Table of contents

Some possible symmetry breaking patterns for unitary group gauge theoretic models based on Higgs scalars in the third-rank totally antisymmetric tensor representations of U(n) are studied. The critical points are expressed in terms of a single parameter xi such that 1/n<or= xi <or=1/3. It is shown that spontaneous symmetry breaking may take place from SU(n) to SU(3)*SU(n-3) for all n, from SU(6) to SU(3)*SU(3), from SU(7) to G₂ from SU(8) to SU(3) and from SU(9) to U(1)*U(1)*U(1)*U(1). The absolute minima are shown to be degenerate for n>or=9, and remarks are made concerning a conjecture of Michel (1979).

The authors introduce a notion of hierarchical potential by iterating a basic elementary geometric construction over an increasing sequence of length scales. They then show that a quantum particle in such a potential exhibits a subdiffusive behaviour characterised by (r²)<or=C(ln t)^beta if the initial state is a normalised wavepacket superposition of states of sufficiently low energy.

A hierarchical model is introduced to describe chemically limited cluster-cluster aggregation. The fractal dimension of the clusters, D, as well as an exponent characterising the number of active sites per cluster are evaluated numerically for d=2,3,4. The relevance of the model to realistic physical situation is discussed.

It is shown that for continuum percolation with overlapping discs having a distribution of radii, the net areal density of discs at percolation threshold depends non-trivially on the distribution, and is not bounded by any finite constant. Results of a Monte Carlo simulation supporting the argument are presented.

The authors study the Ashkin-Teller model in the time-continuous Hamiltonian version. Finite-size scaling is used to calculate the magnetic ( gamma _M), electric ( gamma _P) and thermal ( alpha and nu ) critical exponents for several values of the coupling constant ( lambda ). The results confirm the believed extended scaling relations and suggest a conjecture relating the mass-gap amplitudes and critical indices in the Hamiltonian context.

The authors study the dimensional crossover in directed percolation in three dimensions. Bonds are allowed to have different concentrations along the three cartesian axes of the lattice. Through a position space renormalisation group one obtains the phase diagram where nonpercolating. 1D, 2D and 3D percolating phases are present. The authors find that, contrary to what happens in undirected percolation, the isotropic fixed points are unstable with respect to concentration anisotropy. Numerical estimates are given for the values of critical probabilities and exponents, which are in fairly good agreement with other results, where available.

The authors present a scaling approach to investigate the kinetics of the diffusion-controlled multiparticle reactions A₁+A₂+. . .+A_N to inert, and NA to inert, for a random initial distribution of particles. For the first reaction, if the initial densities of all the particle species are equal, the particle density decays with time t as t^{- alpha} , where alpha =¹/₄d, and d is the spatial dimension. This exponent values is independent of N below an upper critical dimension of d_c=4/(N-1), while for d>or=d_c, alpha assumes the mean-field value of 1/N(N-1). For the decay NA to inert, alpha =¹/₂d, again independent of N, for d<d_c=2/(N-1). These universal decays stem from the reaction kinetics being governed by the decay of spatial fluctuations, an effect which is insensitive to the details of the reaction. The author's predictions are tested by extensive computer simulations. They also examine in detail the reaction A₁+A₂+A₃ to inert for arbitrary initial densities of the three reactants and elucidate a number of interesting asymptotic properties.

The authors present a superfluid N=4 superextension of the Liouville equation with gauge SU(2)*SU(2) symmetry. It is formulated in terms of real quaternionic N=4 superfield subjected to certain Grassmann analyticity constraints and possesses a zero-curvature representation on superalgebra su(1, 1 mod 2). A possible relevance of the obtained system to the SU(2)-superstring is discussed.

For the system in which correlations between quenched defect positions fall as mod x-x' mod ^-a(a=4- epsilon - epsilon _d>0) the epsilon , epsilon _d-expansions of the critical exponents are obtained up to the third order. It is shown that if the task of finding the critical exponents for a system with uncorrelated quenched point defects is solved at order epsilon ^m of the epsilon -expansion, then the critical exponents for the system considered can be simply obtained up to (m+1)th order of the epsilon , epsilon _d expansion. The calculations are carried out completely in the special case of the uniaxial magnet.

For pt.I see ibid., vol.16, p.473 (1983). Finite-dimensional graded tensor representations of OSp(M/N) are enumerated via standard Young diagrams, and their corresponding highest weight and Kac-Dynkin labels are given. A uniform set of conditions on the diagram shape, necessary and sufficient for atypicality, are presented. Weight space techniques are used to provide a complete analysis of an atypical representations for the low-rank cases OSp(2/2), OSp(3/2) and OSp(4/2).

In previous studies of the recursion method, little attention has been paid to the properties of the basis vectors. The authors present the first detailed study of these for the case of a periodic Hamiltonian. In the examples chosen, the probability density scales linearly with n as n to infinity , whenever the local density of states is bounded. Whenever it is unbounded and the recursion coefficients diverge, difference scaling behaviour is found. These findings are explained and a scaling relationship between the asymptotic forms of the recursion coefficients and basis vectors is proposed.

A new kind of Lagrangian symmetry is defined in such a way that the resulting set of Lagrangian symmetries coincides with the set of symmetries of its equations of motion. Several constants of motion may be associated to each of the new symmetry transformations. One example is presented.

The authors calculate semiclassical limiting level spacing distributions P(S) for systems whose classical energy surface is divided into a number of separate region in which motion is regular or chaotic. In the calculation it is assumed that the spectrum is the superposition of statistically independent sequences of levels from each of the classical phase-space regions, sequences from regular regions, having Poisson distributions and those from irregular regions having Wigner distributions. The formulae for P(S) depend on the sum of the Liouville measures of all the classical regular regions, and on the separate Liouville measures of the significant chaotic regions.

A class of time-dependent classical Lagrangians possessing an invariant quadratic in momentum is considered from a quantal point of view. Quantum mechanics is introduced through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out an explicit path integration, that the propagator for such a time-dependent system is related to the propagator of an associated time-independent problem. The expansion of the propagator in terms of the eigenfunctions of the invariant operator is derived and the equivalence of the present theory to that of Lewis and Reisenfeld (1969) is discussed. Explicit analytic forms of propagators are obtained for some cases to illustrate the application of the present approach.

A path integral that reduces to Feynman's checkerboard rule in one space dimension is found for the retarded Dirac propagator in three space dimensions. The only variable are two-component spinors and a binary chirality variable. No action functional is employed. Each spinor together with a chirality corresponds to a spacetime displacement during a time epsilon . A sequence of spinors and chiralities determines a polygonal spacetime path, the first and last spinors specify the initial and final spin states. The transition amplitude for a sequence is given by ( nu _N mod nu _N-1)( nu _N-1 mod . . . mod nu ₂)( nu ₂ mod nu ₁)(i epsilon m)^R where nu _i are the spinors, < mod > is the ordinary inner product in spin space, m is the electron mass and R is the number of times the chirality switches. Integrating over all sequences corresponding to a given displacement yields the Dirac propagator in the limit epsilon to 0. With epsilon >0 this formulation provides an alternative to the point particle model of the electron. In an external electromagnetic potential A_mu the amplitude for a path C is multiplied by exp(-ie integral _cA_mu dx^mu ), requiring spacetime coordinates to specify A_mu (x). The usual perturbation expansion is derived from this rule. These results are extended to nonAbelian gauge potentials. Quantised interaction of particles is not treated.

The result for the quantised charge transport induced by an adiabatically varying substrate potential is generalised to the case in which both substrate disorder and many-body interaction are present. The application of our theory to the problem of the integral and fractional quantised Hall effect is discussed.

The authors explore certain interconnections between density-functional theory and quantum fluid dynamics of many-electron systems, in the relativistic domain following the hydrodynamical approach adopted by Takabayashi for the one-particle Dirac equation. In order to build a 'classical' hydrodynamical interpretation, the spinor formulation is transformed into a tensor formulation by defining a number of density functions (local observables). These lead to six 'classical' fluid dynamic equations, together with two subsidiary conditions, for a complete specification of the system. The various density functions and the hydrodynamical equations are physically interpreted. The relativistic hydrodynamics discussed here correspond to a 'spinning' fluid. The net many-electron fluid consists of components each of which is characterised by fluid dynamic quantities corresponding to each spinor. The net hydrodynamical quantities are obtained by summing over the occupied spinors. Thus, the earlier nonrelativistic 'classical' picture of the many-electron fluid as a collection of individual fluid components is also valid in the relativistic domain.

Using the Caldirola-Kanai Hamiltonian for the quantum dissipative system, the author expresses the propagator as the modified Feynman path integral in the configuration space, which can then be evaluated for the damped harmonic oscillator by Montroll's method. The propagator of the damped harmonic oscillator can also be calculated beyond and at caustics with the help of Horvathy-Feynman formula. The new results are confirmed by investigating the classical paths joining two fixed end-point positions, Finally the author obtains the time-dependent wavefunctions from the propagator of the dynamical system.

In the context of higher-order JWKB approximations for radial problems, the need for modifying the strength of the centrifugal barrier is considered. For spherically symmetric potentials V(r) satisfying the condition r²V(r) to 0 as r to 0, it is shown how to determine the modification required in an arbitrary order n that will ensure that the nth-order JWKB wavefunction has the correct behaviour ( approximately r^l+1) near the origin. The second-order modification of Beckel and Nakhleh (1963) is a special case of the proposed nth-order modification, as are those of Froman and Froman (1974). It is demonstrated that, with the correct modification, the JWKB series truncated at any order n leads to the exact energy spectrum for both the harmonic oscillator and the Coloumb potentials.

The modified effective potential method for treating radial problems in the JWKB approximation is applied to the quartic oscillator defined by the potential V(r)=r⁴. The JWKB quantisation condition for the energy W is shown to be expressible as (2n_r+1) pi =AW³4/+B+CW^-34/+DW^-94/+O(W ^-154/). The l-dependent coefficients, A, B, C and D are determined exactly by taking into account contributions from all orders. On inversion, the above series yields an explicit analytic formula for the energy levels. This formula is easily generalised to d dimensions, and found to reproduce known numerical eigenvalues extremely well.

Using a careful extrapolation method of the exact results for systems with a small number of particles N<or=4, the authors estimate the thermodynamic limit of the free energy density and correlation energy for the Mehta-Dyson one-dimensional plasma with long range logarithmic interaction. The results are then compared with those of the exact solution (the Mehta formula), which is established for every value of the plasma parameter lambda =e²/kT.

The size distribution of clusters is derived analytically, at a given time, in a process of irreversible kinetic aggregation of diffusive clusters which starts from a collection of a great number of individual particles. A general reduced shape is given in the scaling regime, where gelation does not occur. The time evolution is derived by use of Smoluchowski's equation. Then a scaling reasoning allows one to find the exponents in real cases and a quantitative comparison is made with numerical simulations.

The dynamical exponent z is obtained through a Monte Carlo renormalisation group calculation for the two-dimensional three-state Potts model with nonconserved dynamics. The value of z is found to be z=2.7-or+0.4.

The structure of possible equations for a vector-bispinor is examined. A systematic procedure is given for obtaining the covariant equations with a given particle and mass content. It is shown how to derive the particle content and masses for a given equation. The relation between the root method and the method based on spin-projection operators is given.

The authors study the effect of small stochastic perturbations on a simple dynamic system describing a Benard flow.

We investigate diffusion and trapping of excitations in percolation systems by an effective medium approximation. We calculate P₀(t), the probability of being at the origin at time t, on the percolation cluster at the percolation threshold p_c, S_N(t), the mean number of distinct sites visited after time t, at and above p_c and N(t), the survival probability after time t. All of our findings are completely consistent with the previously proposed relations by Rammal and Toulouse. We also consider a more general problem in which the waiting time distribution of an associated continuous-time random walk is long-tailed and propose generalisations of the Rammal-Toulouse conjectures. Our results indicate that the scaling properties of these quantities are sensitive to the details of the system.