropic q-state Potts model in two dimensions is given up to the sixth order in q^-12/. At the critical temperature, it is in agreement with the exact solutions for the partition function and for the latent heat (the latter does not depend on the anisotropy). It does not show any simple factorisation in the parameters of the model.

The plane quadratic lattice gas with nearest-neighbour exclusion and next-nearest-neighbour finite interaction is considered in the framework of the double-site and single-square approximations of the cluster variation method and in the branched lattice approximation. In addition to the disordered and hard-core-ordered phases only one extra phase is shown to exist which arises at low temperatures near the half-maximum density. This phase which is due to the next-nearest-neighbour repulsion exhibits both first- and second-order transitions to the disordered phase.

Calculating, by a Monte Carlo technique, the conductivities of small, two-dimensional, square, bond-percolation networks (4*4 to 50*50) at a limited number of probability values, from the critical region to full conductivity, the authors have been able to show, by an original interpolation technique, that in the critical region the conductivity function does indeed obey a universal, Fisher-type, finite-size scaling function. They further show that this technique permits them to deduce values of the critical exponents for both the conductivity and the correlation length, even from calculations on relatively small networks.

A study is made of the emission effects in quantum field systems contained in the regions of space with a non-stationary boundary. The back reaction of emission on the boundary is taken into account. The classical equations of motion of a system consisting of a free massless scalar field limited in space by a massive reflecting wall (mirror) are investigated in the two-dimensional case. The system is closed, i.e. external forces are absent. This system is quantised. The effect of the reaction of emission on the trajectory of a mirror is studied. The specific peculiarities of the stimulated emission processes in closed quantum systems with boundary are established and investigated. The results are related to hadronic physics.

An approximate expression is derived for the amplitude for the spontaneous emission of a photon by a charged spin-¹/₂ fermion as it is being deflected in an external field. The fermion is coupled to a spin-zero boson field through an interaction which is local and renormalisable. The external field is in the form of an intense pulse of radiation having a well defined direction of propagation. The essential restriction on the field is that it be slowly varying; its spectral composition is otherwise arbitrary. The transition amplitude obtained here depends only on the charge, mass and anomalous magnetic moment of the fermion. This result may be thought of as providing an external-field version of the low-energy theorem for ordinary Compton scattering which has been known for some years.

A simple cubic lattice Lambda of unit spacing with a point of polarisability alpha at each lattice vertex is considered. The response of the lattice to an applied constant electric field and to the electric field due to a charged or dipolar impurity is calculated and asymptotic representations for the polarisation at lattice points far from the surface of the lattice sample and far from the impurities are obtained. Polarisation due to a constant applied external field is shown to depend on the shape of the lattice sample and the dielectric constant of the medium exterior to the lattice sample. Polarisation due to charged or dipolar impurities is shown to be independent of shape and external dielectric constant. The asymptotic representations obtained for the polarisation are compared with model calculations treating the lattice sample as a continuum dielectric of dielectric constant epsilon . This dielectric constant is found to be epsilon =(1+8 pi alpha /3)/(1-4 pi alpha /3) so that the standard Lorentz relation between macroscopic and microscopic electric fields is confirmed for all the cases considered. For the dipolar impurity case the continuum picture must use a rescaled dipole moment which is calculated explicitly. The reaction fields giving this rescaled dipole moment are discussed.

The introduction of properly defined and electric circuits in Fourier space can give a new powerful technique in studying problems of electromagnetic waves, and a novel view of the behaviour of these waves, in different applications. Using Parseval's identity theorem, one is able to obtain all the possible information for electromagnetic waves by its spatial Fourier transforms. Thus inversion in real space can be avoided completely. The authors present the theory and examples of this new technique in various applications of electromagnetic waves. The coincidence with already existing theoretical formulae or results is evidence that the proposed method is completely equivalent to the classical formulation. But, due to its simplicity, this technique could be very useful in studying electromagnetic field problems.

Analytic expressions for the magnetic field of a toroidal stellarator produced by a set of finite-size helical conductors wound on a torus have been obtained by a direct application of the Biot-Savart law. The effects of toroidal curvature have been included as toroidal perturbations which break the helical symmetry of a corresponding straight system. On expressing the problem of field line behaviour as the problem of a conservative Hamiltonian dynamical system under small non-stationary perturbations, approximate toroidal magnetic surfaces have been obtained by an asymptotic method. Numerical examples for simple cases are given to illustrate the effects of toroidal curvature.

For pt.III see ibid., vol.7, p.260 (1974). Axisymmetric disturbances that preserve their form as they move through a Bose condensate are obtained numerically by the solution of the appropriate nonlinear Schrodinger equation. A continuous family is obtained that, in the momentum (p)-energy (E) plane, consists of two branches meeting at a cusp of minimum momentum around 0.140 rho kappa ³/c² and minimum energy about 0.145 rho kappa ^3/c, where rho is density, c is the speed of sound and kappa is the quantum of circulation. For all larger p, there are two possible energy states. One (the lower branch) is (for large enough p) a vortex ring of circulation kappa ; as p to infinity its radius omega approximately (p/ pi kappa )^1/2 becomes infinite and its forward velocity tends to zero. The other (the upper branch) lacks vorticity and is a rarefaction sound pulse that becomes increasingly one dimensional as p to infinity ; its velocity approaches c for large p. The velocity of any member of the family is shown, both numerically and analytically, to be delta E/ delta p, the derivative being taken along the family.

The ground state properties of the Heisenberg antiferromagnetic spin-¹/₂ chain with interactions varying as (distance)^-p are investigated from finite cell calculations. The gap between the ground state and the first excited state as well as the end-to-end spin correlation function have been calculated for finite open chains up to 14 spins and 12 spins respectively. Aided by first-order perturbation calculations near p=0, the authors can extrapolate the gap in the thermodynamic limit, and they find that the gap vanishes for every p not=0. From the analysis of the spin correlation function they find that long-range order exists for p<1 and does not exist for p>2. The existence of long-range order in the range 1<p<2 remains an open question.

The singularity of the heat capacity for the spin-¹/₂ Ising ferromagnet arrayed on simple cubic and diamond lattices is investigated through the analysis of high-precision Monte Carlo data for the internal energy. The analysis incorporates confluent singularities. Two least-squares fitting procedures are considered. In the first one, the leading confluent singularity amplitude is zero and the next confluent singularity is included. For both lattices this procedure gives accordance with: (i) the universal value for the ratio of the leading singularity amplitudes predicted from the epsilon expansion and (ii) the theoretical expectation of equality between the non-singular parts of the heat capacity above and below the critical temperature. In the second least-squares fitting procedure, the leading confluent singularity is finite. This leads to a fit which disagrees with both (i) and (ii). The Monte Carlo results suggest that the failure of the series analysis to accord with (i) and (ii) may be due to neglect of a rather large confluent singularity in the ferromagnetic phase.

A class of new exact solutions is obtained for spherically symmetric and static configurations by considering a simple relation e^nu varies as (1+x)ⁿ. For each integral value of n the field equations can be solved exactly and one gets a new exact solution. For physical relevance of the solutions, the pressure and the density should be finite and positive and the density, P/ rho and dP/d rho should decrease as one goes outwards from the centre to the surface of the structure. Most of the exact solutions known at present are irregular in this respect. The new exact solutions for n=3, 4 and 5 are regular in this respect for a certain range of values of u(=mass/radius). The cases corresponding to n=1 and 2 are already available in the literature, being obtained by other methods. For regular solutions with dP/d rho <or=1, the maximum values of the surface and central redshifts are 0.635 and 1.614 respectively. If one assumes the surface density to be 2*10¹⁴ g cm^-3, a neutron star model corresponding to a mass up to 4.2 M_(.) can be obtained. This is an upper limit for a neutron star model based upon exact solutions with completely regular behaviour and dP/d rho <or=1. In the limiting case when dP/d rho is infinite, the surface and the central redshifts are 1.14 and 7.36 respectively. The variation of density is slow, and for a completely regular solution the maximum value for the ratio of the central to surface densities, that is rho ₀/ rho _s, is 3.0.

The authors investigate some aspects of accretion onto static black holes immersed in a uniform magnetic field. The Ernst metric is employed to find the 'Keplerian' angular momentum distribution and the efficiency of mass-to-energy conversion for a plasma and for test particles. Under almost all physically reasonable conditions for hydrodynamic accretion the effect of the magnetic field is small. However, for test particles the effect can be very important and the efficiency can approach unity.

A recent letter by J. Deenen on 'non-bijective canonical transformations in quantum mechanics' (see ibid., vol.14, p.L273, 1981) is analysed and found to be incomplete. The general representation problem of non-bijective canonical transformations is shown to involve infinite sums of the quotient and tensor product spaces defined by Plebanski and Seligman (1982).