Table of contents

Let G be a Lie group and G_q be its quantum analogue. In general, it is not known how to construct a quantum analogue of the Lie Group TG, the tangent bundle of G. For the case G=GL(n), the quantum group T_hGL(n), h=In q, is described.

On GL_q(2) there exist two types of traces. On GL_q(n), for n>or=3, there are none.

The authors study the influence of finite particle lifetime tau dissipation on the induced Chern-Simons term in a heated and dense medium. They demonstrate that dissipation like temperature suppresses the topological action. A peculiarity in introducing a dissipation into the theory is discussed briefly.

The reaction of a two-component system is studied both under the influence of a stochastic reaction rate and additional fluctuations of the concentrations. There results a noise-induced transition in the case of a coupling between the two kinds of randomness with a characteristic exponent beta =1. The typical time for the dynamics is approximately calculated by means of a perturbation expansion for a mean first passage time.

The Bohr-van Leuwan theorem, relevant to solid state physics, which concerns magnetization in the classical limit, is reviewed. This derivation, which is based on a statistical mechanical evaluation of the free energy, is generalized and applied to the case of a confined plasma. It is shown that an equilibrium, electromagnetically confined classical plasma at a given temperature and occupying finite volume does not exist. The derivation is valid for external as well as self-field confinement.

For pt.I see ibid., vol.25, p.359 (1992). Regarding the discussion of Peres (1989) and Weinberg (1989) concerning a suitable definition of entropy in nonlinear quantum mechanics, two further observations are made. Firstly, by regarding the covariance matrix of the probability distribution over the phase space of wavefunctions as the nonlinear counterpart (p_NL) of the (linear) density matrix (p_L) and employing -Tr p_NL In p_NL, one obtains a limiting transition (as nonlinearities vanish), in which this entropy measure converges to the definition in ordinary quantum mechanics, -Tr p_L In p_L. Secondly, it is argued that Peres' contention that 'nonlinear variants of the Schrodinger equation violate the second law of thermodynamics' is flawed in that it relies upon the entropy of mixing of nonorthogonal states which as Dieks an van Dijk (1988) have indicated is an undefined concept. A proper approach to associating a quantum mechanical entropy with a mixture of a particle into two non-orthogonal states-by first estimating a suitable two-particle density matrix (p) and then employing -Tr p In p-is outlined.

The 1D Heisenberg spin model with an anisotropy of the XXZ type is analysed in terms of the symmetry given by the quantum Galilei group Gamma _q(1). For a chain with an infinite number of sites the authors show that the magnon excitations and the s=1/2, n-magnon bound states are determined by the algebra. In this case the Gamma _q(1) symmetry provides a description naturally compatible with the Bethe ansatz. The recurrence relations determined by Gamma _q(1) permit one to express the energy of the n-magnon bound states in a closed form in terms of Tchebischeff polynomials.

The backbone fractal dimension d^B_f is calculated on two-dimensional percolation clusters at the percolation threshold. Studies are carried out on backbones defined in three different ways: bus bar, point-to-point, and with fully periodic boundaries. The estimates of d^B_f are obtained by measuring the variation of mass with radius for all clusters, and by means of finite size scaling on the bus bar backbones. All cases imply a value of d^B_f=1.64+or-0.01 for the backbone fractal dimension. Because of the high degree of self-consistency of all the results, the authors believe that this estimate represents a considerably improved accuracy.

A lattice gas automaton lacks Galilei invariance, and equilibria of systems moving with a finite speed mod u mod are not simply related by a Galilei transformation to the equilibrium distribution in the rest frame. In the hydrodynamic description of low speed equilibria in lattice gas automata a factor G(p) appears in the nonlinear convective term, Delta . G(p)puu, of the Navier-Stokes equation, that differs from unity due to lack of Galilei invariance. For this non-Galilean factor an expression in terms of fluctuating quantities is derived, a grand ensemble where the total momentum is fluctuating around a zero average. The formula is valid as long as there exists a unique equilibrium state. Consequently, the results can also be used for a direct simulation of G(p) in lattice gas models where the explicit form of the equilibrium distribution is not known, such as in models that violated semi-detailed balance.

The multiscaling structure of the cluster-mass distribution is investigated in a simple cluster-cluster aggregation model with injection in which the volume (or size) of clusters is infinitesimal but the mass of clusters is finite. The model is consistent with the Scheidegger river network model. It is shown that the partition function Z(q) identical to Sigma ^N_n=1 M(t, n)^q scales as Z(q) approximately=t^{zeta (q)} where M(t, n) is the mass of the cluster on the site n at the time t and the summation ranges over all sites. In the limit of a sufficiently large q, zeta (q)/q (or delta zeta (q)/ delta q) gives the exponent gamma of growth of a typical cluster. The exponent also equals to the fractal dimension d_f=1.5 of a typical river in the Scheidegger river network model. The f- alpha spectrum of the normalized mass distribution is derived. It is found that the growth exponent of a typical cluster is exactly given by gamma =1- alpha ( infinity ). The multiscaling of the cluster-mass distribution has a characteristic property for the simple cluster-cluster aggregation system.

For systems whose classical orbits are chaotic, a set of quantum expectation values Q_r is constructed which vanish for all, unlike their classical counterparts C_r which are finite. This behaviour is not paradoxical because Q_r and C_r are moments of time correlation functions, which are dominated by the long-time limit where the quantum and classical evolution disagree.

The authors introduce and investigate via Monte Carlo simulation the finite-size critical behaviour of the body-centred-solid-on-solid (BCSOS) model's sublattice order parameter and related (staggered) susceptibility. They confirm the scaling ansatz of Baxter (1973) for the susceptibility, a quantity which has remained hitherto unexplored. They also verify the Kosterlitz-Thouless universality class in the temperature dependence of both these equilibrium properties. Their extrapolation for the location of the infinite system roughening transition temperature is in good agreement with the exact value.

The authors correlate the symmetry group of the continuous transformations of the Toda lattice to that of the Korteweg-de Vries equation. They show how, by taking into account the continuous limit of the Toda lattice the four-parameter symmetry group of the Toda lattice is contained in that of the KdV equation. By an inverse process, discretization of the symmetry group of the KdV equation, they find a discrete element of the symmetry group of the Toda lattice, which gives, by symmetry reduction, its soliton solution.

The concept of nonlocal Lie-Backlund symmetries can be generalized by including pseudopotentials. For the KdV, HD and AKNS equations, the authors calculate generalized symmetries of such a kind. The solitary wave solutions are obtained through the transformation of trivial solutions.

The author describes implications of spontaneous symmetry breaking (SSB) in the context of far-from-equilibrium transitions. He shows that the SSB of continuous symmetries leads to the appearance of poles in the Fourier transforms of response functions in complete analogy with the Goldstone theorem in Hamiltonian systems.

The author considers the linear instability in homogeneous plasmas that occurs when the effective one-dimensional distribution of electrons is not a 'single maximum' function. The author constructs a quadratic form (,)_R to explain why no linear instability occurs for sufficiently high wavenumbers k, even though the energy quadratic form is indefinite for all k. (,)_R is conserved in the linearized evolution, and positive-definite precisely up to the wavenumber that the instability actually occurs. The author argues against a 'structural instability' due to nonlinear terms.

The authors have studied the critical exponent nu associated with the mean squared end-to-end distance of self-avoiding walks on the two-dimensional X and checkerboard fractal families. By means of exact renormalization group transformations they have calculated nu for the first four members of each family.

Several models have been introduced to construct random packings of identical spheres. The packings are built on a horizontal basal layer using iterative sequential algorithms in which a new added sphere is always in contact with three other spheres in the packing. In the Bennet model the position with the lowest vertical coordinate is selected. In the 'anti-Bennet' model, only positions that are stable under gravity are considered, and that with the highest vertical coordinate is selected. In the Eden model the new sphere is selected at random among all the possibilities and, in the 'stable Eden' model, the random choice is limited to positions that are stable under gravity. In all cases, the density (packing fraction) is determined within an uncertainty of +or-10^-4. The histogram for the number of contacts between spheres, the bond angle distribution and the distance distribution were also determined. The results are compared with corresponding ballistic model results.

The authors analyse the Bethe ansatz equations for the two-particle sector of the spin 1/2 Heisenberg XXX model on a one-dimensional lattice of length N. They show that, beginning at a critical lattice length of N=21.86, new pairs of real solutions develop, whereas complex solutions start to disappear. The integers (that appear in the logarithmic form of the Bethe equations) of the new solutions do not fit into the conventional classification scheme. The total number of solutions in the two-particle sector remains unchanged and is in agreement with the claim that the SU(2) extended Bethe ansatz gives a complete set of 2^N eigenstates.

The authors present detailed numerical results from a computationally efficient cell dynamical system model of domain growth in binary alloys with quenched disorder. Their numerical results suggest that the domain growth law for the disordered case is compatible with (R)(t) approximately (1nt)^x, where x has a weak dependence on the disorder amplitude. However, it is possible that their simulations do not access the true asymptotic regime. They also find that the scaled structure factor for the disordered case is independent of the amplitude of disorder and is the same as that for the pure system.

For networks of formal two-state neurons the sequential learning problem is considered: a set of patterns has been memorized by adjusting a matrix of the synaptic efficacies. Then an extra pattern is presented to the network and should be stored in addition to previous patterns in such a way that the latter are not used. This problem has been solved in the framework of the pseudo-inverse learning rules. The resulting synaptic matrix has the same value as for other variants of the pseudo-inverse learning rule if memorizing of the previous patterns was performed by some pseudo-inverse rule.

Yang-Baxterization of Faddeev-Reshetikhin-Takhtajan algebra (1989) leading to quantum YBE is considered for a class of R-matrices which satisfy Hecke relations. The authors apply such construction of the gl(N) case together with its multiparameter deformations revealing the connection between FRT relations and the extended trigonometric Sklyanin algebra. New realizations of FRT algebra through q-oscillator modes are presented, and their potential importance in the context of quantum-integrable models is discussed. A multiparametrized R-matrix with the inclusion of spectral parameter is constructed as a by-product.

A deformed oscillator, with eigenvalues equal to the eigenvalues of the Schrodinger equation with the Coulomb potential, is constructed. The deformed oscillator algebra has a polynomial realization and the associate deformed operations of integration and differentiation are studied.

The reducibility to differential forms of certain nonlinear integral operators appearing in kinetic theory is discussed in the frame of a stationary problem in gas dynamics with removal events. A wide class of transition kernels allowing such reducibility is characterized, and the validity of these kernels as approximations to real ones is studied in terms of their asymptotic behaviour for large and small velocities. In fact, this behaviour is shown to determine the corresponding limiting velocity dependence in the solutions to the kinetic equation.

The periodic stationary solutions of a model nonlinear evolution equation simulating the propagation of short-wave perturbations in a relaxing medium are studied. Solutions expressed by a multiple-valued function are shown to exist. A method for determining the nonlinear interaction between solitary waves is suggested. An example of a collision of solitons is given.

The exact propagator beyond and at caustics for a pair of coupled and driven oscillators with different frequencies and masses is calculated using the path-integral approach. The exact wavefunctions and energies are also presented. Finally the propagator is re-calculated through an alternative method, using the zeta -function.

The semiclassical propagator for the one-dimensional anharmonic oscillator is investigated, in the configuration space, by means of the so-called Van Vleck formula, which expresses it as a sum over all the denumerably infinite classical paths connecting given points in the same time. Analytical formulae for the paths' contributions are given, together with some numerical results. It is shown that in the general case the amplitudes of the contributions asymptotically approach the same value, while the phases oscillate; the Van Vleck series therefore does not converge, but in the generic case it can be resumed. Finally, the conditions under which one of the paths gives a dominant contribution to the semiclassical propagator are discussed.

Quantum mechanics on the hyperbolic spaces of rank one is discussed using a path interaction technique. Hyperbolic spaces are multi-dimensional generalizations of the hyperbolic plane, i.e. the Poincare upper half-plane endowed with a hyperbolic geometry. The author evaluates the path integral on S₁ equivalent to SO(n, 1)/SO (n) and S₂ equivalent to SU(n,1)/S(U(1)*U(n)) in a particular coordinate system, yielding explicitly the wavefunctions and the energy spectrum. Furthermore, one can exploit a general property of all these spaces, namely that they can be parameterized by a pseudopolar coordinate system.

In the case of a massless spinor field in 4D spacetime it is proved that both Fermi and Bose quantizations can be carried out. The Bose quantization of this field is demonstrated.

The problem of ambiguities in trying to determine a shape by means of scattering experiments, with one or a few illuminating angles and all directions of receivers, is discussed by means of numerical experiments. The model equation gives a good representation of scattering of scalar waves which can taken into account impedance discontinuities inside the scatterer. Physical problems include, for instance, acoustical waves in media where the density and the Lame parameter lambda may vary continuously everywhere except across a finite number of smooth surfaces through which they jump.

The normally ordered Hilbert space image of the complex linear similarity transformation in Grassmann number phase space is derived in fermionic coherent state representation. Though keeping the anticommutator of Fermi operators invariant, the quantum mechanical images of classical transformations are generally not unitary. Remarkably, although kets and bras produced by the non-unitary similarity transformations are not Hermitian conjugates, they still form a complete basis. The derivation of the normally ordered operator which engenders the similarity transformation is facilitated by virtue of the technique of integration within order products.