Table of contents

We give a short geometric proof of the Kochen - Specker no-go theorem for non-contextual hidden variables models.

The long-standing problem of finding coherent states for the (bound state portion of the) hydrogen atom is positively resolved. The states in question (i) are normalized and parametrized continuously, (ii) admit a resolution of unity with a positive measure, and (iii) enjoy the property that the temporal evolution of any coherent state by the hydrogen atom Hamiltonian remains a coherent state for all time.

Ring polymers in three dimensions can be knotted, and the dependence of their critical behaviour on knot type is an open question. We study this problem for polygons on the simple cubic lattice using a novel grand-canonical Monte Carlo method and present numerical evidence that the entropic exponent depends on the knot type of the polygon. We conjecture that the exponent increases by unity for each additional factor in the knot factorization of the polygon.

We present a solution for the stationary state of an asymmetric exclusion model with sequential update and open boundary conditions. We solve the model exactly for random hopping in both directions by applying a matrix-product formalism which was recently used to solve the model with sublattice-parallel update (Hinrichsen H 1995 Weizmann Inst. Preprint cond-mat/9512172). It is shown that the matrix-algebra describing the sequential update and sublattice-parallel update are identical and can be mapped onto the random sequential case treated by Derrida et al (Derrida B, Evans M R, Hakim V and Pasquier V 1993 J. Phys. A: Math. Gen. 26 1493).

An integral solution to the quantum Knizhnik - Zamolodchikov (qKZ) equation with |q|=1 is presented. Upon specialization, it leads to a conjectural formula for correlation functions of the XXZ model in the gapless regime. The validity of this conjecture is verified in special cases, including the nearest-neighbour correlator with an arbitrary coupling constant and general correlators in the XXX and XY limits.

We study kinetic and jamming properties of a space covering process in one dimension. The stochastic process is defined as follows. Seeds are nucleated randomly in space and produce rays which grow with a constant velocity. The growth stops upon collision with another ray. For arbitrary distributions of the growth velocity, the exact coverage, velocity and size distributions are evaluated for both simultaneous and continuous nucleation. In general, simultaneous nucleation exhibits a stronger dependence on the details of the growth velocity distribution in the asymptotic time regime. The coverage in the continuous case exhibits a universal approach to the jammed state, while an inhomogeneous version of the process leads to non-generic decay, with the fraction of right-growing rays.

Graph theory, through the minimal spanning tree (MST), and information theory, through the concept of entropy, are used to define a new parameter which quantitatively characterizes the degree of order (or disorder) in 1D sets of points. Theoretical calculations and results obtained from various computer-simulated distributions of points are compared. For well-chosen conditions which are specified, a good agreement is noted. Finally, a method which is particularly easy to implement is proposed to evaluate the parameter for any real or simulated 1D distribution of points.

We introduce a renormalization scheme for forest-fire models in order to characterize the nature of the critical state and its scale-invariant dynamics. We study one- and two-dimensional models defining a characterization of the phase space that allows us to describe the evolution of the dynamics under a scale transformation. We show the existence of a relevant critical parameter associated with a repulsive fixed point in the phase space. From the renormalization-group point of view these models are therefore critical in the usual sense, because the fixed-point value of the control parameter is crucial in order to get criticality. This general scheme allows us to calculate analytically the critical exponent which describes the approach to the critical point along the repulsive direction and the exponent that characterizes the distribution of forest clusters at the critical point. We obtain , and , , respectively, for the one- and two-dimensional cases, in very good agreement with exact and numerical results.

Generalized Coherent States (GCS) are constructed (and discussed) in order to study quasiclassical behaviour of quantum spin models of the Heisenberg type. Several such models are taken to their semiclassical limits, whose form depends on the spin value as well as the Hamiltonian symmetry. In the continuum approximation, SU(2)/U(1) GCS when applied give rise to the well known Landau - Lifshitz classical phenomenology. For arbitrary spin values one obtains a lattice of coupled nonlinear oscillators. Corresponding classical continuum models are described as well.

We introduce and study frustrated cellular automata (CA) obtained by quenching competing Chaté - Manneville rules. A period-two (P2) rule and a quasi-periodic one with period close to three (QP3) are frozen at random on the lattice sites. We find that the periodic and quasi-periodic cycles are resilient to internal frustration as well as to external unbounded noise. A low concentration of impurities improves the (quasi-)periodicity of the CA, damping the chaotic background noise significantly. Starting from pure QP3 CA, a first phase transition happens at a concentration of rule P2, , leading to a macroscopic fixed point. A second phase transition, at , brings the P2 phase. Although macroscopically stable, the central phase displays a stretched exponential relaxation of the site - site autocorrelations, indicating the presence of a new type of glass with slow dynamics superimposed on the natural cyclic dynamics of the CA rules. These results appear to be quite general and are found for many pairs of rules.

We study an attractive random walk called the SATW in one dimension and show that the walk is diffusive. We show rigorously that asymptotically , where is the average of the square of the number of distinct points visited by the walk in N steps. Since , the average square of the distance to the origin, is of the same order as we get the result that asymptototically.

We investigate the dependence of the retrieval properties of the pseudo-inverse and optimal attractor neural networks on the fraction of stored patterns , the temperature T and the margin parameter . Phase diagrams in the full space of parameters are presented in the regime of extreme dilution, i.e. when the connectivity C satisfies the condition , where N is the number of neurons. Furthermore, we study analytically the neighbourhood of a stored pattern for both models by calculating the average fraction of unstable sites in a pattern that differs by d sites from a given stored pattern. This analysis may shed light on the properties of the basins of attraction of the stored patterns.

I study the product of independent identically distributed random probability matrices. Some exact asymptotic results are obtained. I find that both the left and the right products approach exponentially a probability matrix (asymptotic matrix) in which any two rows are the same. A parameter is introduced for the exponential coefficient which can be used to describe the convergent rate of the products. depends on the distribution of the individual random matrices. I find for D = 2 when each element of the individual random probability matrices is uniformly distributed in [0,1]. In this case, each element of the asymptotic matrix follows a parabolic distribution function. The distribution function of the asymptotic matrix elements can be shown numerically to be non-universal. Numerical tests are carried out for a set of random probability matrices with a particular distribution function. I find that increases monotonically from to as D increases from 3 to 99, and the distribution of random elements in the asymptotic products can be described by a Gaussian function with a mean of 1/D.

In the problem of optimization of pattern stabilization in perceptrons the replica-symmetric ansatz is known to be mathematically unstable for storage capacities greater than some . In this paper we demonstrate that for greater than the one-step replica-symmetry broken (RSB) solution is also unstable. We further show that in this region, full RSB is necessary for an exact solution. Direct evaluation of the two-step RSB solution yields a minimum storage error which is only slightly greater than the one-step RSB, which itself is greater than that given by the (unstable) replica-symmetric ansatz by a much larger amount.

Nonlinear sl(2) algebras subtending generalized angular momentum theories are studied in terms of undeformed generators and bases. We construct their unitary irreducible representations in such a general context. The linear sl(2) case as well as its q-deformation are easily recovered as specific examples. Two other physically interesting applications corresponding to the so-called Higgs and quadratic algebras are also considered. We show that these two nonlinear algebras can be equipped with a Hopf structure.

Using the collective field technique, we give the description of the spin Calogero - Sutherland model (CSM) in terms of free bosons. This approach can be applicable for arbitrary coupling constant and provides the bosonized Hamiltonian of the spin CSM. The boson Fock space can be identified with the Hilbert space of the spin CSM in the large-N limit. We show that the eigenstates corresponding to the Young diagram with a single row or column are represented by the vertex operators. We also derive a dual description of the Hamiltonian and comment on the construction of the general eigenstates.

It is shown that symplectic and orthogonal Schur functions can be realized by the action of the modes of certain vertex operators on the function . The properties of these vertex operators allow one to reproduce the basic propreties of these types of symmetric functions. The vertex operator modes are shown to obey free fermionic relations which provide applications such as calculating products and plethysms as well as generating Hirota-type partial differential equations (PDEs) which have symplectic S-functions as functions.

We examine various realistic generalizations of the basic cellular automaton model describing traffic flow along a highway. In particular, we introduce a slow-to-start rule which simulates a possible delay before a car pulls away from being stationary. Having discussed the case of a bare highway, we then consider the presence of a junction. We study the effects of acceleration, disorder, and slow-to-start behaviour on the queue length at the entrance to the highway. Interestingly, the junction's efficiency is improved by introducing disorder along the highway, and by imposing a speed limit.

We construct a seven-dimensional Dirac equation for quarks by extending the Klein - Gordon equation to seven dimensions, introducing a harmonic-oscillator term in the extra three dimensions, and taking the `square root' of the resulting equation. To facilitate solving the factored equation, we identify an SU(3) algebra in the higher dimensions and use the eigenstates of its Cartan subalgebra as basis functions. We find that the eigenstates of the extended Dirac equation are indeed exact representations of SU(3), one of which we identify with quarks. Indeed, a whole tower of SU(3) Dirac states appears. For principal quantum number , we find a singlet of mass zero; for , we find two 3s of mass , where is an adjustable constant; for , we find two 6s and two 3-bars, all of mass ; etc. Except for the ground state, all SU(3) representations appear in pairs. It is not clear that the pairs can be identified with weak-isospin doublets.

We study a model system composed of interacting quantum spins, with every spin coupled to every other, in the limit where the number of spins K and the angular momentum j of each spin are both large, aiming to explore the effect of large system size on the breakdown of the classical limit of quantum mechanics. We obtain the exact spectrum of the Hamiltonian, and hence the trace of the quantum-mechanical time-evolution operator. We examine the time dependence of , finding a simple approximation which is valid when j and K are large. At a time proportional to , this approximation breaks down, and the long-time behaviour is extremely complex. We use a renormalization scheme to investigate this complexity. The scheme is based upon a generalization of the Gauss continued-fraction map to the complex plane.

We discuss a quantization condition for bound and quasibound states of separable quantum-mechanical systems. Results for simple non-trivial models suggest that the quantization condition gives the poles of the scattering matrix except for those coming from virtual states.

The free quantum states of topologically massive electrodynamics and gravity in 2 + 1 dimensions are found explicitly. It is shown that in both theories the states are described by infrared-regular polarization tensors containing a regularization phase which depends on the spin. This is done by explicitly realizing the quantum algebra on a functional Hilbert space and by finding the Wightman function to define the scalar product on such a Hilbert space. The physical properties of the states are analysed defining creation and annihilation operators.

For both theories, a canonical and covariant quantization procedure is developed. The higher-order derivatives in the gravitational Lagrangian are treated by means of a preliminary Dirac procedure.

The closure of the Poincaré algebra is guaranteed by the infrared-finiteness of the states which is related to the spin of the excitations through the regularization phase. Such a phase may have interesting physical consequences.

We identify the recently studied relativistic orthogonal polynomials in terms of Jacobi polynomials, so all their properties follow from the corresponding properties of Jacobi polynomials through a change of variable.

We show that the Coulomb potential, one- and three-dimensional harmonic oscillator potentials are the only potentials belonging to a certain class which have the following property. The Schrödinger equation has infinitely many eigenvalues belonging to the discrete spectrum with the eigenfunctions having only a finite number of zeros in a complex plane. We show that it is due to this fact that these two potentials are the only potentials, belonging to the class defined in the paper, for which the semiclassical quantization gives an energy spectrum coinciding with the results of an exact quantum mechanical treatment.

One-to-one correspondence between the decay law of von Neumann - Wigner-type potentials and the asymptotic behaviour of the wavefunctions representing `bound states' in the continuum is established.

Using the scattering approach to quantization, we define an appropriate representation of the billiard wavefunction on the scattering Poincaré section. The expectation values of smooth operators in terms of these Poincaré section wavefunctions are expressed as sums over periodic orbits. A special operator is used to define scars on the section and the relation to scars in configuration space is discussed and demonstrated numerically.

A formulation for a non-trivial composition of two classical gauge structures is given: two parent gauge structures of a common base space are synthesized so as to obtain a daughter structure which is fundamental by itself. The model is based on a pair of related connections that take their values in the product space of the corresponding Lie algebras. The curvature, covariant exterior derivatives and associated structural identities all get contributions from both gauge groups. The various induced structures are classified into those whose composition is given just by trivial means, and those which possess an irreducible nature. The pure irreducible piece, in particular, generates a complete super-space of ghosts with an attendant set of super-BRST variation laws, both of which are purely of a geometrical origin.

The variational properties of the scalar so-called `universal' equations are reviewed and generalized. In particular, we note that each member of the Euler hierarchy may have an explicit field dependence. The Euler hierarchy itself is given a new interpretation in terms of the formal complex of variational calculus, and is shown to be related to the algebra of variational symmetries of the first source form.

The variational principle for the real homogeneous Monge - Ampère equation in two dimensions is shown to contain three arbitrary functions of four variables. There exist two different specializations of this variational principle where the Lagrangian is degenerate and furthermore contains an arbitrary function of two variables. The Hamiltonian formulation of these degenerate Lagrangian systems requires the use of Dirac's theory of constraints. As in the case of most completely integrable systems the constraints are second class and Dirac brackets directly yield the Hamiltonian operators. Thus the real homogeneous Monge - Ampère equation in two dimensions admits two classes of infinitely many Hamiltonian operators, namely a family of local, as well as another family non-local Hamiltonian operators and symplectic 2-forms which depend on arbitrary functions of two variables. The simplest non-local Hamiltonian operator corresponds to the Kac - Moody algebra of vector fields and functions on the unit circle. Hamiltonian operators that belong to either class are compatible with each other but between classes there is only one compatible pair. In the case of real Monge - Ampère equations with constant right-hand side this compatible pair is the only pair of Hamiltonian operators that survives. Then the complete integrability of all these real Monge - Ampère equations follows by Magri's theorem. Some of the remarkable properties we have obtained for the Hamiltonian structure of the real homogeneous Monge - Ampère equation in two dimensions turn out to be generic to the real homogeneous Monge - Ampère equation and the geodesic flow for the complex homogeneous Monge - Ampère equation in arbitrary number of dimensions. Hence among all integrable nonlinear evolution equations in one space and one time dimension, the real homogeneous Monge - Ampère equation is distinguished as one that retains its character as an integrable system in multiple dimensions.

The dynamical algebra associated with a family of Isospectral Oscillator Hamiltonians, named distorted Heisenberg algebra because of its dependence on a distortion parameter , has recently been studied. The connection of this algebra with the Hilbert space of entire analytic functions of growth is analysed.

Using the symplectic formalism of classical mechanics, a smooth holonomy effect is identified in a special class of classical ergodic Hamiltonian systems (generalized canonical families) that are cycled adiabatically. It is a smooth shift in a time-resembling observable `tempus' that is canonically conjugated to the phase space volume contained in the energy shell. The curvature 2-form of the adiabatic connection obtained is closely related to Hannay's and the Robbins - Berry 2-form. The measurability of the `tempus' is discussed.

The well known Haldane map from spin chains into the O(3) nonlinear sigma model is generalized to the case of spin ladders. This map allows us to explain the different qualitative behaviour between even and odd ladders, in exactly the same way as it explains the difference between integer and half-integer spin chains. Namely, for even ladders the topological term in the sigma model action is absent, while for odd ladders the parameter, which multiplies the topological term, is equal to , where S is the spin of the ladder. Hence even ladders should have a dynamically generated spin gap, while odd ladders with half-integer spin should stay gapless and physically equivalent to a perturbed Wess - Zumino - Witten model in the infrared regime. We also derive some consequences from the dependence of the sigma model coupling constant on the ladder Heisenberg couplings constants.

A mathematical description of the Kirkwood gap in the 2/1 resonance with Jupiter is discussed in detail, providing a new insight into a traditionally controversial issue. The discussion is further extended to demonstrate how, under slightly different circumstances, the same kind of resonance can create the opposite effect of asteroid clustering.