Table of contents

Journal of Physics A: Mathematical and General invites your manuscript submission on topics connected with the Second Symposium on Quantum Theory and Symmetries (18--21 July 2001, Krakow, Poland) for publication in June 2002 in a Special Issue entitled Foundations of Quantum Theory. Participants at that meeting, as well as other researchers working in this field, are invited to submit a research paper to this issue. More information about the Symposium and its coverage can be found at the webpage http://qts2.ifj.edu.pl.

During the past five years, the foundations of quantum theory have once again attracted considerable attention arising from progress achieved in such areas as quantum field theory, gauge theory, the non-stationary Casimir effect as a mechanism for creating squeezed photons and other particles from vacuum, the symmetry approach, quantum groups, path-integral and Moyal quantizations, the tomographic formulation of quantum mechanics, the nonlinear extension of quantum mechanics, the EPR and other quantum paradoxes, the quantum superposition principle and entanglement, the star-product and different commutation relations of observables consistent with the same quantum dynamics. Foundations of Quantum Theory will be a basis for theoretical and technical progress in the quantum information process.

As Guest Editors of this Special Issue, we encourage contributions in all aspects of the foundations of quantum theory. The arrangements for submission and publication are:

• Manuscripts submitted to this Special Issue will be refereed according to the usual practice and high standards of the journal.

• Editorial procedures will be undertaken by the new Editorial Office of Institute of Physics Publishing at the P. N. Lebedev Institute in Moscow in association with the main Publishing Office in Bristol.

• Contributions to the Special Issue should be submitted, preferably by e-mail and preferably in standard LaTeX form, to the Guest Editors atIOPP@sci.lebedev.ru, quoting `Special Issue/Foundations of Quantum Theory'.

• Authors unable to submit by e-mail may send hard copy contributions to IOPP Division, P. N. Lebedev Physical Institute, Leninskii Prospect 53, 117924 Moscow, Russia.

• The deadline for receipt of contributed papers is 1 November 2001, and the deadline for acceptance of articles is 1 March 2002, to facilitate publication in June 2002.

• Further advice on publishing your work in Journal of Physics A: Mathematical and General may be found at www.iop.org/Journals/ja.

• This Special Issue will appear in both the paper and online editions of Journal of Physics A: Mathematical and General. The corresponding author of each contribution will receive a complimentary copy of the issue, in addition to the usual 25 free offprints.

We look forward to receiving and processing your contribution to this Special Issue.

Andrzej Horzela and Edward KapuscikH. Niewodniczanski Institute of Nuclear Physics, Krakow, PolandVladimir I. Man'koP. N. Lebedev Physical Institute, Moscow, RussiaGuest Editors

Localized oscillations appear both in ordered nonlinear lattices (breathers) and in disordered linear lattices (Anderson modes). Numerical studies on a class of two-dimensional systems of the Klein-Gordon type show that there exist two different types of bifurcation in the path from nonlinearity-order to linearity-disorder: inverse pitchforks, with or without period doubling, and saddle-nodes. This was discovered for a one-dimensional system in a previous work of Archilla, MacKay and Marin. The appearance of a saddle-node bifurcation indicates that nonlinearity and disorder begin to interfere destructively and localization is not possible. In contrast, the appearance of a pitchfork bifurcation indicates that localization persists.

The well known uncertainty product of communication theory for a signal in the time domain and its Fourier transform in the frequency domain is studied for a `composite signal', i.e. a `pure' signal to which a time-delayed replica is added. This uncertainty product shows the appearance of local maxima and minima as a function of the time delay, leading to the following conjecture:

the uncertainty product of a non-Gaussian composite signal can be smaller than that of the `pure' signal.

As an example this conjecture will be proven for the derivative of the Gaussian signal and for the Cauchy distribution. The effect on the uncertainty product of adding a delayed scaled replica of a signal to the original signal in the time domain leads to an important possibility for interpretation in the study of the reverberation phenomenon in echo-location signals of dolphins.

We report a combined numerical approach to study the localization properties of the one-dimensional tight-binding model with potential modulated along the prime numbers. A localization-delocalization transition was found as a function of the potential intensity; it is also argued that there are delocalized states for any value of the potential intensity.

We investigate the competition between barrier slowing down and proliferation induced superdiffusion in a model of population dynamics in a random force field. Numerical results in d = 1 suggest that a new intermediate diffusion behaviour appears. We introduce the idea of proliferation assisted barrier crossing and give a Flory-like argument to understand qualitatively this non-trivial diffusive behaviour. A renormalization group analysis close to the critical dimension d_c = 2 confirms that the random force fixed point is unstable and flows towards an uncontrolled strong coupling regime.

Phenomenological scaling arguments suggest the existence of universal amplitudes in the finite-size scaling of certain correlation lengths in strongly anisotropic or dynamical phase transitions. For equilibrium systems, provided that translation invariance and hyperscaling are valid, the Privman-Fisher scaling form of isotropic equilibrium phase transitions is readily generalized. For non-equilibrium systems, universality is shown analytically for directed percolation and is tested numerically in the annihilation-coagulation model and in the pair contact process with diffusion. In these models, for both periodic and free boundary conditions, the universality of the finite-size scaling amplitude of the leading relaxation time is checked. Amplitude universality reveals strong transient effects along the active-inactive transition line in the pair contact process.

We investigate systems of real scalar fields in bidimensional spacetime, dealing with potentials that are small modifications of potentials that admit supersymmetric extensions. The modifications are controlled by a real parameter, which allows the implementation of a perturbation procedure when such a parameter is small. The procedure allows one to obtain the energy and topological charge in closed forms, up to first order in the parameter. We illustrate the procedure with some examples. In particular, we show how to remove the degeneracy in energy for the one-field and the two-field solutions that appear in a model of two real scalar fields.

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in the formal variational calculus. It is well known that the radical of a finite-dimensional Novikov algebra is transitive. In this paper, we prove that a kind realization of Novikov algebras given by S Gel'fand is transitive and we give a deformation theory of Novikov algebras. In two and three dimensions, we find that all transitive Novikov algebras can be realized as the Novikov algebras given by S Gel'fand and their compatible infinitesimal deformations.

In this paper, we discuss the Lie symmetries, symmetry algebra and symmetry reductions of the equation which describes constant mean curvature surfaces via the generalized Weierstrass-Enneper formulae. First we point out that the equation admits an infinite-dimensional symmetry Lie algebra. Then using symmetry reductions, we obtain two integrable Hamiltonian systems (one autonomous, the other nonautonomous) with two degrees of freedom. The autonomous one was obtained by Konopelchenko and Taimanov by other means. Our method provides a new approach for construction of constant mean curvature surfaces.

We develop a Magnus expansion well suited for Floquet theory of linear ordinary differential equations with periodic coefficients. We build up a recursive scheme to obtain the terms in the new expansion and give an explicit sufficient condition for its convergence. The method and formulae are applied to an illustrative example from quantum mechanics.

We further investigate the strange spectra of the Orr-Sommerfeld operator using the plane Poiseuille flow as a basic stationary flow for normal fluids in the two-fluid system of helium II by a verified preconditioned complex-matrix solver. The strange spectra are composed of one pair of eigenvalues with the same phase speed (real part) but different amplification factors (imaginary part) corresponding to the specific Reynolds number and wavenumber we select. These kinds of degeneracy disappear for Reynolds number around 400, where the `drifting' of the complete spectra imposes much more complexity on the searching.

The equations of motion for a relativistic extended object loaded with a superconducting edge are found in terms of geometrical quantities defined on the worldsheet. The results are applied to the study of a domain wall bounded by a superconducting string. Several cases for attaining equilibrium configurations are discussed.

We show that the (1+1)-dimensional quantum superplane introduced by Manin is a quantum supergroup, according to the Faddeev-Reshetikhin-Takhtajan approach, when it is extended by the inverse of the bosonic variable. We then give its supermatrix element, its corresponding R-matrix and its Hopf structure. This new point of view allows us, first, to realize its dual Hopf superalgebra starting from postulated initial pairings. Second, we construct a right-invariant differential calculus on it and then deduce the corresponding quantum Lie superalgebra which as a commutation superalgebra appears classical, and as Hopf structure is a non-cocommutative q-deformed one.

In this paper it is proved that the quantum relative entropy D(ρ||σ) can be asymptotically attained by the relative entropy of probabilities given by a certain sequence of positive-operator-valued measures (POVMs). The sequence of POVMs depends on σ, but is independent of the choice of ρ.

It is well known that the third-order Lorentz-Dirac equation admits `runaway' solutions wherein the energy of the particle grows without limit, even when there is no external force. These solutions can be denied simply on physical grounds, and on the basis of careful analysis of the correspondence between classical and quantum theory. Nonetheless, one would prefer an equation that did not admit unphysical behaviour at the outset. Such an equation - an integro-differential version of the Lorentz-Dirac equation - is currently available either in only one dimension or in three dimensions (3D) in the non-relativistic limit.

It is shown herein how the Lorentz-Dirac equation may be integrated without approximation, and is thereby converted to a second-order integro-differential equation in 3D satisfying the above requirement, i.e. as a result, no additional constraints on the solutions are required because runaway solutions are intrinsically absent. The derivation is placed within the historical context established by standard works on classical electrodynamics by Rohrlich, and by Jackson.

The vacuum excitation and squeezing of two harmonic oscillators with delta-kicked interactions for four types of elementary coupling are studied. The exact quantum motion for the Heisenberg operators and the explicit form of squeezing operators are found. The variances are calculated to study the squeezing properties for position, momentum and generalized quadrature operators.

Integrable Hamiltonians with velocity-dependent potentials, including those of Fokker-Planck Hamiltonians H = ½(p_x² + p_y²) + k_xp_x + k_yp_y, are constructed from integrable Hamiltonians of type H = ½(p_x² + p_y²) + V(x,y). In order to carry out the analytical investigations, we convert the problem into that of two coupled anisotropic quartic anharmonic oscillators using certain canonical transformations; afterwards we give a complete description of the real phase space topology of the system. We give also an explicit periodic solution for singular common-level sets of the first integrals. All generic bifurcations of Liouville tori were determined analytically and numerically.

A discussion of dimensional reductions, which are not classical symmetry reductions, is made for the BKP and CKP hierarchies of integrable evolution equations. A novel direct method for testing Pfaffian solutions to bilinear identities is presented and applied to these reductions.

The tomographic-probability distribution for a measurable coordinate and spin projection is introduced to describe quantum states as an alternative to the density matrix. An analogue of the Pauli equation for the spin-½ particle is obtained for such a probability distribution instead of the usual equation for the wavefunction. Examples of the tomographic description of Landau levels and coherent states of a charged particle moving in a constant magnetic field are presented.

The relativistic generalization of the Kepler map describing diffusive excitation of the relativistic hydrogen-like atom in a monochromatic field is derived. It is shown that the trajectories which are regular in the non-relativistic case may become chaotic in the relativistic case under the same conditions.

The main difficulty of quantum field theory is the problem of divergences and renormalization. However, realistic models of quantum field theory are renormalized within the perturbative framework only. It is important to investigate renormalization beyond perturbation theory. However, known models of constructive field theory do not contain such difficulties as infinite renormalization of the wavefunction. In this paper an exactly solvable quantum mechanical model with such a difficulty is constructed. This model is a simplified analogue of the large-N approximation to the Φφ^aφ^a-model in six-dimensional spacetime. It is necessary to introduce an indefinite inner product to renormalize the theory. The mathematical results of the theory of Pontriagin spaces are essentially used. It is remarkable that not only the field but also the canonically conjugated momentum become well defined operators after adding counterterms.

A harmonic oscillator subject to a parametric pulse is examined. The aim of the paper is to present a new theory for analysing transitions caused by parametric pulses. The new theoretical notions which are introduced relate the pulse parameters in a direct way with transition matrix elements.

The harmonic-oscillator transitions are expressed in terms of the asymptotic properties of a companion oscillator, the Milne (amplitude) oscillator. A traditional phase-amplitude decomposition of the harmonic-oscillator solutions results in the so-called Milne's equation for the amplitude, and the phase is determined by an exact relation to the amplitude. This approach is extended in the present analysis with new relevant concepts and parameters for pulse dynamics of classical and quantal systems.

The amplitude oscillator has a particularly nice numerical behaviour. In the case of strong pulses it does not possess any of the fast oscillations induced by the pulse on the original harmonic oscillator. Furthermore, the new dynamical parameters introduced in this approach are closely related to the relevant characteristics of the pulse.

The relevance to quantum mechanical problems such as reflection and transmission from a localized well and the mechanical problem of controlling vibrations is illustrated.

A generalization of the Stieltjes relations for the fourth Painlevé (PIV) transcendents and their higher analogues determined by dressing chains is proposed. It is proven that if a rational function from a certain class satisfies these relations it must be a solution of some higher PIV equation. The approach is based on the interpretation of the Stieltjes relations as local trivial monodromy conditions for certain Schrödinger equations in the complex domain. As a corollary a new class of Schrödinger operators with trivial monodromy is constructed in terms of the PIV transcendents.