Table of contents

The generalization of the Polyakov-Wiegmann method (1983) of fermionization to a two-dimensional sigma -model related to the one on an arbitrary homogeneous space G/H is presented. The generating functional for correlation functions of the model is expressed in the fermion path integral variables. The corresponding fermionic model is a constrained gauge model of a four-fermion interaction.

The explicit forms of the irreducible representation matrices for the quantum Sl_q(3) enveloping algebra are computed by a new technique.

The SU_q(1,1) quantum algebra is used for the description of vibrational spectra of diatomic molecules for which SU(1,1) is used in the classical approach. While in the classical case only the first two terms of the empirical Dunham expansion occur, in the quantum case all terms are obtained. The improved description of the empirical data is obtained with q being a phase (and not a real number), in agreement with the situation occurring in the q-rotor (having the symmetry SU_q(2)) used for the description of rotational spectra of molecules and nuclei.

The q-deformed differential calculus is proposed and analysed in the framework of quantum plane. The q-deformed differential operator algebra is investigated and applied in the investigation of the quantum group SU_q(2) and its representations. The generalization to n-dimensional differential calculus is made and shown to be a new solution to quantum plane by providing a new solution of the Yang-Baxter equation.

The author gives a new entropic analogue of the recently reported information theoretic limits to knowledge of states. A natural relationship between the quantum correlation information and the quantum mechanical entropy is thereby revealed. In addition some progress is made towards a rigorous proof of both results and a complete solution to the problem of asymptotic optimal measurement. In particular the author employs elementary convex analysis to prove that the optimal operator valued measure must be a rank-one projection valued measure.

The phaseshift experienced by a neutral particle with a magnetic moment as it encircles an infinite line of charge (Aharonov-Casher effect) (1984) is derived as a special case of geometrical phases, i.e. the standard Berry phase and the gauge-invariant Yang phase.

A typical quantum measurement process involves two steps. Step one is to separate the initial wavefunction into its spatial components which are then guided towards detectors at different locations. The second step involves the direct detection of the particle by a detector. Such a measurement process presupposes the existence of an interaction which can produce step one. The authors give a general mathematical formulation of the first step and establish the existence of a unitary evolution group which produces the separation. The results obtained support the view that quantum measurements are reducible to local position measurements.

The author computes the critical exponent eta , which relates to the anomalous dimension of the electron, in the large N_f expansion of QED at leading order in arbitrary dimensions in the Landau gauge. Subsequently transcendental numbers will arise at fifth order and beyond in the corresponding renormalization group function in four dimensions.

The boundary conditions are constructed for the general XXZ quantum integrable spin chain such that the resulting system is invariant under the quantum algebra U_q(sl(2)) action. The result generalizes that due to Pasquier and Saleur (1990), obtained for spin-1/2 and spin-1.

Hierarchical models are introduced to discuss the anomalous multifractal measure of the current fraction distribution on the percolating cluster analytically. The deterministic fractal model proposed by de Arcangelis, Redner and Coniglio (1986) is extended to take into account the faster decreasing minimum current fraction than power law. It is shown that, in the hierarchical models constructed from the generator of a ladder configuration with m plaquettes, the minimum current fraction has the dependence on L, i_min approximately=exp(-cm(1n L)²), where L is the system size and c a constant. The other hierarchical model showing i_min approximately=exp(-c(1n L)(In(In L))) is also found.

A simple growth model with screening is presented to mimic the forest formation. The tree grows independently if it is not screened by other trees but the growth of the tree stops when it is screened by other trees. The height distribution of trees is found to scale as h^{- gamma} (1( gamma (2) by computer simulations. It is shown that the pattern of the forest is a self-affine fractal. The scaling exponent gamma is also calculated by using a Monte Carlo renormalization-group method. The values of the exponent gamma agree with those of the simulation.

The connection between solutions of massless Dirac and Maxwell equations is established. It is shown that the massless Dirac equation is invariant under three different representations of the Poincare algebra corresponding to spins 1/2 and 1 and 0, and under three superalgebras. All generators of these symmetry algebras and superalgebras are local (differential operators of first order). A system of two Dirac equations with masses m and -m has analogous symmetry properties. Invariant nonlinear generalizations of this system are described. The authors construct the complete set of P(1, 3)-inequivalent ansatze of codimension 1 for all representations of Poincare algebra discussed. These ansatze are used for reduction and finding exact solutions of some nonlinear Dirac equations.

As an application of q-deformed algebras to standard quantum mechanics, the author shows that the SU(1, 1) dynamical symmetry of the quantum harmonic oscillator is deformed, in the first order of approximation, to the dynamical symmetry defined by the quantized universal enveloping algebra of SU(1, 1) when the harmonic potential is perturbed with a potential in x⁴. The resolution of the anharmonic oscillator is carried out algebraically in terms of generalized lowering and rising operators.

Using the representations of the Heisenberg-Weyl relations the authors develop a systematic scheme for constructing finite and infinite dimensional representations of the elements of the quantum groups GL_q(n), where the deformation parameter q is a primitive root of unity. Explicit results for the examples GL_q(2), GL_q(3) and GL_q(4) are discussed.

Starting from the commutativity and associativity properties which characterize an abstract Abelian hypergroup, a system of simultaneous, dually eigenvector-eigenvalue quantities is constructed, in terms of the matrices defining the hypergroup and of a set of generators of an elementary algebra, which possesses trivial representations. It is observed that the particular (symmetric) expressions of Rehren (1989) for such simultaneous eigenvectors-derived in the context of algebraic QFT-actually render supplementary restrictions to Verlinde's algebra (1988) corresponding to the modular invariance of the partition function of two-dimensional conformal field theory.

The authors show how one can construct hierarchies of nonlinear differential difference equations with n-dependent coefficients. Among these equations they present explicitly a set of inhomogeneous Toda lattice equations which are associated with a discrete Schrodinger spectral problem whose potentials diverge asymptotically. Then they derive a new Darboux transformation which allows them to get bounded solutions for the equations presented before and apply it in a specially simple case when the solution turns out to be expressed in terms of Hermite polynomials.

An iterated scattering map for three-particle-rearrangement scattering is constructed by forming sequences of scattering trajectories. This map works in a similar way to the well known Poincare map for bound trajectories. It provides useful information for investigating the integrability properties of the system. A numerical example is shown for the Calogero-Moser system with hyperbolic potential functions.

The classical-limit scattering matrix (CLSM) formalism was developed by Miller as a simple semiclassical procedure for the analyses of complex scattering and reaction processes. It invokes simple classical dynamics to define trajectories and superposition procedures to obtain interference effects. In one-dimensional systems, Miller (1974) has shown that the CLSM is equivalent to the standard WKB approximation. In this paper, it is shown that the equivalence between the CLSM and WKB extends to general systems. This equivalence between the WKB and CLSM formalisms is attributed to the general equivalence of the h(cross)-expansion procedure in the WKB with the stationary phase approximation used in the CLSM.

The author analyses the vacuum-polarization interaction energy (VPIE) of electrically neutral systems represented by Wilson loops (WLs). Although small, the VPIE is a long-range interaction. This fact suggests that, in the absence of the classical dipole-dipole interaction, the cloud of virtual electron-positron pairs around one WL can disturb the other. Hence, the VPIE may be included in the physical explanation of forces, between neutral systems, with induced dipole behaviour. This VPIE is a static effect for separations beyond 2a₀ (a₀ is the Bohr radius). Consequently, it is a complementary correction to the (retarded) Casimir-Polder atom-atom interaction which becomes important for separations beyond 137a₀.

The Kaluza-Klein monopole system is quantized via path integrals in parabolic coordinates. The wavefunctions and energy spectrum of the discrete and continuous spectrum are explicitly evaluated.

The problem of objectification of measurement is investigated in orthodox quantum mechanics, that is assuming the validity of the Schrodinger equation (and the other postulates) for the composite object-plus-measuring-instrument system. It is argued that by giving physical meaning to basic entities, the positivist and the realist positions must be explicitly distinguished. It is shown that the objectification problem is satisfactorily solved within quantum mechanics for the positivist, but not for the realist. For the latter, a partially satisfactory solution is obtained by assuming that the quantum mechanical description of individual systems is incomplete and, by assuming further, that the pointer observable has definite values prior to any measurement in any quantum state (a so-called beable).

The last in a series of three papers in which it is shown how the radial part of non-relativistic and relativistic hydrogenic bound-state calculations involving the Green functions can be presented in a unified manner. The work presented here is concerned with the reduced Green functions which arise in second-order stationary state perturbation theory. Using a simple linear transformation of the four radial parts of the relativistic reduced Green function it is shown how the non-relativistic and relativistic functions are special instances of the solution of a general second-order differential equation. The general solution of this equation is exhibited in the form of a Sturmian expansion, and complete solutions in both cases are presented. Recursion relations are deduced for the radial parts of both reduced Green functions and their matrix elements are examined in detail. As a test of the given functions the second-order effect of a perturbation of the nuclear charge is calculated and is shown to agree exactly with the value expected from a simple Taylor expansion of the hydrogenic energy formula.

The authors carry out a detailed analysis of the spectral rigidity for integrable systems with a view towards understanding the deviations from the universal behaviour (L/15) in systems with degeneracies in orbit actions. In the process, they also investigate the saturation region. Their studies show that the energy dependence of Delta ₃( infinity ) is a better indicator of the underlying universality in integrable systems.

A method to compute the curvature of the unstable manifold is introduced and applied to Henon map and Duffing attractor, showing that it allows the authors to locate the homoclinic tangencies and, in turn, to construct a generating partition. The probability distribution of curvature-values is investigated, showing a power-law decay. Finally, the shape of the multifractal spectrum of effective Liapunov exponents in nonhyperbolic systems is discussed.

Given a one-dimensional cellular automaton rule f, a block transform of f is a rule T_bf such that there exists between the limit sets of both rules a bijection that replaces each site value x in a configuration belonging to the limit set of f with a string x^b=xx. . .x of length b in the corresponding configuration belonging to the limit set of T_bf. If f is totalistic, there exists a unique totalistic block transform and a large number of nontotalistic block transforms T_bf. If f is not totalistic, there are no totalistic block transforms but there still exists a large number of non-totalistic block transforms. Their number increases very rapidly with the block size b and the range of f. The range of T_bf may be any integer greater than or equal to rb. Many block transforms are studied. The evolution according to rule T_bf towards its limit set is discussed in terms of the annihilation of defects. These defects are often simply related to the defects characterizing the evolution according to rule f.

The authors use phenomenological renormalization group techniques to study the bulk and surface properties of self-avoiding walks on a Manhattan lattice. They find that, as is the case for the bulk exponents, the underlying bond directionality does not change the universality class of the surface exponents relative to undirected lattices. They also obtain an estimate for the position of the binding transition.

The fractal dimension of Ising and Potts clusters are determined via the fixed scale transformation approach, which exploits both the self-similarity and the dynamical invariance of these systems at criticality. The results are easily extended to droplets. A discussion of the interrelationships between the present approach and renormalization group methods as well as Glauber-type dynamics is provided.

Diffusion-limited aggregates are among many important fractal shapes that involve deep indentations usually called fjords. To estimate the harmonic measure at the bottom of a fjord seems a prohibitive task, but the authors find that a new mathematical equality due to Beurling, Carleson and Jones makes it easy. They find that the harmonic measure at the bottom of a fjord, as a function of its Euclidean depth, can exhibit a wide range of behaviours. They introduce an infinite family of model fjords, for which the equality takes a very simple form. In this family the decay of the harmonic measure at their bottoms can be, for example, power law, semi-exponential, stretched exponential and exponentially stretched exponential. They show that self-affinity or randomness can lead to faster than power law decays of the minimal growth probability on boundaries.

A modified self-avoiding walks model previously proposed for the square lattice is applied to the triangular and honeycomb lattices. In the model, the walker is restricted not to take any turn which will put the walker in a direction rotated by more than a certain angle, Phi _max, from any of the directions previously taken. The generating functions are obtained, and various quantities are evaluated exactly. For all three (square, triangular and honeycomb) lattices, it is found that the model exhibits the characteristics reminiscent of one-dimensional self-avoiding walks in all directions, while strongly retaining the anisotropic effect of the direction of the first step. For the honeycomb lattice, the authors have studied the model for Phi _max= pi (A-model) and Phi _max=3/2 pi (B-model). Unlike other walks, the B-model exhibits a special property: the oscillations between even and odd steps in various quantities such as the number of N-step walks do not decay as the number of steps increases.

The growth of order in Ising models with nonconserved order parameter is considered for quenches to final temperatures T_f=0 and T_f=T_c. The results of numerical simulations in spatial dimension d=2 are presented. In all cases a scaling regime is entered for sufficiently long times, where the characteristic length scale is the 'domain size' L(t) approximately t¹2/, for T_f=0, and the 'nonequilibrium correlation length', xi (t) approximately t¹z/, for T_f=T_c. The equal-time correlation function has the expected scaling forms f(r/L(t)) and r^{-(d-2+ eta )}f_c(r/ xi (t)) for T_f=0 and T_c respectively. The scaling function f_c(x) has interesting short-distance behaviour which is elucidated using scaling arguments and by in - and 1/n-expansions. The T=0 scaling function f(x) depends on whether the spin correlations present in the initial conditions are of long or short range, as does the exponent which describes the decay of the autocorrelation function, A(t)=((S_i(t))S_i(O)) approximately L(t)^-. Results for a quench from the equilibrium critical state to T_f=0 are consistent with theoretical predictions.

The authors have performed a Monte Carlo study of the dynamical phase diagram of the two-dimensional p-state clock models for 4<or=p<or=10, by comparing the time evolution of two configurations subjected to the same thermal noise. For p=4, they find the expected two phase structure, similar to the Ising case, with a transition temperature which coincides with the known static one within the error bars. The dynamical critical exponents are measured. For p>or=5 they observe, between the low-temperature frozen phase and the high-temperature disordered one, two new phases, which are not present for p<or=4. The first is connected to the expected emergence of a spin-wave phase, and the other may be a purely dynamical effect although its connection to the soft phase might suggest the existence of a related equilibrium property. The transition temperatures are determined and are found to satisfy a phenomenological duality-like relation.

The authors report calculations of the critical capacity of perceptrons that are subject to pattern-dependent noise. They also calculate the capacity of a perceptron whose weights take values on a shifted sphere, and show how this system resembles 'noisy' behaviour in some limits.

The author presents models involving an anti-Hermitian scalar field Phi , endowed with a symmetry breaking potential. Like the Skyrme model, these are invariant under the global unitary transformation Phi to A Phi A^-1 and have topologically stable localized solutions. Unlike the Skyrme model, these are defined on every R_d, and a subclass of these models possess minimal action explicit solutions with winding number n.

The authors show that the 'mKdV with one-half degree of nonlinearity', proposed by Xiao (1991), does not pass the Painleve test, contrary to the author's claim.