Table of contents

The geometric interpretation, due to Thierry-Mieg (1980) of the ghost field as certain components of an Ehresmann connection on the total space of a principal bundle, and of the BRS transformation as part of the exterior derivative on the total space, is discussed and amended to incorporate the spacetime dependence of the ghost field. Using this interpretation it is shown that non-trivial solutions to the Wess-Zumino consistency condition for gauge anomalies are related to non-trivial de Rham cohomology of the Lie group.

For a time-periodic Hamiltonian H(p,q,t) of period T, the area crossing a collection of curves at time 0 spanning two homotopic orbits of common period nT, in a time T, is shown to be the difference between the actions, contour integral pdq-Hdt, of the orbits. Similarly in an autonomous Hamiltonian system of two degrees of freedom the flux of energy surface volume per unit time through a surface spanning two homotopic orbits of the same energy is given by the difference between the actions, contour integral p.dq, of the orbits. Analogous results hold for pairs of orbits which converge together in both directions of time.

The bound state energies of the screened Coulomb potential-exp(- lambda r) cos( mu r)/r can be approximated on account of a scaling variational principle and in such a way that no expansion in the screening parameters is required.

The authors consider a microscopic quantum finite dynamical system from which a macroscopic description is obtained. It is shown that the mean values of observables corresponding to macroscopic states (states obtained from an operation known as a coarse-graining) are almost periodic functions of time.

Monte Carlo results are presented for the spreading of 3D percolation, where the seed consists of an (infinite) straight line. This line can either be in the bulk of the material into which spreading is possible, on a planar surface of the material, or on a rectangular edge. In the last two cases, spreading occurs only into angular regions with 180 degrees (resp 90 degrees ). While the mean distance grows at p=p_c in all three cases with the same exponent, the average number of growth sites grows (resp vanishes) in all three cases with different exponents. In particular, this implies that at p=p_c, the sites on the surface of a big cube belonging to the maximal cluster within that cube have fractal dimension <1.

The scaling amplitudes of the spin-spin and energy-energy correlation lengths for the Hamiltonian limit of the 3D Ising model are computed on a square lattice. Both periodic and antiperiodic boundary conditions are considered. The ratios of the scaling amplitudes are found to be universal.

For two-dimensional Eden clusters grown on flat substrates the way the perimeter per unit length and the surface width depend on the orientation of the substrate with respect to the lattice is measured. The evolution of the perimeter per unit length during growth is shown to be independent of the substrate length. For the surface width strong corrections to scaling are observed. The anisotropy of three-dimensional Eden clusters and its implications for the scaling of the surface width are investigated. The largest two and three-dimensional clusters studied are more than one order of magnitude larger than in previous simulations.

The authors show that the patterns in diffusion-limited aggregation (DLA) on a lattice emerge from the interplay of lattice anisotropy and fluctuations. These fluctuations can be damped by Monte Carlo averaging. Increasing its amount, the effective anisotropy becomes larger and a crossover from tip splitting typical for continuum DLA to stable tips is observed, in analogy with a number of recent experiments. The simulations suggest the following scenario for the transitions which take place as a function of the increasing effective anisotropy: disordered patterns to dendritic structures to needle crystals. It is shown that DLA clusters go through the same sequence of transitions as a function of their size. Therefore, diffusion-limited aggregates on a lattice are asymptotically not fractals.

A model of cluster aggregation with and without loops is introduced where clusters can both aggregate and fragment. In the steady state equilibrium, the clusters have a fractal dimension D=1.57+or-0.06 (2.03+or-0.05) in two (three) dimensions. The cluster-size distribution has scaling form and depends on the kinetics. The results are compared with irreversible growth and static cluster models.

A model of diffusion in one dimension, with a hierarchical pattern of hopping rates, is studied by an exact renormalisation method. Non-universal time scaling exponents are obtained for the autocorrelation function, range and average diffusion distance.

The conductivity and permeability exponents for the Swiss cheese model of continuum percolation are calculated numerically, using the transfer matrix method. In two dimensions, the conductivity exponent, t, is shown to equal its universal value t_un=1.24, while the permeability exponent e=2.53 is considerably larger. The same exponents in three dimensions are also determined and found to be t=2.46 and e=4.1, greatly exceeding the universal value t_un=1.95. The results are in fair agreement with earlier theoretical predictions.

Non-equilibrium percolating-non-percolating phase transitions in kinetic growth of percolation clusters in the presence of an external bias (biased percolation) describing the spread of forest fires with wind, etc, are investigated theoretically. A real space renormalisation group technique is used to calculate a phase diagram, critical exponents and spreading velocities explicitly in a square lattice. The effects of the bias on phase transitions and critical phenomena are clarified. The bias causes the directed spread belonging to a different universality class from that of the unbiased process. New critical exponents for spreading velocities of directed and reverse directed percolation processes are introduced and evaluated explicitly.

Forest fires under directional constraints, such as wind or local topography, generalise the bond percolation problem of symmetrical fires. The authors analyse the mass M and the radius of gyration R_g of the burning tree clusters as a function of time. M approximately t^alpha and R_g approximately t^beta , with alpha and beta being the fractal exponents.

Landau and Lifshitz, (1958) showed that phenomenological equations of extended nonequilibrium thermodynamics, reciprocity included, can be cast in Lagrangian form, so long as the kinetic equations are linear in time derivatives of the even variables. It is shown that this formalism can be extended to the general nonlinear case.

Defect-phase dynamics is incorporated to obtain a coupled set of equations of motion for many topological defects (edge-type dislocations) in a Rayleigh-Benard roll structure. An illustrative application of the theory is presented.

The authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations. The physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system. The emphasis is, however, on the mathematical aspects, i.e. the formal procedure necessary to find single solitary wave solutions. By means of examples it is shown how various cases of pulse-type and kink-type solutions are to be obtained by this method. An exhaustive list of equations so treated is presented in tabular form, together with the particular intermediate relations necessary for deriving their solutions. The extension of the technique to construct N-soliton solutions and indicate connections with other existing methods is outlined.

Dirac's conjecture is proved for a general class of systems having only first-class constraints. In other words it is shown for that kind of problem that all the primary or secondary first-class constraints generate equivalence transformations between physical states.

Cocycles for P₊ up arrow (2+1) with values in a Hilbert space are studied. It is found that cocycles for irreducible representations (apart from the case of vanishing momentum) are trivial. Further, cocycles for the action V(.)V^-1 with values in the representation space of V are investigated, and found to be trivial. Some consequences for physics are obtained.

A planar domain D contains a single line of magnetic flux Phi . Switching on Phi breaks time-reversal symmetry (T) for quantal particles with charge q moving in D, whilst preserving the geometry of classical (billiard) trajectories bouncing off the boundary delta D. If delta D is such that these classical trajectories are chaotic, the authors predict that T breaking will cause the local statistics of quantal energy levels to change their universality class, from that of the Gaussian orthogonal ensemble (GOE) of random-matrix theory to that of the Gaussian unitary ensemble (GUE). In the semiclassical limit this transition is abrupt; for statistics involving the first N levels, GUE behaviour requires that the quantum flux alpha identical to q Phi /h>>0.13N^-14/. The special flux alpha =1/2 corresponds to 'false T breaking' and for this case GOE statistics are predicted. These predictions are confirmed by numerical computation of spectral statistics for a classically chaotic billiard without symmetry, for which delta D is a cubic conformal image of the unit disc.

The authors extend the classification of symmetries necessary to predict the universality class of spectral fluctuations of quantal systems whose classical motion is chaotic, by explaining that a system with neither time-reversal symmetry (T) nor geometric symmetry may display the spectral statistics of the Gaussian orthogonal ensemble (GOE), rather than those of the Gaussian unitary ensemble (GUE), provided it possesses instead some combination of symmetries which includes T. Such combinations constitute invariance under anti-unitary transformations (whose classical analogue are called anticanonical). For a particle in a magnetic field B plus scalar potential V, an example is TS_x where S_x is a mirror reflection under which B and V are invariant. The authors illustrate this numerically for a single flux line in a hard-walled enclosure (Aharonov-Bohm quantum billiards), which also provides an example of an anti-unitary symmetry of non-geometrical origin; the spectral fluctuations are, as predicted, GOE rather than GUE.

The bound state solutions of Schrodinger's equation for the anharmonic oscillator potentials V=x²+ lambda x^2k (k=2,3, . . . ) have been investigated, using elementary techniques of low-order variational perturbation theory. For the quartic oscillator (k=2) a scaled harmonic potential provides a remarkably accurate model for all lambda . Although this model is slightly less satisfactory for higher-order anharmonicities (k>or=3), the perturbation procedures remain effective, and can be applied successfully provided that higher-order terms are calculated.

Exact eigenfunctions, which simultaneously diagonalise the Hamiltonian of a 2:1 resonant, two-dimensional harmonic oscillator and an additional constant of the motion, cubic in the cartesian displacement coordinates and momenta, are found by direct solution of the Schrodinger equation in parabolic coordinates. The connection with the usual harmonic-oscillator cartesian basis is established and used in the formulation of a second-order perturbation theory for the oscillator with a particular form of nonlinear coupling. Uniform semiclassical quantisation of the unperturbed oscillator is discussed.

A renormalised version of inner product theory is used to give accurate energies for six states of a perturbed two-dimensional oscillator and to obtain the Rayleigh-Schrodinger energy perturbation series.

The formal manipulation with delta -potentials for modelling contact interactions is compared with the recent mathematical theory of this phenomenon.

The fine and hyperfine structure energy terms of a hydrogenic atom in spherical space are obtained as functions of quantum numbers. It is found that the degenerate fine structure energy levels are split by the curvature correction +or-Ba₀²(j+¹/₂)/2R².

The geometry associated with the double-scattering contribution to elastic 3-3 collisions is investigated in detail. This makes it possible to separate clearly the dynamical content of the leading asymptotic term contributed by on-mass-shell intermediate states and to calculate by quadratures the next-to-leading term. The relevance of the present results is discussed in connection with flux calculations.

The authors study the Z(4) spin model in two dimensions in the region of parameters where a critical phase may occur. The authors results, obtained by Monte Carlo simulations, Monte Carlo renormalisation group technique and a study of the elementary excitations near the SOS model, indicate that there is no critical phase in the Z(4) model.

Improved mean-field-like approximations to critical behaviour at surfaces are considered. By using the high-temperature expansion of the 2D Ising model susceptibility, good estimates for critical values of the 3D semiinfinite Ising model; are obtained. As an example, the critical surface coupling enhancement is found to be Delta _C=0.44, which must be compared with the standard mean-field approximation value Delta _C^MF=0.25 and the recent Monte Carlo result Delta _C^MC=0.50+or-0.03.

The percolation problem in the semi-infinite plane is discussed in terms of the two competing length scales xi (p), the correlation length, and d, the distance from the surface. A crossover hypothesis for the percolation probability P(d, xi (p)) is proposed and two limiting regimes are identified. For xi (p)<<d the critical behaviour is governed by the usual exponent beta while for xi (p)>>d a new critical exponent, beta _s, is required. Using simple RSRG procedures, a sequence of approximations to beta _s/ beta are obtained and these are seen to show good agreement with a recent Monte Carlo simulation of the system. Finally the authors indicate how the techniques are applied to the semiinfinite three-dimensional case.

The mean-field theory of the discrete cubic and chiral cubic systems is shown to be exact in the N to infinity limit.

A numerical study of the SOS interface in two dimensions extends the asymptotic analysis of van Leeuwen and Hilhorst (1981). Global properties, one-point functions, and two-point correlations computed earlier, including the direct correlation function and the susceptibilities, are parameterised and their dependence on the external field is analyzed in detail. Intrinsic properties of the interface, including the new longitudinal correlation length are discussed.

The authors investigate the statistical properties of uniform star polymers with f branches, modelled on lattices in two and three dimensions. It is shown that the growth constant exists and is equal to mu ^f, where mu is the self-avoiding walk limit. The f dependence of the corresponding critical exponent gamma (f) is studied using exact enumeration and Monte Carlo techniques and the results are compared with the predictions of Miyake and Freed (1983) obtained using chain conformation space renormalisation group method.

The authors show that the reflection coefficient r of a succession of N random optical layers has a well defined, but sample-dependent, limit when N to infinity . The probability distribution or r at total reflection is found to depend only on (Imr)/ xi ^2/3 where xi ² is the noise variance. The analytical expressions appearing in this localisation problem are found to be the same as the universal functions describing intermittency with noise.

The general form of the massless spin-5/2 wave equation for a symmetrical tensor-bispinor is given. It is shown that there exists a whole class of massless equations. The general form of gauge transformation and source constraint is given, and the invariant scalar product and Lagrangian are also presented. The gamma -tracelessness of a gauge field is not needed. It appears that the Lagrangian dose not determine the equation uniquely; the knowledge of the invariant scalar product is also needed.

A distinction is drawn between construction rules leading to inhomogeneous and homogeneous random fractal configurations.

We reply to the comments made by Melrose on our earlier paper.

For original paper see Glaus, ibid., vol.18, p.L609 (1985). It is pointed out that the dimensional reduction for tree branched polymers may be derived without the replica trick.

It is demonstrated that the conformally covariant, spin-3/2 and spin-2 equations are reducible.