Table of contents

A result of Hojman and Harleston (1981) has been used to extend the known relationship between conserved quantities and non-Noether symmetries. It is shown that N constants of the motion may be derived from each non-Noether symmetry of an N-dimensional Lagrangian system; this applies whether the symmetry is continuous or discrete. In addition, the properties of the Cartan form are utilised to to give a new proof of the Hojman-Harleston theorem.

A method for handling the inverse problem of the calculus of variations is proposed.

Presents exact solutions and eigenvalues for the Schrodinger equation with the x² + λx²/(1 + gx²) interaction on the half axis x >or= 0. The solutions are given in the form of definite integrals and the eigenvalues by means of a well defined limiting procedure.

The coordinate freedom at the post-Newtonian level is used to obtain a parametrised post-Newtonian metric from which the dominating nonlinear term is absent for all gravitational theories within this framework. This absence makes explicit the fact that gravitational nonlinearities are not observable in the post-Newtonian approximation.

Shows that quantum gravity in the first-order form is equivalent to the quantised Einstein theory only in the phases where (e_mu ^a)=e_{0 mu} ^a, det e₀ not=0. If det e₀=0 one must take into account configurations with non-zero torsion. The analogy is discussed with the first-order formulation for gauge theories and the connection-dependent counter-terms for the spinor theory.

The authors introduce a suitable set of volume-dependent coherent states and use correlation inequalities to treat the condensation problem of bosons in an external field.

A real space renormalisation group method is developed for the transfer matrix, and applied to the anisotropic two-dimensional Ising model as an example. It is possible to maintain self-duality and so to obtain the exact critical line.

To describe a polymer in a random medium, the author considers a self-avoiding walk (SAW) on a lattice whose bonds are randomly favourable (with probability 1-p) or unfavourable (with probability p). The size of a SAW of N steps is calculated when the lattice is a strip by performing products of random matrices. When the weight of unfavourable bonds tends to 0, there exist critical concentrations p_c where the size of the SAW undergoes first-order transitions. The origin of these transitions is explained as well as the fact that the limits p to 0 or p to 1 are discontinuous. Possible consequences for higher-dimensional systems are discussed.

A grand canonical ensemble of continuous chains and rings is introduced for describing polymerisation equilibrium like that of sulphur. Equivalence is shown to O(n) field theory, with n continuous, n to 1.

The critical temperature for a quenched Ising model on the square lattice, with vertical random interactions, is found exactly. The detailed phase diagram is obtained for the bond-dilute model and for the model with mixed ferro- and antiferromagnetic delta distribution of the bonds. For the class of the models with symmetric distribution of the bonds (P(J)=P(-J)) it is shown that the critical temperature is always equal to zero. An immediate consequence of this is the theorem about the absence of the second-order phase transitions for the square, fully random, quenched Ising model with symmetric distribution of the bonds.

Using the separation technique developed by Osborne and Stuart (1980), travelling wave solutions are written down for the nonlinear reaction-diffusion equation of Schlogl's second model, assuming that the necessary conditions for the static phase-coexistence are satisfied.

It is shown that Lewis's empirical, linear relationship between the average area of a cell and the number of its sides in two-dimensional mosaics corresponds to maximal arbitrariness in the cellular distribution. An expression for the distribution is given in the general case.

For pt.II see ibid., vol.14, no.7, p.1543-56 (1981). The Clebsch-Gordan coefficients for single- and double-valued corepresentations for even and odd bases of 44 anti-unitary Shubnikov (magnetic) point groups of tetragonal, orthorhombic, monoclinic and triclinic (crystal) systems have been calculated. Basis functions, multiplication tables and compatibility tables are presented. A method based on the Racah lemma has been applied. The coefficients can be applied in crystal field theory and solid state spectroscopy (selection rules, Wigner-Eckart theorem, etc.).

For pt.III see ibid., vol.15, no.3, p.711-24 (1982). The Clebsch-Gordan coefficients for single- and double-valued corepresentations for even and odd basis functions of 35 anti-unitary Shubnikov (magnetic) point groups of hexagonal and trigonal systems have been calculated. Basis functions, multiplication tables and compatibility tables are also presented. A method based on the Racah lemma has been applied. The coefficients are necessary in solving problems of crystal field theory and solid state spectroscopy (selection rules, Wigner-Eckart theorem, etc.).

Extends the author's previous work on the description of virtual relativistic particles by unitary irreducible representations of the Weyl group to the case of particles with zero on-mass-shell mass. The structure of the d-dimensional Weyl group Lie algebra on the light-cone is that of the group of inhomogeneous Galilei transformations and non-relativistic dilatations on a Euclidean space plus time of (d-2) space dimensions. The 'dilatation' generators of the two non-relativistic algebras are D+or-M^0(d-1), and an infinite transformation of these operators takes the momentum operator P^mu into (P⁺/ square root 2, O, +or-P^+or-/ square root 2) respectively, where P^+or-=(P⁰+or-P^(d-1))/ square root 2. The transformations therefore take a state of an off-mass-shell massless particle, of arbitrary momentum, transformation of these operators takes the momentum, onto the mass-shell. It is calculated how the spin and position operators transform under the infinite transformations, and express the Weyl group generators in terms of the transformed operators. A canonical form is also constructed for the transformed operators and generators, and a discussion given of the transformation properties of the single-particle states under a unitary operator of the Weyl group. The case d=4 is discussed in detail and, in particular, Weinberg's theorem on massless irreducible representations of the d=4 Poincare group is extended to the case of the Weyl group.

A general formula has been obtained for calculating the partial spatial moments of a distribution function directly from its Fourier transform. Examples to illustrate the technique are given based upon neutron diffusion and transport theory. The extensions to other areas of physics and engineering are indicated.

The authors discuss some properties of a physically interesting nonlinear integro-differential equation with periodic boundary conditions. It is the natural periodic analogue of the intermediate long wave equation, and it provides a periodic analogue of the Benjamin-Ono equation in the appropriate limit. Due to the speciality of the integral operator, the equation admits a Backlund transformation, an infinity of motion constants, etc. Two simple periodic solutions are exhibited. The equation may be transformed into more than one kind of bilinear equation.

The gauge-invariant formulation of quantum mechanics is applied to the charged harmonic oscillator in an electromagnetic field in the electric dipole approximation to obtain the probability that the oscillator is in a particular state at time t. This probability is compared with the corresponding 'probability' calculated from the conventional approach to the interaction of radiation with matter using the interaction A.p. The probabilities do not agree with each other. From the principle of gauge invariance it is concluded that the probability calculated from the gauge-invariant formulation is correct in this case, and that the conventional approach in general is incorrect.

The primitive semiclassical (EBK) theory of bound states is applied to finite non-relativistic systems with a discrete symmetry, particularly systems of identical particles. The symmetry properties of invariant tori are obtained, and illustrated for simple models with a pair of identical particles, each with one freedom. Examples without and with interaction between the particles, and with vibration and rotation are all considered. Particular care has to be taken with interacting particles that rotate around a ring.

The amplification of the unstable photon orbits in the Schwarzschild metric near r=3M by stimulated emission is discussed. The behaviour of these orbits is studied in detail, and it is shown that orbits which circle the black hole a large number of times can be greatly amplified by stimulated emission. The effects of the Doppler and gravitational red shifts are calculated. A mechanism by which a population inversion can be created in infalling atoms is suggested. An upper bound on the energy released by this process is discussed, and it is shown that the energy output by stimulated emission is a very small fraction of the total energy released by accretion of matter onto a black hole.

Considers a distribution of charged dust in rigid motion where the electromagnetic potential vector A^mu =kv^mu , v^mu being the velocity vector of the dust. Such a relation makes the magnetic field vector everywhere coincident in direction with the vorticity tensor. Without introducing any specific symmetry assumption. Maxwell's equations and some of the Einstein equations yield some constraints on the ratio of charge to mass density. The investigation further leads to some coordinate-independent formulae from which it seems possible to construct hitherto unknown solutions of the Einstein-Maxwell equations for distributions of charged dust for both constant and non-constant k.

A model is considered in which the gravitational action effectively depends upon the symmetry-broken vacuum state associated with a twisted scalar field theory. It is suggested that the author's observation of a small cosmological constant, despite predicted vacuum contributions typically of the order 10^-6 cm^-2, is a consequence of our location in the Universe.

Static spherically symmetric solutions of a particular model of gravitation, including quadratic curvature and torsion terms, are discussed and compared with those found earlier in a model described by a different action. Considerable emphasis is placed on a Lorentz group covariant formulation of the models and the role of local diffeomorphisms of space-time carefully distinguished from the structure group for gravitation. It is argued that the 'modified Poincare gauge' approach of Hehl (1980) and co-workers mixes these concepts and that their model for gravity is dynamically specified by a locally Lorentz invariant action. The logical distinction between a space-time transformation generated by symmetries of particular solutions to a model and gauge covariant symmetries that apply to all solutions is demonstrated.

It is proved that there is no possible set of field redefinitions that give linearised off-shell versions of N>or=3 extended supergravity, assuming that the associated global supersymmetry algebras have no central charges and that fermionic auxiliary fields are always monogamous. The results also prove the absence of off-shell verions of eleven-dimensional simple supergravity.

The dynamics of vibrational energy relaxation by collisions in molecular beams and free jet expansions are examined. Within the stochastic approach afforded by the use of the master equation, the incomplete relaxation process may be modelled by assuming a time-dependent transition rate matrix. For non-degenerate levels and weak interactions the state distribution is Boltzmannian if the transition rate matrix is of the Landau-Teller type. The ramifications of this result on the analysis of recent studies of vibrational relaxation in seeded beams is briefly discussed.

The stationary equilibrium of turbulent flow is a balance between the external energy input and internal viscous dissipation. The authors introduce, a natural way, a maximum chaoticity principle for turbulent equilibrium that leads to the prediction of correct energy spectra in a number of different dynamical situations. The principle has been applied to three- and two-dimensional turbulence, intermittency corrections, thermal convection, helicity turbulence and three- and two-dimensional magnetohydrodynamics. Various analogies have been pointed out with the statistical mechanics of conservation systems to which a maximum entropy principle applies.

Functional equations are derived for the eigenvalues of the row-to-row transfer matrix of the generalised hard hexagon model. These equations are exact for a lattice with N columns and are solved in the limit N to infinity . The partition function per site is rederived without the previous analyticity assumptions. The authors are also able to calculate the interfacial tension and the correlation length; the associated critical exponents are mu = nu = nu '=⁵/₆ in agreement with the scaling relations.

The electrostatic potential at point (¹/₆, ¹/₆, ¹/₆) in the unit cell of side length 2 of the NaCl lattice is found to be square root 3. An n-dimensional generalisation of a dipole sum, evaluated exactly by Glasser and Zucker (1980) in the case n=2, is given.

The axial next-nearest-neighbour Ising (ANNNI) and four-state asymmetric clock (4 ASC) models are investigated by perturbing about points at which they reduce the simple nearest-neighbour Ising models. These limits are (a) for the ANNNI model-(i) small nearest-neighbour interaction, J₁, in the x direction and (ii) small next-nearest-neighbour interaction, J₂, in the x direction, and (b) for the 4 ASC model small asymmetry, Delta . In the limit J₁ to 0, the phase boundary is located to first order in J₁ and found to be Ising-like. In the limit J₂ to 0, the behaviour to second order in J₂ is more complicated but is identical to that found to second order in Delta for the 4 ASC model. These results are discussed in the light of other recent work.

Obtains upper bounds to spin correlation functions in the thermodynamic limit of the zero external field limit for the Ising model of general spins and for the n-vector model assuming that the exchange integrals are quenched random variables and that their probability distributions are continuous. The upper bounds involve, as a factor, the corresponding spin correlation function of a uniform ferromagnetic Ising model of spin +or-1, in which the exchange integral is determined by the distributions of the random exchange integrals of the original system. Using these upper bounds, the authors find a sufficient condition on the probability distributions of the exchange integrals for the disappearance of the ferromagnetic and antiferromagnetic states in the system.

The self-avoiding walk approximation is used to give upper bounds to the zero-field spin-pair correlation function, zero-field susceptibilities and the spontaneous magnetisation for the systems of the random-bond Ising model of general spin S and the random-bond n-vector model, where the exchange integrals are assumed to be random variables. Using these bounds, the authors find a sufficient condition for the disappearance of the long-range orders. The condition is stronger for some cases than the one obtained previously by the authors. The present one applies to some cases for which the preceding one does not.

A detailed investigation of the energy deposition in, and surface reflection from, an infinite half-space has been made. Two types of source are considered: the first is a line source embedded in the medium perpendicular to the surface and the second is an incident pencil, of arbitrary direction, incident at a point. The resulting problem involves three dimensions in space and therefore requires description by a transport equation in the appropriate coordinates. The physical problem considered is that of a beam of incident ions or a line ion source in the medium. Only the fate of the foreign incident ions is considered and no attempt is made to follow the recoil atoms generated. Progress is made in the analytical solution of the problem by assuming an energy-independent mean free path and the transport approximation for the scattering kernel. The Wiener-Hopf method is used together with Fourier transforms for transverse directions. Considerable success has been achieved in obtaining exact solutions for some special limiting cases, and the numerical results which emerge are tabulated.

The statistical mechanical problem of two polymer molecules subject to a linking number constraint is solved in the so-called 'pre-averaged approximation', whereby a function of the coordinates of one of the molecules is replaced by its averaged value. For this molecule a critical winding number m_c(L,s) can be identified, depending on the total length L of the molecules and the arc length scale s over which the configurational properties are being investigated. The authors find that m_c approximately N^3/4(L/s)^1/4 and that for winding numbers m<m_c(L,L) the original entropy of the chain still dominates and the configuration remains approximately Gaussian. For m>m_c(L,L), the constraint dominates and evidence is presented to suggest that a partially collapsed solenoidal-like state is adopted, but one which is nevertheless still a Gaussian random coil on a smaller arc length scale s<s*, with s*=L(m_c/m)⁴.

Variational principles for electromagnetic fields in anisotropic inhomogeneous media for general time dependence are derived. The natural boundary conditions resulting from the formulations are investigated. By including proper boundary integrals, additional conditions may be satisfied. This leads to fewer restrictions on the trial fields and consequently a wider range of applicability.

The Fourier transforms of the lowest two orders of bridge graphs in the Abe-Meeron expansion for the one-component plasma are evaluated numerically after using the Goursat-Feynman relations to reduce the dimensionality of the integrals involved. After inverse Fourier transformation, the results are used to form approximants for the complete set of bridge graphs. These are then incorporated into modified hypernetted chain (HNC) theories which are expected to be useful in the intermediate-coupling regime. The results of the modified HNC equations improve upon the usual HNC results in terms of virial-compressibility consistency and they suggest that the excess energies found by Monte Carlo simulation may not be very reliable in the intermediate-coupling regime.

A study is made concerning the growth of discontinuities in a very hot relative plasma at temperatures of 10⁵K or above in intense magnetic fields. The effects of radiation pressure and radiation energy density have been taken into account, while the profiles structured by the radiant heat transfer are assumed to be embedded in the discontinuities. The modes of propagation of weak MHD waves have been determined. The fundamental growth equation governing the growth and decay of weak wavefronts propagating into a hot relativistic results are found to be in full agreement with earlier results. The curvature effects on the global behaviour of wave amplitude have been investigated and a finite time is determined for the formation of caustics due to focusing. The problem of the breakdown of weak waves and the consequent formation of shock waves has also been completely solved and a finite critical time t_c is determined when a weak wave will terminate into a shock wave due to nonlinear steepening. The critical amplitude of the initial wave has been determined such that any compressive wave with an initial amplitude greater than the critical one always develops into a shock wave, while an initial amplitude less than the critical one results in a decay of the wave. The relativistic, magnetic and radiation effects on the global behaviour of weak discontinuities have also been investigated and illustrated graphically.

The nonlinear interaction between waves in plasmas or other nonlinear media is studied using a general formalism based on two time scales and allowing for the presence of negative-energy waves when necessary. The use of coupled-mode theory, based on selection rules between four frequencies and four wavenumbers, allows a description of the interaction for conservative media, where the remainder of the waves outside the selected four acts as a noise in the system through an equivalent frequency mismatch. The solutions for the interacting amplitudes are given via elliptic functions, indicating a slow oscillatory exchange of energy or an explosive instability. The mere presence of negative-energy waves is not sufficient to obtain an explosive instability and a discussion is given of the necessary conditions for such an instability. For some special cases the threshold and growth rate for the instability are calculated, and it is found that the threshold increases linearly with the total equivalent frequency mismatch.

The authors briefly discuss the experimental test for detection of the anisotropic effect of the one-way velocity of light suggested by Chang (1980). In the authors' opinion Chang's proposition is rather inadequate to detect such phenomena.

Reviews the theorem that the Maxwell action, and not just the field equation, is invariant under duality transformations.