Table of contents

This is a call for contributions to a special issue ofJournal of Physics A: Mathematical and Theoretical dedicated to the subject of the conference `IDAQUIS 2007', 23–27 July 2007 (http://www.ualg.pt/idaquis/welcome.html). Invited speakers and participants at that meeting and other researchers working in the field are invited to submit a research paper to this issue.

The Editorial Board has invited Andreas Fring, Petr Kulish, Nenad Manojlovic, Zoltan Nagy, Joana Nunes da Costa and Henning Samtleben to serve as Guest Editors for the special issue. Their criteria for acceptance of contributions are as follows:

•The subject of the paper should relate to the subject of the workshop (see the website of the conference http://www.ualg.pt/idaquis/welcome.html).

•Contributions will be refereed and processed according to the usual procedure of the journal.

•Conference papers may be based on already published work but should either contain significant additional new results and/or insights or give a survey of the present state of the art, a critical assessment of the present understanding of a topic, and a discussion of open problems.

•Papers submitted by non-participants should be original and contain substantial new results.

The guidelines for the preparation of contributions are the following:

•The DEADLINE for submission of contributions is 31 October 2007. This deadline will allow the special issue to appear in June 2008.

•There is a nominal page limit of 30 printed pages per contribution for invited speakers and 10 printed pages for all other contributors. For papers exceeding this limit, the Guest Editors reserve the right to request a reduction in length.

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

•Contributions to the special issue should, if possible, be submitted electronically by web upload at www.iop.org/Journals/jphysa or by e-mail to jphysa@iop.org, quoting `JPhysA Special Issue—IDAQUIS'. Submissions should ideally be in standard LaTeX form. Please see the website for further information on electronic submissions. All contributions should be accompanied by a read-me file or covering letter giving the postal and e-mail addresses for correspondence. The Publishing Office should be notified of any subsequent change of address.

•This special issue will be published in the paper and online version of the journal. Each participant at the workshop and the corresponding author of each contribution will receive a complimentary copy of the issue.

Dyson's integration theorem is widely used in the computation of eigenvalue correlation functions in random matrix theory. Here, we focus on the variant of the theorem for determinants, relevant for the unitary ensembles with Dyson index β = 2. We derive a formula reducing the (n − k)-fold integral of an n × n determinant of a kernel of two sets of arbitrary functions to a determinant of size k × k. Our generalization allows for sets of functions that are not orthogonal or bi-orthogonal with respect to the integration measure. In the special case of orthogonal functions Dyson's theorem is recovered.

Many dynamical systems, such as the Lotka–Volterra predator–prey model and the Euler equations for the free rotation of a rigid body, are symmetric. The standard and well-known real solutions to such dynamical systems constitute an infinitessimal subclass of the full set of complex solutions. This paper examines a subset of the complex solutions that contains the real solutions, namely those having symmetry. The condition of symmetry selects out complex solutions that are periodic.

The torsion-free de Sitter reducible metric linear connection on a five-dimensional manifold is studied in conjunction with a method of constructing spacetime via a partially restricted reference cross section of the bundle of linear frames described in earlier publications. It leads to the equation R_αβ = (1/l)g_αβ where l is a fundamental length, assumed to be the Planck length. It also yields the Einstein vacuum equations on the constructed spacetime. Using the Feynman formulation of quantum mechanics together with a dimension reduction by unrestricted transformation of the reference cross section, a geometrical model for quantum non-locality is described. The dimension reduction is then applied to the Schwarzschild solution of Einstein's equations which is found to contain the quantum plane rotation associated with a particle at rest.

This paper reviews a recently discovered link between integrable quantum field theories and certain ordinary differential equations in the complex domain. Along the way, aspects of -symmetric quantum mechanics are discussed, and some elementary features of the six-vertex model and the Bethe ansatz are explained.

We construct equilibrium networks by introducing an energy function depending on the degree of each node as well as the product of neighboring degrees. With this topological energy function, networks constitute a canonical ensemble, which follows the Boltzmann distribution for given temperature. It is observed that the system undergoes a topological phase transition from a random network to a star or a fully connected network as the temperature is lowered. Both mean-field analysis and numerical simulations reveal strong first-order phase transitions at temperatures which decrease logarithmically with the system size. Quantitative discrepancies of the simulation results from the mean-field prediction are discussed in view of the strong first-order nature.

We investigate the initial-value problem of the nonlinear Liouville hierarchy. For the general form of the interaction potential we construct a solution in terms of an expansion over particle clusters whose evolution is described by the corresponding-order cumulant of evolution operators of a system of finitely many particles. For the initial data from the space of integrable functions the existence of a strong solution of the Cauchy problem is proved.

We study the intersection points of a fixed planar curve Γ with the nodal set of a translationally invariant and isotropic Gaussian random field Ψ(r) and the zeros of its normal derivative across the curve. The intersection points form a discrete random process which is the object of this study. The field probability distribution function is completely specified by the correlation G(|r − r'|) = ⟨Ψ(r)Ψ(r')⟩. Given an arbitrary G(|r − r'|), we compute the two-point correlation function of the point process on the line, and derive other statistical measures (repulsion, rigidity) which characterize the short- and long-range correlations of the intersection points. We use these statistical measures to quantitatively characterize the complex patterns displayed by various kinds of nodal networks. We apply these statistics in particular to nodal patterns of random waves and of eigenfunctions of chaotic billiards. Of special interest is the observation that for monochromatic random waves, the number variance of the intersections with long straight segments grows like Lln L, as opposed to the linear growth predicted by the percolation model, which was successfully used to predict other long-range nodal properties of that field.

We study the time evolution of the quantum-classical correspondence (QCC) for the well-known model of quantized perturbed cat maps on the torus in the very specific regime of semi-classically small perturbations. The quality of the QCC is measured by the overlap of the classical phase-space density and corresponding Wigner function of the quantum system called quantum-classical fidelity (QCF). In the analysed regime the QCF strongly deviates from the known general behaviour discussed in Horvat et al (2006 Nonlinearity19 1–23 (Preprint quant-ph/0601139)), in particular it decays faster than exponential. Here we study and explain the observed behaviour of the QCF and the apparent violation of the QCC principle.

The invariants of solvable triangular Lie algebras with one nil-independent diagonal element are studied exhaustively. Bases of the invariant sets of all such algebras are constructed using an original algebraic algorithm based on Cartan's method of moving frames and the special technique developed for triangular and closed algebras in Boyko et al (J. Phys. A: Math. Theor. 2007 40 7557). The conjecture of Tremblay and Winternitz (J. Phys. A: Math. Gen. 2001 34 9085) on the number and form of elements in the bases is completed and proved.

It is well known that the eigenvalues of tridiagonal matrices can be identified with the zeros of polynomials satisfying three-term recursion relations and being therefore members of an orthogonal set. A class of such polynomials is identified some of which feature zeros given by simple formulae involving integer numbers. In the process certain neat formulae are also obtained, which perhaps deserve to be included in standard compilations, since they involve classical polynomials such as the Jacobi polynomials and other 'named' polynomials.

A general type of almost linear second-order differential equations, which are directly related to several interesting physical problems, is characterized. The solutions of these equations are obtained using the factorization technique, and their non-autonomous invariants are also found by means of scale transformations.

A function f, defined on the vertices of a graph G, induces nodal domains on the graph. Nodal domains of discrete and metric graphs are of growing interest among physicists and mathematicians. In this paper, several results regarding the nodal domain counts of discrete graphs are derived. One such result is a global upper bound for the number of nodal domains of G, in terms of its chromatic number. Another result is a criterion of resolution of (Laplacian) isospectral graphs via their nodal counts. Several additional results are also shown.

The Conte–Musette method has been modified for the search of only elliptic solutions to systems of differential equations. A key idea of this a prior restriction is to simplify calculations by means of the use of a few Laurent-series solutions instead of one and the use of the residue theorem. The application of this approach to the quintic complex one-dimensional Ginzburg–Landau equation (CGLE5) allows us to find elliptic solutions in the wave form. Restrictions on coefficients, which are necessary conditions for the existence of elliptic solutions for the CGLE5, have been found as well. We demonstrate that to find elliptic solutions the analysis of a system of differential equations is preferable to the analysis of the equivalent single differential equation.

We study the two-spin entanglement distribution along the infinite S = 1/2 chain described by the XY model in a transverse field; closed analytical expressions are derived for the one-tangle and the concurrences C_r, with r being the distance between the two possibly entangled spins, for the values of the Hamiltonian parameters close to those corresponding to factorized ground states. The total amount of the entanglement, the fraction of such entanglement which is stored in pairwise entanglement and the way such fraction distributes along the chain are discussed, with attention focused on the dependence on the anisotropy of the exchange interaction. Near factorization a characteristic length scale naturally emerges in the system, which is specifically related with the entanglement properties and diverges at the critical point of the fully isotropic model. In general, we find that anisotropy rules a complex behavior of the entanglement properties, which results in the fact that more isotropic models, despite being characterized by a larger amount of the total entanglement, present a smaller fraction of the pairwise entanglement: the latter, in turn, is more evenly distributed along the chain, to the extent that, in the fully isotropic model at the critical field, the concurrences do not depend on r.

We analyse in mathematical detail, within the framework of the QMUPL model of spontaneous wavefunction collapse, the von Neumann measurement scheme for the measurement of a 1/2 spin particle. We prove that, according to the equation of the model, (i) throughout the whole measurement process, the pointer of the measuring device is always perfectly well localized in space; (ii) the probabilities for the possible outcomes are distributed in agreement with the Born probability rule; (iii) at the end of the measurement the state of the microscopic system has collapsed to the eigenstate corresponding to the measured eigenvalue. This analysis shows rigorously how dynamical reduction models provide a consistent solution to the measurement problem of quantum mechanics.

We establish a relation between topological and quantum entanglement for a multi-qubit state by considering the unitary representations of the Artin braid group. We construct topological operators that can entangle a multi-qubit state. In particular we construct operators that create quantum entanglement for multi-qubit states based on the Segre ideal of complex multi-projective space. We also discuss in detail and construct these operators for two-qubit and three-qubit states.

A real-valued symplectic group underlies the dynamics of quantum-mechanical systems. Exploring the embedding of unitary groups clarifies the relation of non-Hermitian - and -symmetric quantum theories of recent interest. Symmetries of the full dynamical framework are quite rich and reveal new viewpoints on many topics in quantum theory. Transformations mixing 'upper' and 'lower' components of certain symplectic multiplets are indistinguishable from coupling antimatter degrees of freedom. Quantities long identified with physical observables are valid canonical coordinates of the theory, without needing support from measurement doctrine. Dirac's canonical quantization is derived from consistency, and would be redundant as a new postulate. A second-order dynamical framework exists in which observables are just the same as the underlying degrees of freedom.

Based on the nonlinear coherent states method, a general and simple algebraic formalism for the construction of 'f-deformed intelligent states' has been introduced. The structure has the potentiality to apply to systems with a known discrete spectrum as well as the generalized coherent states with known nonlinearity function f(n). As some physical appearance of the proposed formalism, a few new classes of intelligent states associated with 'center-of-mass motion of a trapped ion', 'harmonious states' and 'hydrogen-like spectrum' have been realized. Finally, the nonclassicality of the obtained states has been investigated. To achieve this purpose the quantum statistical properties using the Mandel parameter and the squeezing of the quadratures of the radiation field corresponding to the introduced states have been established numerically.

Spin–charge separation in pure SU(2) Yang–Mills theory was recently found to involve the dynamics of an O(3) nonlinear sigma model and, seemingly, a Grassmannian nonlinear sigma model. In this paper we explicitly construct the Grassmannian sigma model of the form appearing in the spin–charge separated SU(2) theory through a quaternionic decomposition of the manifold, thus verifying its relevance in this context. The coupling between this model and the O(3) nonlinear sigma model is further commented upon.

We study an N = 1 supersymmetric model in an S¹ × R³ spacetime. We find that by choosing appropriate boundary conditions for the contributing fields supersymmetry can be preserved. However, if we add a hard supersymmetry breaking term, we observe that for small values of the length of the S¹ dimension, supersymmetry remains unbroken and breaks spontaneously when the length exceeds a critical value. The final picture resembles the first-order phase transition picture. A toy cosmological application is discussed.

We will present an extension of the standard model of particle physics in its almost-commutative formulation. This extension is guided by the minimal approach to almost-commutative geometries employed by Iochum et al (2004 J. Math. Phys.45 5003 (Preprint hep-th/0312276)), Jureit and Stephan (2005 J. Math. Phys.46 043512 (Preprint hep-th/0501134)), Schücker (2005 Preprinthep-th/0501181), Jureit et al (2005 J. Math. Phys.46 072303 (Preprint hep-th/0503190)) and Jureit and Stephan (2006Preprint hep-th/0610040), although the model presented here is not minimal itself. The corresponding almost-commutative geometry leads to a Yang–Mills–Higgs model which consists of the standard model and two new fermions of opposite electromagnetic charge which may possess a new colour-like gauge group. As a new phenomenon, grand unification is no longer required by the spectral action.