Table of contents

The authors prove the existence of a topological horseshoe in the Henon mapping for values of the parameters such that a strange attractor is numerically observed.

The exact N-dimensional joint eigenvalue distribution is used to show that the single-eigenvalue distribution P(E) is of the form f(E-M₀/N) where M₀/N is the mean value of matrix elements. It is further shown that the distribution can be written as Wigner's distribution centred at M₀/N plus correction terms of the order of N^-1.

Low-temperature series expansions for the zero-field partition function and order parameter of the three-state Potts model have been extended to u³¹. Estimates for alpha ' and beta show rather poor convergence but are consistent with the values alpha '=¹/₃, beta =¹/₉ which characterise the 'hard hexagons' lattice gas.

Derives exact s-wave orthonormal eigenfunctions for an exponential potential in the non-relativistic case and show that the condition G(r)>or=0 for r>or=0 where G(r)= delta ( integral ₀^r(u(r))²dr) delta m does not hold in general.

For a non-local separable potential it is shown that, in general, a positive energy bound state cannot be interpreted as a zero-width resonance.

It is shown that the twisting type-N equations for empty space derived by Hauser are a special case of the equations given by Robinson et al. (1969).

The authors study a correlated-site percolation model on a square lattice in which a site is occupied if all four bonds surrounding it are occupied; this model is of possible relevance to supercooled H₂O and D₂O. Using Monte Carlo methods, they find the critical concentration of sites rho _c=0.562+or-0.001, and the critical exponents nu =1.33+or-0.07 and gamma =2.56+or-0.27. These results indicate that this correlated percolation problem is in the same universality class as uncorrelated percolation, while the critical threshold is decreased by about 5% relative to its value for the corresponding random-site percolation problems.

Using a generalised duality transformation and symmetry considerations, the authors obtain the phase diagram for Z(N) spin models. Using known properties of the Villain model, they conclude that for N>or=4 there are at least three phases, one of them being soft. For N a prime number, N>3, there are only three types of phases, two being characterised by symmetry arguments, whereas the third one is soft and has all powers of the order and disorder parameters vanishing. For N not a prime number, N>4, there are in addition to this soft phase, phases characterised by non-vanishing powers of the order or disorder parameter, with Z(N') symmetries being broken, where N' is a divisor of N.

The authors give a simplified discussion of 'secondary' and further Faddeev-Popov ghost fields occurring in covariantly quantised gauge theories with differentially constrained gauge transformations. Several examples of this new phenomenon are presented.

Free tensor potentials of arbitrary rank in covariant 'Gupta-Bleuler' and non-covariant 'Coulomb' gauges are shown to be unitarily equivalent as in the case of quantum electrodynamics. The corresponding Euclidean potentials can be constructed and some of their properties are discussed.

A new method for investigating the behaviour of lattice Hamiltonian field theories is described. The method uses finite-size scaling to extrapolate finite-lattice results to the infinite chain limit. The technique is illustrated by application to the transverse Ising model and the (O(N)-Heisenberg Hamiltonians (N=2,3) in (1+1) dimensions. The accuracy of the method appears comparable to or better than existing approaches.

The interaction of the Benjamin-Ono solitons is studied in great detail employing the exact N-soliton solution presented by Matsuno (1979). For the two-soliton case, the nature of the interaction is shown to be characterised by the ratio of the amplitudes of the two solitons. Furthermore, the initial-value problem of the linearised Benjamin-Ono equation is solved analytically and the asymptotic form of the solution for large time is given.

The authors present arguments for the detailed form of Lagrangian to be selected in gauging Z₂-graded Lie superalgebras, with particular reference to SU(n mod m). On physical grounds the use of the graded trace is unacceptable. The ordinary tract is not invariant under the whole superalgebra, though they find that a condition of 'first-order' gauge invariance can be satisfied. Certain restrictions on the gauge potentials then result, which they enumerate most specifically for SU(2/1).

The properties of a previously algebraically derived fifth label generating operator S for quadrupole-phonon states are discussed. This algebraic operator is connected with the operators following from pure group theoretical principles. A method is developed by which it is possible to calculate numerically the eigenvalues of the operator S.

Considers the covariant Laplacian in Rⁿ with Dirichlet boundary conditions on the boundary of a 'regular' region Lambda , in an arbitrary Yang-Mills field A of class C¹. The author proves that its Green function G_Lambda (A) obeys //G_Lambda (A)//<or=//G_Lambda (0)// for all A. The proof is based on a comparison theorem with finite-difference operators, and a result for gauge fields on a lattice.

A complete decomposition of the space of tensors of rank k over a vector space of dimension n into basis vectors transforming canonically under S_k*GL(n) is performed using only the properties of the symmetric group. The basis arises in a natural way (via Frobenius reciprocity) by identifying each subspace of tensors of a fixed weight lambda with the permutation representation of S_k induced by the subgroup S_lambda =S_{lambda 1}*S_{lambda 2}*...*S_{lambda n}.

An approximate procedure, first introduced by Lucke (1978) for calculating the velocity correlation function in stationary homogeneous turbulence, is here applied to the Duffing equation driven by white noise. The approximation in question is based on the application of Feynman's variational principle to the functional integral representation of a generator for the correlation functions. The spectral function for a statistically stationary state is calculated for various values of the damping and non-linear coupling constants and compared with accurate results previously obtained by Bixon and Zwanzig (1971) and by Morton and Corrsin (1970). The agreement is found to be poor except for the case of strong damping and weak non-linearity. Moreover, a simpler approximation obtained by statistical linearisation of the Duffing equation gives much better results.

A generalisation of the recurrence method of series analysis is developed which permits the analysis of power-law confluent singularities in a function from its expansion in a power series. This method yields directly critical points and associated exponents for each element of an array of Kth-order, inhomogeneous, differential equation approximants (M₀,M₁,...M_K;L). Biased approximants are also discussed. Tests are presented which show that the method can be superior to the Dlog Pade approximant for determining the dominant critical exponent in a function known to have confluent singularities, and can yield good approximations for the leading confluent exponent. An application to the problem of determining correction-to-scaling exponents in 3D spin- infinity Ising models yields results in agreement with other studies.

The products of exponentials containing various angular momentum operators are expressed on a single rotation operator. This gives new expressions for the traces of products of angular momentum operators in terms of hypergeometric functions. Since hypergeometric functions are related to Jacobi polynomials, the traces can also be expressed in terms of these polynomials.

The collision between two solitary waves in a one-dimensional Lennard-Jones chain of identical masses with nearest-neighbour interaction has been studied numerically. The authors consider two solitary waves of high energy travelling in the same direction such that they approach each other and collide. After collision there emerge again two solitary waves. A (small) disturbance, however, stays behind and the energy and momenta of the outcoming solitary waves are not equal to the corresponding quantities of the incoming ones. The effects are relatively small, however. Calculations are performed for different ratios of the velocity of the faster solitary wave to the velocity of the slower one. The effects mentioned above ultimately diminish if this ratio tends to unity. Further they calculate the influence of shifting the initial position of one of the solitons for fixed initial position of the other one. The results are interpreted in terms of the individual motion of the particles.

The authors discuss the 1/N expansion of an O(N) quartic single-mode model. The imposition of boundary conditions by the 1/N expansion is explicitly displayed, using the exact solutions. In particular, it is shown that in the negative mass case the 1/N expansion is Borel summable whereas the lambda (and h(cross)) expansions are not. The applicability of these results to more realistic models (i.e. as the first term of a strong coupling approximation scheme) is discussed.

A gauge theory for spin-; fields is discussed.

By mapping the perturbation scheme onto the ladder operator formalism, the field of application of the Schrodinger-Infeld-Hull factorisation method is enlarged. It is shown how, at each order of the perturbation, perturbed ladder operators can be constructed. Thus, without having to calculate explicitly either the excited unperturbed functions or any matrix element, one obtains analytical expressions of the perturbed eigenvalues in terms of the quantum numbers of the factorisable unperturbed problem. A three-terms recurrence relation, valid at any rank of the perturbation, is derived and leads to closed form expressions of the perturbed eigenfunctions. Consequently, a closed form expression of any matrix element on the basis of the perturbed eigenfunctions is easily obtained from the calculation of one unique particular integral.

A method is formulated to extend the field of application of the energy sudden (adiabatic) and centrifugal sudden approximations in scattering calculations. The method consists in applying the approximations in limited radial intervals and only to the coefficients of linearly independent solutions of uncoupled equations. This gives rise to a transformation on each of the channel wavefunctions at the points separating the intervals. The authors have tested the method in a two-channel model with good results.

The density matrix, rather than the wavefunction describing the system of a fixed number of non-relativistic identical particles, is subject to the second quantisation. Here the bilinear operators which move a particle from a given state to another appear and satisfy the Lie algebraic relations of the unitary group SU(p) when the dimension p to infinity . The drawing into consideration of the system with a variable number of particles implies the extension of this algebra into one of the simple Lie algebras of classical (orthogonal, symplectic or unitary) groups in the even-dimensional spaces. These Lie algebras correspond to the para-Fermi-, para-Bose- and para-uniquantisation of fields, respectively.

A harmonic quantum oscillator and the motion of charged particles in a magnetic field have been considered, the dependence on time of the magnetic field and the oscillator frequency being such that they permit us to keep an exact record of non-adiabaticity. For these systems, exact solutions, motion integrals, coherent states and Green functions have been constructed and proper amplitudes and transition probabilities calculated. One of the possible practical applications of the results obtained has been pointed out, namely their use for the interpretation of experiments with electronic vibrational transitions in molecules.

The Kerr metric admits a Killing tensor which yields a second-order constant of motion for classical trajectories. We find an explicit expression for this constant using a particular coordinate system, the oblate spheroidal. Owing to the separability of the Hamilton-Jacobi equation in this system, it is easy to show that the second-order constant is, in fact, the square of the total angular momentum of the particle at infinity, corrected with terms which arise from the non-inertial character of the coordinate system.

The Plebanski type of the Ricci tensor for all metrics admitting a four-parameter group of motions acting on non-null (space-like or time-like) hypersurfaces is determined by using the Newman-Penrose formalism and a classification method due to Ludwig and Scanlan. For each metric, all possible degeneracies of the Plebanski type are considered.

The Plebanski type of the Ricci tensor is given for all Bianchi type metrics admitting a four-parameter group of motions acting on null hypersurfaces using the Ludwig-Scanlan classification method and the Newman-Penrose formalism. A class of solutions is found for which the Ricci tensor has a null eigenvector.

Massive spherical configurations with an isothermal core and an inverse-square variation of density in the envelope have been studied. It is found that such structures are pulsationally stable. Keeping in view the relativistic restriction on the speed of a signal (v_s<or=c), it is found that K_max=0.80 for such structures. The maximum surface and central red shifts are found to be 0.502 and 28.7 respectively. An application of such structures to the case of a neutron star gives the maximum mass and size of the star to be 2.31 M(.) and 12.2 km.

A simple recursion formula is presented for calculating stationary axisymmetric (asymptotically flat) Einstein fields with any number of constants. It generates Kerr (Schwarzschild) particles from the vacuum (Minkowski space), and stationary asymptotically flat solutions from static ones.

For pt.II see ibid., vol.12, p.1061 (1979). Suggests a conserved 'energy-momentum tensor' for the Robinson-Trautman solutions of the Einstein and the Einstein-Maxwell vacuum field equations which, in the linear approximation discussed in earlier papers, is shown to yield to first order a rate of radiation of four-momentum equal to the loss of four-momentum by the source. In the case of a charged source the Larmor formula is obtained.

Considers the linearised gravitational and electromagnetic fields of an axially symmetric charged source which is rotating about its symmetry axis and moving with arbitrary acceleration along its symmetry axis when viewed in the flat background space-time. It is established that if the linearised field of the body and the linearised twist of the degenerate principal null direction of the Weyl tensor are 'wire' singularity-free, then the body must either move with zero acceleration or perform runaway motion from an unaccelerated state in the infinite past, and its mass and charge are constant at lowest order. Also, its rotation is either uniform or singular in the infinite past.

The authors introduce effective Hamiltonians for Goldstone modes of the Euclidean group, representing fluctuations in the surface of a critical droplet or in the interface between two phases. The Euclidean invariance is non-linearly realised on the Goldstone fields. The Hamiltonians are non-renormalisable in more than one dimension, showing that the disappearance of a phase transition in one dimension for systems with a discrete symmetry may be interpreted in terms of the infrared instabilities induced by these modes. The existence and form of these Hamiltonians indicates the universality of the essential singularity at a first-order phase transition in models with Euclidean invariance.

Investigates the influence of spontaneous symmetry breaking on the dynamical cluster properties of correlation functions of a quantum statistical ensemble, that is clustering with respect to time and space-time. While the impact of phase transitions on pure space-like clustering, i.e. the static results, is well known, less seems to be known about the dynamical behaviour of response functions and susceptibilities. Precise statements are made which show that, contrary to the static case, the decay properties depend sensitively on the type of Goldstone excitation.

The momentum space renormalisation group is applied to a Hamiltonian which describes site-bond correlated percolation, which in turn models the sol-gel transition when solvent effects are present. Mean field theory is used to determine the qualitative features of the phase diagram which includes sol and gel phases in either one- or two-phase regions. Recursion relations in d=6- epsilon dimensions are derived and are shown to have a fixed point corresponding to simultaneous gelation and phase separation. At this fixed point, the critical exponents describing phase separation are the same as for the Ising model nu ₀=¹/₂, eta =0 for d>4 whereas those describing gelation differ from those for the usual percolation problem: nu ₁=2/(d-2), eta ₁=0 for 4<d<6.

Exact values of the numbers of connected clusters of n sites, each site having valence no larger than nu , are presented for the simple cubic lattice for nu =2, 3, 4, 5 and 6 for small values of n. Assuming a plausible asymptotic form for the dependence of these numbers on n and nu the authors show rigorously that the exponent tau characterising the dominant singularity in the generating function is negative for nu =2 but positive for nu >or=3. Series analysis techniques suggest that the same value of tau obtains for all nu >or=3.

The first-order cumulant expansion of the real-space renormalisation group is applied to the Ising model on d-dimensional hypercubic lattices in d=2, 3, 4 and 5. These calculations enable one to study the effect of dimensionality on the first-order cumulant expansion. Monte Carlo methods are used to make the calculations possible. One result is that the scaling power a_H shows the effect of the critical dimension, while the scaling power a_epsilon does not.

It is proven by Peierls' argument in connection with reflection positivity that the lattice gas on the FCC lattice with nearest-neighbour repulsion (with interaction energy a) exists in an ordered state at low enough temperature provided the chemical potential, mu , satisfies 0< mu <4a, 4a< mu <8a or 8a< mu <12a. This result immediately carries over to the antiferromagnetic Ising model and the lattice gas with nearest-neighbour exclusion on the FCC lattice, both of which will also exist in an ordered state under suitable circumstances. In particular, the existence of a phase transition at zero magnetic field is confirmed.

A self-avoiding walk on a lattice may be characterised by the 'excluded volume ratio' V=closest approach of two centres/stop length. Walks to other than nearest-neighbour sites on a simple cubic lattice have high coordination numbers and low values of the excluded volume ratio. Some general results are presented for a class of these walks. Exact enumerations and Monte Carlo simulations have been made of the total number C_N and the mean square length (R_N²) for two examples of this class. These measurements are used to test the 'universality hypothesis' which contends that C_N approximately N^1/6 mu ^N and (R_N²) approximately N^6/5 as N to infinity , irrespective of the value of V. The data are in reasonable agreement with these statements, and the universality hypothesis is found to provide a good basis for the description of a self-avoiding walk.

The critical dynamics of the one-dimensional Ising system with conserved magnetisation is studied by the real-space renormalisation group approach. The relaxation of odd and even spin-operator perturbations is calculated. The critical exponent obtained by an exact transformation is z=3 in agreement with the conventional theory.

Clusters which just span finite lattices of various sizes have been generated using a Monte Carlo method. These have been analysed to form estimates of the mean values of cyclomatic index, valence and perimeter. In addition the shortest spanning self-avoiding walk has been characterised. The ramified nature of these clusters is discussed in terms of these properties.

The general properties of the indefinite metric Fock quantisation are studied. Some applications of the abstract construction and examples (including the four-dimensional pure gauge model) are discussed.

The signature of the metric of extended space-time is investigated in order that spontaneous symmetry breaking is allowed in electroweak interactions embedded in SU(2 mod 1). The only possibility appears to be (+---++), in the case of one lepton. Each additional lepton (or pair of flavours) adds an extra pair of time-like dimensions provided each such lepton has mass less than 53 GeV.

The authors consider a system of coagulating monomers together with an external source of monomer with arbitrary production rate p(t). The moments M_k(t)= Sigma _n=1n^k nu _n(t), where nu _n(t) is the concentration of n-mer at time t (and therefore also the weight and z average of the system), are obtained exactly as a function of p(t). The special case where the external source is the conversion of a substrate molecule to a monomer is discussed in more detail. If p(t) is an exponential, the concentrations nu _n(t) can actually be evaluated by a simple contour integral.

The electrons produced by cathode emission in a plane diode generate ions by collisions with background gas atoms. Poisson's equation is solved numerically in the steady state for non-relativistic particle motion assuming that, at the cathode surface, there is zero field and an abundant supply of zero-energy electrons. A suitable choice of non-dimensional variables enables the interelectrode potential distribution, space charge distribution and electron current density to be presented quite generally as a function of the gas filling parameters. The calculations are carried out for anode potentials up to 30 kV and for diode currents up to 1.7 times the Child-Langmuir vacuum limit. Depletion of neutral particles defines two modes of diode operation to which these calculations are applicable. The first is the pulsed mode on a time scale over which the neutral depletion is negligible and the second mode is the final steady state in which the ion flux is balanced by an opposing self-diffusion flux of neutral particles. Finally, the calculations are applied to diodes with a xenon gas filling and it is shown that the above currents can be generated with less than 1% gas scattering of the electron beam.

The magnetic properties of spin glasses are studied in a recently proposed mean field theory; in this approach the replica symmetry is broken and the order parameter is a function (q(x)) on the interval 0-1. Exact results at the critical temperature and approximated results at all the temperatures are derived. The comparison with the computer simulations is briefly presented.

A direct renormalisation method, previously applied to the self-avoiding random walk problem, is shown to give exact results for the one-dimensional random walk problem for every choice of the cell size.