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.

We show that an arbitrary probability distribution can be represented in an exponential form. In physical contexts, this implies that the equilibrium distribution of any classical or quantum dynamical system is expressible in a grand canonical form.

We derive explicit expressions for Green functions and some related characteristics of the Rashba and Dresselhaus Hamiltonians with a uniform magnetic field.

A solvable nonlinear (system of) evolution PDEs in multidimensional space, involving elliptic functions, is identified, and certain of its solutions are exhibited. An isochronous version of this (system of) evolution PDEs in multidimensional space is also reported.

The transient fluctuation theorem for stochastic processes was first put forward by D J Searles and D J Evans (1999). In the present paper, it is rigorously proved that the transient fluctuation theorem (TFT) of sample entropy production, which is previously defined by Jiang et al (2003) and Reid et al (2005) and also called the general action functional up to boundary terms by Lebowitz and Spohn (1999), holds for general stochastic processes without the assumption of Markovian, homogeneous or stationary properties. Then the condition of our theorem is verified for various stochastic processes, including homogeneous, inhomogeneous Markov chains and general diffusion processes. Among these cases, the transient fluctuation theorems for inhomogeneous Markov chains and general diffusion processes are rigorously derived for the first time.

We analyse a criterion, introduced by Joshi and Lafortune, for the integrability of cellular automata obtained from discrete systems through the ultradiscretization procedure. We show that while this criterion can be used in order to single out integrable ultradiscrete systems, there do exist cases where the system is nonintegrable and still the criterion is satisfied. Conversely we show that for ultradiscrete systems that are derived from linearizable mappings the criterion is not satisfied. We investigate this phenomenon further in the case of a mapping which includes a linearizable subcase and show how the violation of the criterion comes to be. Finally, we comment on the growth properties of ultradiscrete systems.

Premetric electrodynamics is a useful formalism for unified description of a wide range of modified electrodynamics models. It is also applicable in describing of the electromagnetic processes in anisotropic media. In the current paper, we present a covariant gauge-independent derivation of the generalized dispersion relation for electromagnetic waves in a medium with a local, linear constitutive law. Moreover, we derive a generalized photon propagator (Green function in the momentum representation). For Maxwell constitutive tensor, the standard light cone structure and the standard Feynman propagator are reinstated.

The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor.40 F75), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian -symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily -symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.

We study structural properties of growing networks where both addition and deletion of nodes are possible. Our model network evolves via two independent processes. With rate r, a node is added to the system and this node links to a randomly selected existing node. With rate 1, a randomly selected node is deleted and its parent node inherits the links of its immediate descendants. We show that the in-component size distribution decays algebraically, c_k ∼ k^−β as k → ∞. The exponent β = 2 + (r − 1)⁻¹ varies continuously with the addition rate r. Structural properties of the network including the height distribution, the diameter of the network, the average distance between two nodes and the fraction of dangling nodes are also obtained analytically. Interestingly, the deletion process leads to a giant hub, a single node with a macroscopic degree whereas all other nodes have a microscopic degree.

A general pattern theorem for weighted self-avoiding polygons (SAPs) and self-avoiding walks (SAWs) in is obtained. The pattern theorem for SAPs fits into the general framework of the pattern theorem for lattice clusters introduced by Madras (1999 Ann. Comb.3 357–84). Note that, unlike other pattern theorems proved for SAPs, this pattern theorem does not rely on first establishing a relationship between SAPs and SAWs. These results are applied to obtain pattern theorems for self-interacting SAPs and self-interacting SAWs.

Starting from the basic-exponential, a q-deformed version of the exponential function established in the framework of the basic-hypergeometric series, we present a possible formulation of a generalized statistical mechanics. In a q-nonuniform lattice we introduce the basic-entropy related to the basic-exponential by means of a q-variational principle. Remarkably, this distribution exhibits a natural cut-off in the energy spectrum. This fact, already encountered in other formulations of generalized statistical mechanics, is expected to be relevant to the applications of the theory to those systems governed by long-range interactions. By employing the q-calculus, it is shown that the standard thermodynamic functional relationships are preserved, mimicking, in this way, the mathematical structure of the ordinary thermostatistics which is recovered in the q → 1 limit.

We introduce a new quantum heat engine, in which the working medium undergoes quantum quasi-adiabatic and non-unitary processes to extract work in the cycle. Two kinds of working medium are considered. The first one consists of a three-level quantum system, whereas the second is a quantum system with a discrete level and a continuum. Net work done by this engine is calculated and discussed. The results show that this quantum heat engine behaves like the two-level quantum heat engine in both the high-temperature and low-temperature limits, but it operates differently at intermediate temperatures. The efficiency of this quantum heat engine is also presented and discussed.

A simple numerical scheme based on the local equilibrium theory is developed to compute the density and pressure profiles of a weakly repulsive Bogoliubov gas. From these profiles the local velocity of sound has been calculated. The numerical scheme avoids the divergence problems encountered in some cases and the results agree well with those of previous workers. The effect of interactions on bosons confined to a small region of space by a bounded harmonic potential of extent r₀ has also been studied.

A new formalism is presented for high-energy analysis of the Green function for Fokker–Planck and Schrödinger equations in one dimension. Formulae for the asymptotic expansion in powers of the inverse wave number are derived, and conditions for the validity of the expansion are studied through the analysis of the remainder term. This method is applicable to a large class of potentials, including the cases where the potential V(x) is infinite as x → ±∞. The short-time expansion of the Green function is also discussed.

We apply stochastic quantization to a system of N interacting identical bosons in an external potential Φ, by means of a general stationary-action principle. The collective motion is described in terms of a Markovian diffusion on , with joint density and entangled current velocity field , in principle of non-gradient form, related to one another by the continuity equation. Dynamical equations relax to those of canonical quantization, in some analogy with Parisi–Wu stochastic quantization. Thanks to the identity of particles, the one-particle marginal densities ρ, in the physical space , are all the same and it is possible to give, under mild conditions, a natural definition of the single-particle current velocity, which is related to ρ by the continuity equation in . The motion of single particles in the physical space comes to be described in terms of a non-Markovian three-dimensional diffusion with common density ρ and, at least at dynamical equilibrium, common current velocity v. The three-dimensional drift is perturbed by zero-mean terms depending on the whole configuration of the N-boson interacting system. Finally, we discuss in detail under which conditions the one-particle dynamical equations, which in their general form allow rotational perturbations, can be particularized, up to a change of variables, to the Gross–Pitaevskii equations.

Model ecosystems with quenched, symmetric interspecies interactions have been extensively studied using the replica method of the statistical mechanics of disordered systems. Here, we consider a more general scenario in which both the species abundances and the interspecies interactions change with time, albeit in widely separated timescales. The equilibrium of the coupled dynamics is studied analytically within the partial annealing framework, in which the number of replicas n takes on positive as well as negative values. In the case n > 0, which describes ecosystems characterized by the cooperative interspecies interactions, we find a discontinuous transition to a regime of zero diversity, whereas in the case where competition prevails, n < 0, we find that the species diversity is maximum.

We have analytically obtained all the density matrix elements up to six lattice sites for the spin-1/2 Heisenberg XXZ chain at Δ = 1/2. We use the multiple integral formula of the correlation function for the massless XXZ chain derived by Jimbo and Miwa. As for the spin–spin correlation functions, we have newly obtained the fourth- and fifth-neighbour transverse correlation functions. We have calculated all the eigenvalues of the density matrix and analyse the eigenvalue distribution. Using these results the exact values of the entanglement entropy for the reduced density matrix up to six lattice sites have been obtained. We observe that our exact results agree quite well with the asymptotic formula predicted by the conformal field theory.

A grinding process with classification is considered. An integral grinding equation connecting the final particle size distribution function to the particle size distribution function before the grinding process is studied. Geometric partition models are used to obtain the breakage function. The results are compared with experimental data.

Darboux transformations and explicit solutions to Ablowitz–Ladik (AL) equations with self-consistent sources (ALESCS) are studied. Based on the Darboux transformation (DT) for the AL problem, we construct three types of non-auto-Bäcklund transformations connected with AL systems with different numbers of sources. The degenerate cases of DT and their applications to the reduced systems of ALESCS, for instance, discrete nonlinear Schrödinger with self-consistent sources (D-NLSSCS) and discrete mKdV equation with self-consistent sources (D-mKdVSCS), are discussed. Many types of solutions of ALESCS, D-NLSSCS and D-mKdVSCS, including solitons, positons, negatons can be derived from DTs and their degenerate cases.

In a previous paper, we have proposed a new integrable Hamiltonian describing two interacting particles in a harmonic mean field in D = 1 dimensional space. Here, we generalize this Hamiltonian to the D ⩾ 2 dimensional space. We show that the system is exactly solvable for a certain domain of the interaction constant, and that the size of this domain increases with D. We also show this D-dimensional Hamiltonian to be supersymmetric and shape invariant, very much as in the D = 1 case.

The equivalence transformation algebra L_E and some of its differential invariants for the class of equations u_t = (h(u)u_x)_x + f(x, u, u_x) (h ≠ 0) are obtained. Using these invariants, we characterize subclasses which can be mapped by means of an equivalence transformation into the well-studied family of equations v_t = (v^kv_x)_x.

A non-Hermitian complex symmetric 2 × 2-matrix toy model is used to study projective Hilbert space structures in the vicinity of exceptional points (EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseux-expanded in terms of the root vectors at the EP. It is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behaviour in the EP-limit can be resolved by projectively extending the original Hilbert space. The complementary normalization conditions correspond then to two different affine charts of this enlarged projective Hilbert space. Geometric phase and phase-jump behaviour are analysed, and the usefulness of the phase rigidity as measure for the distance to EP configurations is demonstrated. Finally, EP-related aspects of -symmetrically extended quantum mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated.

This paper provides the generalization of the work by Floreanini et al (1993 J. Phys. A: Math. Gen.26 611–4) who generated bibasic hypergeometric functions from (p, q)-oscillators. We consider a six-parameter deformed oscillator algebra realized from the (p, q)-deformed boson oscillators. We build the corresponding Fock space representation in an infinite-dimensional subspace of the Hilbert space of a harmonic oscillator. We also discuss the properties of a discrete spectrum of the Hamiltonian of the deformed harmonic oscillator corresponding to this system. We then define a realization of the deformed algebra in terms of a generalized derivative and investigate the relation between this representation and generalized bibasic Laguerre functions and polynomials.

We investigate the PT-symmetry of the quantum group invariant XXZ chain. We show that the PT-operator commutes with the quantum group action and also discuss the transformation properties of the Bethe wavefunction. We exploit the fact that the Hamiltonian is an element of the Temperley–Lieb algebra in order to give an explicit and exact construction of an operator that ensures quasi-Hermiticity of the model. This construction relies on earlier ideas related to quantum group reduction. We then employ this result in connection with the quantum analogue of Schur–Weyl duality to introduce a dual pair of C-operators, both of which have closed algebraic expressions. These are novel, exact results connecting the research areas of integrable lattice systems and non-Hermitian Hamiltonians.

We study character generating functions (character generators) of simple Lie algebras. The expression due to Patera and Sharp, derived from the Weyl character formula, is first reviewed. A new general formula is then found. It makes clear the distinct roles of 'outside' and 'inside' elements of the integrity basis, and helps determine their quadratic incompatibilities. We review, analyse and extend the results obtained by Gaskell using the Demazure character formulae. We find that the fundamental generalized-poset graphs underlying the character generators can be deduced from such calculations. These graphs, introduced by Baclawski and Towber, can be simplified for the purposes of constructing the character generator. The generating functions can be written easily using the simplified graphs, and associated Demazure expressions. The rank-two algebras are treated in detail, but we believe our results are indicative of those for general simple Lie algebras.

The algebra and calculus of generalized differential forms are reviewed and developed. Bases of minus one-forms are studied and used in the investigation of groups of generalized forms and generalized connections. Different representations of generalized forms are discussed. Physical and mathematical applications of generalized forms are presented in a number of examples.

We study the A_k generalized model of the O(1) loop model on a cylinder. The affine Hecke algebra associated with the model is characterized by a vanishing condition, the cylindric relation. We present two representations of the algebra: the first one is the spin representation, and the other is in the vector space of states of the A_k generalized model. A state of the model is a natural generalization of a link pattern. We propose a new graphical way of dealing with the Yang–Baxter equation and q-symmetrizers by the use of the rhombus tiling. The relation between two representations and the meaning of the cylindric relations are clarified through the rhombus tiling. The sum rule for this model is obtained by solving the q-KZ equation at the Razumov–Stroganov point.

We develop a density tensor hierarchy for open system dynamics that recovers information about fluctuations (or 'noise') lost in passing to the reduced density matrix. For the case of fluctuations arising from a classical probability distribution, the hierarchy is formed from expectations of products of pure state density matrix elements and can be compactly summarized by a simple generating function. For the case of quantum fluctuations arising when a quantum system interacts with a quantum environment in an overall pure state, the corresponding hierarchy is defined as the environmental trace of products of system matrix elements of the full density matrix. Whereas all members of the classical noise hierarchy are system observables, only the lowest member of the quantum noise hierarchy is directly experimentally measurable. The unit trace and idempotence properties of the pure state density matrix imply descent relations for the tensor hierarchies, that relate the order n tensor, under contraction of appropriate pairs of tensor indices, to the order n − 1 tensor. As examples to illustrate the classical probability distribution formalism, we consider a spatially isotropic ensemble of spin-1/2 pure states, a quantum system evolving by an Itô stochastic Schrödinger equation and a quantum system evolving by a jump process Schrödinger equation. As examples to illustrate the corresponding trace formalism in the quantum fluctuation case, we consider the tensor hierarchies for collisional Brownian motion of an infinite mass Brownian particle and for the weak coupling Born–Markov master equation. In different specializations, the latter gives the hierarchies generalizing the quantum optical master equation and the Caldeira–Leggett master equation. As a further application of the density tensor, we contrast stochastic Schrödinger equations that reduce and that do not reduce the state vector, and discuss why a quantum system coupled to a quantum environment behaves like the latter. The descent relations for our various examples are checked in a series of appendices.

The Klein paradox is one of the cornerstones in the development of quantum mechanics, and its consequences were used in various branches of physics, ranging from elementary particles to solid state. Yet its mathematical derivation is questionable in a number of steps, resulting in the wrong solution of Dirac equation. In this paper, the paradox is analysed in more detail, and its mathematical content is emphasized in order to show that in the study of extreme conditions of matter simple arguments may result in erroneous predictions.

In the framework of non-relativistic quantum mechanics and with the help of Green's functions formalism, we study the behaviour of weakly bound states in a non-central two-particle potential as they approach the continuum threshold. Through estimating Green's function for positive potentials we derive rigorously the upper bound on the wavefunction, which helps us to control its falloff. In particular, we prove that for potentials whose repulsive part decays slower than 1/r² the bound states approaching the threshold do not spread and eventually become bound states at the threshold. This means that such systems never reach supersizes, which would extend far beyond the effective range of attraction. The method presented here is applicable in the many-body case.

A simple algorithm is given to treat perturbed oscillator bound states and resonances. The method is applied to a bound state problem for which the energy is a non-analytic function of a perturbation parameter and also to two resonance problems, one of which has a spectrum with an unusual dual nature.

Recently relativistic quantum information has received considerable attention due to its theoretical importance and practical application. In particular, quantum entanglement in non-inertial reference frames has been studied for scalar and Dirac fields. As a further step along this line, we here shall investigate quantum entanglement of electromagnetic field in non-inertial reference frames. In particular, the entanglement of the photon helicity entangled state is extensively analysed. Interestingly, the resultant logarithmic negativity and mutual information remain the same as those for inertial reference frames, which is completely different from that previously obtained for the particle number entangled state.

By analysing the concept of contextuality (Bell–Kochen–Specker) in terms of pre- and post-selection, it is possible to assign definite values to observables in a new and surprising way. Physical reasons are presented for restrictions on these assignments. When measurements are performed which do not disturb the pre- and post-selection (i.e. weak measurements), then novel experimental aspects of contextuality can be demonstrated. We also prove that every PPS-paradox with definite predictions directly implies 'quantum contextuality' which is introduced as the analogue of contextuality at the level of quantum mechanics rather than at the level of hidden variable theories. Finally, we argue that certain results of these measurements (e.g. eccentric weak values outside the eigenvalue spectrum) cannot be explained by a 'classical-like' hidden variable theory.

We investigate the effect of phase shift on the entanglement transfer in two parallel 1D spin chains. We calculate the concurrence, measures for two-qubit entanglement, as a function of time. We find the maximum achievable entanglement in the cases with and without phase shift. Although the entanglement transfer becomes slow in most cases with phase shift, the significant enhancement of the maximum achievable entanglement is obtained which suggests its potential usefulness in quantum information processing.

Using a recently developed method, based on a generalization of the zero curvature representation of Zakharov and Shabat, we study the integrability structure in the Abelian Higgs model. It is shown that the model contains integrable sectors, where integrability is understood as the existence of infinitely many conserved currents. In particular, a gauge invariant description of the weak and strong integrable sectors is provided. The pertinent integrability conditions are given by a U(1) generalization of the standard strong and weak constraints for models with a two-dimensional target space. The Bogomolny sector is discussed, as well, and we find that each Bogomolny configuration supports infinitely many conserved currents. Finally, other models with U(1) gauge symmetry are investigated.

I show that reflection positivity implies that the force between any mirror pair of charge-conjugate probes of the quantum vacuum is attractive. This generalizes a recent theorem of Kenneth and Klich (2006 Phys. Rev. Lett.97 160401 (Preprint quant-ph/0601011)) to interacting quantum fields, to arbitrary semiclassical bodies and to quantized probes with non-overlapping wavefunctions. I also prove that the torques on charge-conjugate probes tend always to rotate them into a mirror-symmetric position.

We investigate the transition from second- to first-order systems. Quantum mechanically, this transforms configuration space into phase space and hence introduces noncommutativity in the former. This transition may be described in terms of spectral flow. Gaps in the energy or mass spectrum may become large which effectively truncates the available state space. Using both operator and path integral languages we explicitly discuss examples in quantum mechanics, (light-front) quantum field theory and string theory.

In this paper, we compare the classical conservative properties of five difference schemes applied to the coupled Klein–Gordon–Schrödinger equations in quantum physics, and investigate the numerical behaviour of the schemes in the implementation. Numerical results reveal merits and shortcomings of the schemes, and show that all five schemes are stable in the classical conservation laws. In the sense of preservation of classical conservative properties, conservative schemes are better than others.

Quaternion analysis of time-dependent Maxwell's equations in the presence of electric and magnetic charges has been developed and the solutions for the classical problem of moving charges (electric and magnetic) are obtained in a unique, simple and consistent manner.

We consider a massive relativistic particle in the background of a gravitational plane wave. The corresponding Green functions for both spinless and spin- cases, previously computed by Barducci and Giachetti (2005 J. Phys. A: Math. Gen.38 1615), are reobtained here by alternative methods, as for example, the Fock–Schwinger proper-time method and the algebraic method. In analogy with the electromagnetic case, we show that for a gravitational plane-wave background a semiclassical approach is also sufficient to provide the exact result, though the Lagrangian involved is far from being a quadratic one.

It was recently noted that the dispersion relation for the magnons of planar SYM can be identified with the Casimir of a certain deformation of the Poincaré algebra, in which the energy and momentum operators are supplemented by a boost generator J. By considering the relationship between J and , we derive a q-deformed super-Poincaré symmetry algebra of the kinematics. Using this, we show that the dynamic magnon representations may be obtained by boosting from a fixed rest-frame representation. We comment on aspects of the coalgebra structure and some implications for the question of boost covariance of the S-matrix.

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