Table of contents

This volume of Journal of Physics: Conference Series contains the Proceedings of the 5th International Symposium in Quantum Theory and Symmetries (QTS5), held in Valladolid, Spain, 22–28 July 2007. This is the fifth of a series of conferences previously held in Goslar (Germany) 1999, QTS1; Cracow (Poland) 2001, QTS2; Cincinnati (USA) 2003, QTS3, and Varna (Bulgaria) 2005, QTS4.

The QTS5 symposium gathered 181 participants from 39 countries working in different fields on Theoretical Physics. The spirit of the QTS conference series is to join researchers in a wide variety of topics in Theoretical Physics, as a way to make accessible recent results and the new lines of different fields. The QTS5 conference offered the following list of topics:

1. Symmetries in String Theory, Quantum Gravity and related

2. Symmetries in Quantum Field Theories, Conformal and Related Field Theories, Lattice and Noncommutative Theories, Gauge Theories

3. Quantum Computing, Information and Control

4. Foundations of Quantum Theory

5. Quantum Optics, Coherent States, Wigner Functions

6. Dynamical and Integrable Systems

7. Symmetries in Condensed Matter and Statistical Physics

8. Symmetries in Particle Physics, Nuclear, Atomic and Molecular

9. Nonlinear Quantum Mechanics

10. Time Asymmetric Quantum Mechanics

11. SUSY Quantum Mechanics, PT symmetries and pseudo-Hamiltonians

12. Mathematical Methods for Symmetries and Quantum Theories

13. Symmetries in Chemistry Biology and other Sciences

Papers accepted for publication in the present issue are based on the contributions from the participants in the QTS5 conference after a peer review process. In addition, a special issue of Journal Physics A: Mathematical and Theoretical contains contributions from plenary speakers, some participants as well as contributions from other authors whose works fit into the topics of the conference.

The organization of the conference had the following pattern. In the morning there were five plenary or general sessions for all the participants, which aimed to give a state of the art view of different topics. The afternoon was devoted to 6 parallel sessions, each one including the most afine topics. Also there was a poster session that could be visited in the intermediates and during the coffe breaks. A total of 23 plenary conferences, 130 parallel communications and 25 posters were presented in QTS5. All the sessions were given in the Palace of Conferences Conde Ansúrez which among other facilities also had a computer room. As social activities we had a guided walk around the historical part of the city, a visit to the nearby city of Segovia, as well as a welcome party and farewell dinner. These activities were intended to foster contact and informal discussions among the participants in the meeting.

We are much indebted to several institutions that without their support the organization of the QTS5 symposium would not have been possible. In this respect we greatly acknowledge to Ministerio de Educación of Spain and Junta de Castilla y León for general financial support, to Fundación Universidades de Castilla y León for a number of grants to young researchers who otherwise would not have attended the conference, to the European Physical Society that provided a number of grants for eastern countries and to the University of Valladolid where the event took place.

We want to thank all the members of the Conference Board and the International Advisory Committee of QTS5 for honoring us with their help and support.

Here we must mention our gratitude to the other members of the Local Organizing Committe of QTS5, Manuel Gadella, José Manuel Izquierdo and Sengül Kuru, who are not Editors of this JPCS issue but shared with us the complex organization of this event.

We acknowledge IOP Publishing for the facilities in the publication of these Proceedings.

Finally, on behalf of the Local Organizing Committee, we would like to thank all the participants in the QTS5 conference for their presence, contributions, and for the good atmosphere achieved during their stay. We hope that the experience of spending these days in Valladolid has been most fruitful for all of them.

O Arratia, J A Calzada, F Gómez-Cubillo, J Negro and M A del Olmo Universidad de Valladolid, Spain Editors of the QTS5 Proceedings

Conference Board

1. S T Ali (Montreal)

2. L L Boyle (Canterbury)

3. M A del Olmo (Valladolid)

4. V K Dobrev (Sofia)

5. H D Doebner (Clausthal), Chair

6. E Kapuscik (Cracow)

7. V I Man'ko (Moscow)

8. G Marmo (Naples)

9. G S Pogosyan (Yerevan and Dubna)

10. T H Seligman (Cuernavaca)

11. A I Solomon (Paris and Open University)

12. P Suranyi (Cincinnati)

13. L C R Wijewardhana (Cincinnati)

International Advisory Committee

1. L Accardi, (Roma)

2. M Asorey, (Zaragoza)

3. M T Batchelor, (Canberra)

4. C M Bender, (Washington)

5. A Bohm, (Texas)

6. E Celeghini, (Firenze)

7. I Cirac, (Garching)

8. S Ferrara, (CERN)

9. J P Gazeau, (Paris)

10. G Goldin , (Rutgers)

11. F Iachello, (Yale)

12. T Janssen, (Nijmegen)

13. J Klauder, (Gainesville)

14. P Kulish, (St Petersburg)

15. B Mielnik, (Mexico DF)

16. W Miller, (Minneapolis)

17. M Plyushchay, (Santiago de Chile)

18. O Ragnisco, (Roma)

19. S Randjbar-Daemi, (ICTP)

20. M Santander, (Valladolid)

21. G Sierra, (Madrid)

22. P Townsend, (Cambridge)

23. S Twarock, (York)

24. F Wilczek, (Boston)

25. P Winternitz, (Montreal)

26. K B Wolf, (Cuernavaca)

Local Organizing Committee (University of Valladolid)

1. Oscar Arratia

2. Juan A Calzada

3. Manuel Gadella

4. Fernando Gómez-Cubillo

5. José Manuel Izquierdo

6. Sengül Kuru

7. Javier Negro

8. Mariano A del Olmo (Chairman)

Ladies and Gentlemen

Dear Friends and Colleagues

I welcome you at the 5th International Symposium `Quantum Theory and Symmetries, QTS5' in Valladolid as Chairman of the Conference Board of this biannual series.

The aim of the series is to arrange an international meeting place for scientists working in theoretical and mathematical physics, in mathematics, in mathematical biology and chemistry and in other sciences for the presentation and discussion of recent developments in connection with quantum physics and chemistry, material science and related further fields, like life sciences and engineering, which are based on mathematical methods which can be applied to model and to understand microphysical and other systems through inherent symmetries in their widest sense. These systems include, e.g., foundations and extensions of quantum theory; quantum probability; quantum optics and quantum information; the description of nonrelativistic, finite dimensional and chaotic systems; quantum field theory, particle physics, string theory and quantum gravity.

Symmetries in their widest sense describe properties of a system which could be modelled, e.g., through geometry, group theory, topology, algebras, differential geometry, noncommutative geometry, functional analysis and approximation methods; numerical evaluation techniques are necessary to connect such symmetries with experimental results. If you ask for a more detailed characterisation of this notion a hand waving indirect answer is: Collect titles and contents of the contributions of the proceedings of QTS4 and get a characterisation through semantic closure.

Quantum theory and its Symmetries was and is a diversified and rapidly growing field. The number of and the types of systems with an internal symmetry and the corresponding mathematical models develop fast. This is reflected in the content of the five former international symposia of this series: The first symposium, QTS1–1999, was organized in Goslar (Germany) with 170 participants and 89 contributions in the proceedings; it was centred on the foundations and extensions of quantum theory, on quantisation methods and on q-algebras. In QTS2–2001 in Cracow (Poland) with 175 participants and 81 contributions; the main topics were applications of quantum mechanics, representations of algebras and group theoretical techniques in physics. In the symposium QTS3–2003 in Cincinnati (USA) with 145 participants and 92 contributions, quantum field theory, loop quantum gravity, string and brane theory was discussed. The focus in QTS4–2005 in Varna (Bulgaria) with 228 participant and 105 contributions, was on conformal field theory, quantum gravity, noncommutative geometry and quantum groups. Three proceedings volumes were published with World Scientific and one volume with Heron Press. The promising and interesting programme for QTS5–2007 in Valladolid (Spain) attracted more than 200 participants; the contributions will be published in a special issue of Journal of Physics A: Mathematical and Theoretical and a volume of Journal of Physics: Conference Series. This shows the wide scope of symmetry in connection with quantum physics and related sciences.

In the background of the symposia series is the Conference Board with presently 13 members. The Board encourages scientists and Institutions to present detailed proposals for a QTS symposium; it agrees to one proposal and is prepared to assist in matters of organisation; the local organisers are responsible for the scientific programme and for the organisation, including the budget. The Board decided that the next symposium QTS6 will be held 2009 at the University of Kentucky in Lexington (USA); Alan Shapere is the chairman of the Local Organizing committee.

In the name of all of you I express my appreciation and my thanks to the members of the Local Organizing Committee of QTS5, especially to Mariano del Olmo. The programme is outstanding; it covers recent and new developments in our field. The organization is very effective and complete. We have all the necessary condition for a successful and smooth meeting. Thank you again Mariano.

H-D Doebner Chairman of the Conference Board of QTS5

We show that N = 1+1 supersymmetric extension of 3D gravity with torsion, with suitable boundary conditions, has the asymptotic superconformal symmetry. The existence of exact supersymmetries implies the stability conditions for the black hole solutions.

Amplitudes for fermion-fermion, boson-boson and fermion-boson interactions are calculated in the second order of perturbation theory in the Lobachevsky space. An essential ingredient of the model is the Weinberg's 2(2j + 1)— component formalism for describing a particle of spin j. The boson-boson amplitude is then compared with the two-fermion amplitude obtained long ago by Skachkov on the basis of the Hamiltonian formulation of quantum field theory on the mass hyperboloid, p²₀ — p² = M², proposed by Kadyshevsky. The parametrization of the amplitudes by means of the momentum transfer in the Lobachevsky space leads to same spin structures in the expressions of T — matrices for the fermion case and the boson case. However, certain differences are found. Possible physical applications are discussed.

Two types of gauge transformations of noncommutative pure gauge theory are discussed. It is shown that Yang-Mills theory with the so called twisted gauge symmetry is consistent provided it also enjoys the standard noncommutative *-gauge symmetry.

The structure of integrable field theories in the presence of jump defects is discussed in terms of boundary functions under the Lagrangian formalism. Explicit examples of bosonic and fermionic theories are considered. In particular, the boundary functions for the N = 1 and N = 2 super sinh-Gordon models are constructed and shown to generate the Backlund transformations for its soliton solutions. As a new and interesting example, a solution with an incoming boson and an outgoing fermion for the N = 1 case is presented. The resulting integrable models are shown to be invariant under supersymmetric transformation.

A new formulation of the Electroweak Model with 3-dimensional spherical geometry in the target space is suggested. The free Lagrangian in the spherical field space along with the standard gauge field Lagrangian form the full Higgsless Lagrangian of the model, whose second order terms reproduce the same fields with the same masses as the Standard Electroweak Model. The vector bosons masses are automatically generated, so there is no need in special mechanism of spontaneous symmetry breaking.

A possible nonlinear completion of massive gravity of the Fierz-Pauli type is proposed. The theory describes a system consisting of a massive tensor field of the Fierz-Pauli type and an additional massive vector field. Massless limit as well as flat-spacetime limit can be taken smoothly. Constructing a nonlinear version of the physical-state condition which drives an extra scalar ghost from physical states is still unsettled.

The model is considered in non canonical bases SU(3) ⊃ SO(3) for the state vectors. The rational map ansatz are used to describe the soliton with topological number bigger than one. The canonical quantization generates flve difierent momenta of inertia in Hamiltonian and quantum mass corrections, which can stabilize the quantum soliton solution. The explicit expressions of the Lagrangian and Hamiltonian are derived for this model.

The deformation quantization procedure can introduce the quantum effect by including the deformation of complex structures on a bi-dimensional Riemann surface Σ without boundary. This is the platform of bi-dimensional conformal models like string theory and two-dimensional gravity. Then, the star product can become a quasiconformal mapping on this surface. This makes the road toward a Teichmuller theory of the Moyal-Weyl star product through the quasiconformal mappings as lifting of universal covering surfaces of any Riemann surface.

We give an analyical meaning to a vertex operator by using an analog of the Hida Fock space

We use the representation theory of SO(2; n) to determine the renormalized vacuum energy for a massless scalar field in the n-dimensional Einstein universe subject to Dirichlet boundary conditions on a sphere of maximum radius. The problem is an exactly solvable one. This is in remarkable contrast to the analogous problem in flat n dimensional Minkowski space where, except for the lowest dimensional case (n = 2), there is no known exactly solvable method of solution for any radius of the spherical boundary. For n = 4 our results agree with those of Bayen and Özcan, Class. Quant. Grav., 10 (1993) L115-L121. We use our results to obtain some qualitative information about the Casimir effect for spherical boundaries of smaller radii, and we comment on how one may apply these results to obtain information about the corresponding problem in Minkowski space.

We review the developing effective action framework for field-theoretical models in the presence of branes: describe general scheme based on Dirichlet-to-Neumann reduction and present some applications and results.

We consider an approach similar to field theory for the application of Noether theorem to extended paticles. We obtain Euler-Lagrange equations for the extended particles as well as an equation binding the internal and external currents. The concrete case of spin 1/2 is considered in detail.

A new quantum architecture for multiplying signed integers is presented based on Booth's algorithm, which is well known in classical computation. It is shown how a quantum binary chain might be encoded by its flank changes, giving the final product in 2's-complement representation.

In this paper we propose a new realist, non-collapse interpretation of quantum mechanics, which moves away from the prevailing trend in the subject by paying special attention to the physical relevance of the interpretation. In particular, our proposal endows the Hamiltonian of the system, systematically ignored in the traditional interpretations, with a central role: it distinguishes between systems and subsystems and is the main ingredient in the selection of the definite-valued observables. We show how this interpretation solves the measurement problem, both in the ideal and in the non-ideal version, and we argue for the physical relevance of the new definite-value assignment.

The properties of the commutator function of the free electron Dirac field, zeros of which respond to the independent measurements of the electron charge density in two different points of the space-time, are discussed. The established properties of one-space dimensional commutator function D(x; t) in the discrete representation, show the dependency of the number and location of its zeros and points of discontinuity on the value of normalizing factor. In this connection, the following new property of the commutator function is found: the length of the time interval containing at least one zero of D(x; t) depends on the relation between the value of the normalizing factor and the Compton wavelength of the electron.

Here, the ordinary gauge transformations in configuration space are generalized to the phase space. The corresponding gauge functions are separable in terms of those of the configuration and the momentum spaces. Furthermore, we consider the gauge transformations in phase space for which their gauge functions are not, in general, separable. It is shown that, the assumption of the general global gauge symmetry in phase space immediately yields the quantization of the electric charge. We obtain this result without assuming the concept of the magnetic monopole that Dirac did in his approach to the problem of electric charge quantization and which is not detected yet.

It is conventionally believed that Bohm's quanturn theory yields all the predictions of standard quantum theory. However, in this paper, we show that the transmission and reflection coefficients (i.e., probabilities) predicted by Bohm's quanturn theory can be different from those predicted by standard quantum theory for energy eigenstates of the step barrier.

Semigroups of stochastic and bistochastic matrices constructed by means of spin tomograms or tomographic probabilities and their relations to the problem of Bell's inequalities and entanglement are reviewed. The probability determining the quantum state of spins and the probability densities determining the quantum states of particles with continuous variables are considered. Entropies for semigroups of stochastic and bisctochastic matrices are studied, in view of both the Shannon information entropy and its generalization like Rényi entropy. Qubit portraits of qudit states are discussed in the connection with the problem of Bell's inequality violation for entangled states.

We set up the Maxwell's equations and subsequently the classical wave equations for the electromagnetic waves which together with their generating source, an oscillatory charge of zero rest mass in general travelling, make up a particle travelling similarly as the source at velocity ν in the field of an external scalar and vector potentials. The direct solutions in constant external field are Doppler-displaced plane waves propagating at the velocity of light c; at the de Broglie wavelength scale and expressed in terms of the dynamically equivalent and appropriate geometric mean wave variables, these render as functions identical to the space-time functions of a corresponding Dirac spinor, and in turn identical to de Broglie phase waves previously obtained from explicit superposition. For two spin-half particles of a common set of space-time functions constrained with antisymmetric spin functions as follows the Pauli principle for same charges and as separately indirectly induced based on experiment for opposite charges, the complete wave functions are identical to the Dirac spinor. The back-substitution of the so explicitly determined complete wave functions in the corresponding classical wave equations of the two particles, subjected further to reductions appropriate for the stationary-state particle motion and to rotation invariance when in three dimensions, give a Dirac equation set; the procedure and conclusion are directly extendible to arbitrarily varying potentials by use of the Furious theorem and to particle motions in three dimensions by virtue of the characteristics of de Broglie particle motion. Through the derivation of the Dirac equation, the study hopes to lend insight into the connections between the Dirac wave functions and the electrodynamic components of simple particles under the government by the well established basic laws of electrodynamics.

Using Lie algebra methods, we find a new class of Bogoliubov transformations which generalize the notion of squeezed states. The Hamiltonians for the simple harmonic and anharmonic oscillators, turn out to be the generators of a Lie group, whose other generators may be found exactly, or up to any desired order of the perturbation parameter. An element of this Lie group, which is realized as the multi-photon operator, transforms the anharmonic Hamiltonian to the harmonic one. The transformation of the ordinary annihilation and creation operators under this unitary transformation leads to the introduction of multi-photon coherent states. We specifically consider four-photon coherent states in detail and study the time dependent position and momentum uncertainties in these states.

We consider a four-qubit system, initially in a coherent state along the z-axis, and study its time dependence under the influence of the well-known two-axis countertwisting Hamiltonian. Analyzing the time dependence of the spin operators, we observe its squeezing properties, according to the criterion given by Kitagawa et al. We also perform similar calculations for the same Hamiltonian in the presence of an external fleld and compare its squeezing properties with that of the former. It is observed that the details of the squeezing properties are difierent in the two cases. Plots of the squeezing parameters as a function of time and the rotation angle for both cases are presented.

In this paper, we study a new quantum reversible effect for a three-level atom which interacts with electromagnetic cavity field. We investigate the reversible conditions for a three-level atom that is in resonance with two cavity modes and in off resonance with a single mode of cavity. The possibility of the reversible factorization between the atom and cavity radiation is the subject of this work. It is found the reversible condition for which these quantum subsystems can (or not) restore their diagonal moments.

Coherent states are constructed for systems generated by supersymmetry from an initial Hamiltonian with a purely discrete spectrum such that the levels depend analytically on their subindex. The technique is illustrated by means of the trigonometric Pöschl-Teller potentials.

A symplectic connection on a cotangent bundle T* M induced by a Levi-Civita connection on a configuration space M is constructed. General properties of an Abelian connection built from the induced symplectic connection are presented. An example of a finite Abelian connection determined by the induced symplectic connection has been found.

The concepts of generalized zero curvature conditions and integrability in higher dimensions are briefly reviewed, where integrability in this context always means the existence of infinitely many conservation laws. It turns out that the conservation laws provided by the generalized integrability are, under certain additional assumptions, generated by the volume-preserving diffeomorphisms on target space. The possible conservation laws for field theories with a three-dimensional target space are classified. Further, an explicit example is discussed in some detail.

A class of quantum superintegrable Hamiltonians defined on a two-dimensional hyperboloid is considered together with a set of intertwining operators connecting all of them. It is shown that such intertwining operators close a su(2; 1) Lie algebra and determine the Hamiltonians through the Casimir operators. The physical states are characterized as unitary representations of su(2; 1).

We present a general review of the dynamics of topoligical solitons in 1 and 2 dimensions and then discuss some recent work on the scattering of various solitonic objects (such as kinks and breathers etc) on potential obstructions.

We consider a quantum mechanical system consisting of a linear chain of harmonic oscillators coupled by a nearest neighbor interaction. The system configuration can be closed (periodic boundary conditions) or open (non-periodic case). We show that such systems can be considered as Wigner Quantum Systems (WQS), thus yielding extra solutions apart from the canonical solution. In particular, a class of WQS-solutions is given in terms of unitary representations of the Lie superalgebra gl(1|n). In order to determine physical properties of the new solutions, one needs to solve a number of interesting but dificult representation theoretical problems. We present these problems and their solution, and show how the new results yield attractive properties for the quantum system (energy spectrum, position probabilities, spacial properties).

The geometry of Stäckel systems is employed to find, for a given (classical) natural Hamiltonian, another natural Hamiltonian which is integrable by separation of variables and in some sense close to the first one. The study is intended as a first step for a perturbative analysis of a given Hamiltonian system. The method proposed here, still in development, is in large part coordinate independent and effective mainly on manifolds of constant curvature. Examples are given for the quadrupole field and Hénon-Heiles systems. Stäckel systems are associated with quadratic in the momenta first integrals which play a fundamental role in quantization of classical systems.

The analysis of gene expression allow to study the functions of genes and their roles in difierent processes in the cell of a living system, including the cell cycle. Clustering is widely used in the analysis of high-throughput gene expression data to flnd patterns of similarity that enable related gene groups and functions to be identifled. Clustering algorithms are very sensitive to the choice of initial conditions and optimal number of clusters. In this paper, we investigate the impact of metrics and cluster parametrisation for three clustering models and propose a method for optimisation of cluster parameters based on cluster compactness and separation. A case study presents the analysis of gene expression data for E.coli bacteria.

To describe thermal pairing systems, an approach that combines the modified BCS method and the Lipkin-Nogami particle-number projection method is proposed. The expressions of the thermal pairing gap and internal energy are derived. The latter are numerically studied as a function of the temperature within the Richardson model.

The algebraic Interacting Vector Boson Model (IVBM) is extended by exploiting three new subgroup chains in the reduction of its highest symplectic dynamical symmetry group Sp(12, R) to the physical angular momentum subgroup SO(3). The corresponding exactly solvable limiting cases are applied to achieve a description of complex nuclear collective spectra of even-even nuclei in the rare earth and actinide regions up to states of very high angular momentum.

First we exploit two reductions in which collective modes can be mixed, and obtain successful descriptions of both positive and negative parity band conflgurations. The structure of band-head conflgurations, whose importance is established in the flrst two limits, is examined in a third reduction, that also provides important links between the subgroups of the other limits.

Born-Infeld theory is a non-linear formalism which has many applications in string and electromagnetic theories. Although, the existence of magnetic monopoles and dyons are suggested by Born-Infeld theory, but this theory is not invariant under the dual transformations. In this theory electric fields for point charged particles are not singular at origin (r = 0), but magnetic fields and vector potentials are still singular. In this paper we show that the vanishing of dual symmetry is responsible for these singularities. Furthermore, we present the dual symmetric Born-Infeld theory, by a symmetric definition of electromagnetic fields in terms of new scalar and vector potentials, as well as the ordinary ones. All singularities of vector potential and magnetic field are removed as an immediate consequence of this symmetry.

We use the relativistic mean field framework to analyse the reliability of the explanation of the pseudospin symmetry (PSS) that has been accepted, quite generally, by the scientific community, in the last decade. We make a comparative analysis of the mechanisms responsible for the breaking of the spin and pseudospin symmetries that shows the different nature of these symmetries. We propose an explanation of the PSS, also valid in the nonrelativistic limit, in which the effect of the deviation of the single-particle central potential from a harmonic oscillator on the breaking of the degeneracy of pseudospin doublets is partially compansated by the effect of the spin-orbit interaction.

A nonlinear Schrodinger equation, that had been obtained within the context of the maximum uncertainty principle, has the form of a difference-differential equation and exhibits some interesting properties. Here we discuss that equation in the regime where the nonlinearity length scale is large compared to the deBroglie wavelength; just as in the perturbative regime, the equation again displays some universality. We also briefly discuss stationary solutions to a naturally induced discretisation of that equation.

A quantum mechanical theory with time asymmetry intrinsic to states (or observables) features the concept of an initial time of the state and thus a preparation time of the physical system represented by the state. This special time is investigated in the context of scattering theory, where, in standard quantum mechanics, the physical meaning of a preparation time has remained obscure. In an experiment, the preparation time corresponds to an ensemble of times of scattering marking the times in the laboratory when one scattering projectile interacts with one target quantum.

We consider a model for an unstable quantum oscillator. The energy levels, the level widths and wave functions of the unstable states have been explicitly found. The obtained results can be applied for the description of the properties of such systems as quantum wells or series of molecular or nuclear resonances.

In this note, we discuss some mathematical properties of Gamow vectors. We start with the definition and properties of the approximate Gamow vectors that are Hilbert space vectors with similar properties to Gamow vectors and are defined by integration over the Hamiltonian spectrum and not over the whole real line, as is the case with ordinary Gamow vectors. We also discuss some properties of Gamow dyads, which are density operators constructed with Gamow vectors.

The work reviews some mathematical aspects of spectral properties, eigenfunction expansions and scattering theory in rigged Hilbert spaces, laying emphasis on Lippmann-Schwinger equations and Schrödinger operators.

On the basis of Dirac's causality, we will show that the time evolution is limited to a semigroup. The abstract vector space for states and (yes-or-no) observables are then not the entire Hilbert space but its particular dense subspaces, called Hardy spaces. The Hardy spaces and their functional spaces together make the Hardy rigged Hilbert spaces, which is also called the time-asymmetric boundary condition (TABC). We will illustrate the usage of the TABC with the neutral kaon decay experiment.

Three dimensional exactly solvable quantum potentials for which an extra term of form 1/r² has been added are shown to maintain their functional form which allows the construction of the Hamiltonian hierarchy and the determination of the spectra of eigenvalues and eigenfunctions within the Supersymmetric Quantum Mechanics formalism. For the specific cases of the Harmonic Oscillator and the Coulomb potentials, known as Pseudo-Harmonic Oscillator and Pseudo-Coulomb potentials, it is shown here that the inclusion of the new term corresponds to rescaling the angular momentum and it is responsible for maintaining their exact solvability.

The eigenvalue problem of the spherically symmetric oscillator Hamiltonian is revisited in the context of canonical raising and lowering operators. The Hamiltonian is then factorized in terms of two not mutually adjoint factorizing operators which, in turn, give rise to a non-Hermitian radial Hamiltonian. The set of eigenvalues of this new Hamiltonian is exactly the same as the energy spectrum of the radial oscillator and the new square-integrable eigenfunctions are complex Darboux-deformations of the associated Laguerre polynomials.

The second-order supersymmetry transformations are used to generate Hamiltonians with known spectra departing from the trigonometric Poschl-Teller potentials.

Complex potentials are constructed as Darboux-deformations of short range, radial nonsingular potentials. They behave as optical devices which both refracts and absorbs light waves. The deformation preserves the initial spectrum of energies and it is implemented by means of a Gamow-Siegert function (resonance state). As straightforward example, the method is applied to the radial square well. Analytical derivations of the involved resonances show that they are 'quantized' while the corresponding wave-functions are shown to behave as bounded states under the broken of parity symmetry of the related one-dimensional problem.

The solution of non-central PT -symmetric potentials is discussed by the separation of the variables in polar and angular coordinates. Conditions are formulated to guarantee the separation of the variables and the PT symmetry of the potential. The original eigenvalue equation is separated into one-dimensional Schrödinger-type differential equations. The importance of the boundary conditions, especially that of the periodic boundary condition of the azimuthal equation is pointed out. Further conditions leading to exact solutions of the whole problem are also formulated. An example combining the harmonic oscillator and the Scarf I potential in the radial and polar equation is discussed in detail, and the bound-state wave functions and the energy eigenvalues are derived. The spectrum exhibits partial degeneracies similar to those observed in the spectrum of the isotropic harmonic oscillator.

A generalization of the concept of PT—symmetric Hamiltonians H = p² + V (x) is presented. For the usual analytic potentials V (x) (with singularities) and for the recently widely accepted 'PT—symmetric' asymptotic boundary conditions for wave functions ψ(x) (selected inside a pair of complex wedges generalizing the usual x → ±∞ asymptotics), non-equivalent quantum toboggans are defined as integrated along topologically different paths C of coordinates x ε Cl.

A classification of Hermitian representations for the recently introduced fuzzy torus algebra is presented. This is carried out by regarding the fuzzy torus algebra as a q-deformation of parafermion. In addition to the known representations, new representations of both finite and infinite dimension are found. Using the infinite dimensional representation, coherent state for the fuzzy torus is constructed. Dirac operator on commutative torus is also discussed.

We present a generalization of a recently developed method for accelerated Monte Carlo calculation of path integrals of generic many particle systems. Using this method and the derived hierarchy of efiective actions we calculate the energy spectra of a two particle model with quartic interaction for several values of coupling and demonstrate agreement with analytical results governing the increase in efficiency of the new method.

Applications of algebras in physics are related to the connection of measurable observables to relevant elements of the algebras, usually the generators. However, in the determination of the generators in Lie algebras there is place for some arbitrary conventions. The situation is much more involved in the context of quantum algebras, where inside the quantum universal enveloping algebra, we have not enough primitive elements that allow for a privileged set of generators and all basic sets are equivalent. In this paper we discuss how the Drinfeld double structure underlying every simple Lie bialgebra characterizes uniquely a particular basis without any freedom, completing the Cartan program on simple algebras. By means of a perturbative construction, a distinguished deformed basis (we call it the analytical basis) is obtained for every quantum group as the analytical prolongation of the above defined Lie basis of the corresponding Lie bialgebra. It turns out that the whole construction is unique, so to each quantum universal enveloping algebra is associated one and only one bialgebra. In this way the problem of the classification of quantum algebras is moved to the classification of bialgebras. In order to make this procedure more clear, we discuss in detail the simple cases of su(2) and su_q(2).

In the present note we consider a class of second order parabolic equations with position dependent coefficients; such equations describe a diffusion of (quasi) particles with a variable mass. We represent a solution of Cauchy-Dirichlet problem for such class of equations in a bounded domain in the form of a limit of finite dimensional integrals of elementary functions. Such kind of a representation is usually called Feynman formula and can be used for calculations. Finite dimensional integrals in our Feynman formula give approximations for a functional integral over a probability measure on a set of trajectories in the domain where the solution of the considered problem is investigated; this measure is generated by a diffusion process with variable diffusion coefficient and absorption on the boundary, hence, to get Feynman formula also means to get a representation of the solution of the considered problem with the help of a functional integral (such kind of a representation is usually called Feynman-Kac formula).

The spin-statistics theorem in quantum field theory relates the spin of a particle to the statistics obeyed by that particle. Here we investigate an interesting correspondence or connection between curvature (κ = ±1) and quantum statistics (Fermi-Dirac and Bose-Einstein, respectively). The interrelation between both concepts is established through vacuum coherent configurations of zero modes in quantum field theory on the compact O(3) and noncompact O(2; 1) (spatial) isometry subgroups of de Sitter and Anti de Sitter spaces, respectively. The high frequency limit, is retrieved as a (zero curvature) group contraction to the Newton-Hooke (harmonic oscillator) group. We also make some comments on the physical significance of the vacuum energy density and the cosmological constant problem.

We show that the Inönü-Wigner contraction naturally associated to a reduction chain s ⊃ s' of semisimple Lie algebras induces a decomposition of the Casimir operators into homogeneous polynomials, the terms of which can be used to obtain additional mutually commuting missing label operators for this reduction. The adjunction of these scalars that are no more invariants of the contraction allow to solve the missing label problem for those reductions where the contraction provides an insuflcient number of labelling operators.

The classical and quantum position-dependent mass harmonic oscillators are constructed by means of the supersymmetric approach. The correspondence between the classical and the quantum Hamiltonians is used to fix the ordering of the kinetic term in the quantum framework. Some examples are given for different types of mass function.

I review our recent work on contractions of affine Kac-Moody algebras (KMA) and present new results. We study generalized contractions of KMA with respect to their twisted and untwisted KM subalgebras. As a concrete example, we discuss contraction of D⁽¹⁾₄ and D⁽³⁾₄, based on Z₃-grading. We also describe examples of 'level-dependent' contractions, which are based on Z-gradings of KMA. Our work generalizes the Inönü-Wigner contraction of P. Majumdar in several directions. We also give an algorithm for constructing Kac-Moody-like algebras ĝ for any Lie algebra g.

We present a new Lie algebraic method to study the eigenvalues and eigenfunctions of quantum anharmonic oscillators. We consider the Hamiltonians for the simple harmonic and anharmonic oscillator as the two generators of a Lie algebra, whose other generators may be found exactly or up to any order of the parameter involved. Speciflcally, the closed commutator algebra for the quartic anharmonic oscillator is established in a perturbation sense. An element of this Lie group, turning out to be the four-photon operator, transforms the quartic anharmonic oscillator Hamiltonian to the harmonic one in a perturbation sense; thus, facilitating the calculation of the eigenvalues and eigenfunctions of the former.

In this paper we present the first steps for obtaining a discrete Quantum Mechanics making use of the Umbral Calculus. The idea is to discretize the continuous Schrödinger equation substituting the continuous derivatives by discrete ones and the space-time continuous variables by well determined operators that verify some Umbral Calculus conditions. In this way we assure that some properties of integrability and symmetries of the continuous equation are preserved and also the solutions of the continuous case can be recovered discretized in a simple way. The case of the Schrödinger equation with a potential depending only in the space variable is discussed.

In this work, a method to extract continuous syrnmetries of general second-order linearl ordinary differential equation is presented. The formalism is illustrated by the examples.

Classical Lie point symmetry techniques are employed to time dependent nonlinear heat diffusion equations describing thermal energy storage in a medium subjected to a convective heat transfer to the surrounding environment at the boundary through a variable heat transfer coefficient. Exponential temperature-dependent thermal conductivity and heat capacity are assumed. Group classification for the source term is performed and some exciting large symmetry algebras are admitted. It turns out that the principal Lie algebra extends when the source term vanishes and when it is given as the exponential function of temperature. Reduction by one of the independent variables is performed for some realistic choices of the source term. In some case the resulting nonlinear ordinary differential equation with appropriate corresponding conditions are solved using Adomian decomposition method.

During recent years, exact solutions of position-dependent mass Schrödinger equations have inspired intense research activities, based on the use of point canonical transformations, Lie algebraic methods or supersymmetric quantum mechanical techniques. Here we highlight the interest of another approach to such problems, relying on quadratic algebras. We illustrate this point by constructing spectrum generating algebras for a class of d-dimensional radial harmonic oscillators with d ≥ 2 (including the one-dimensional oscillator on the line via some minor changes) and a specific mass choice. This provides us with a counterpart of the well-known su(1, 1) Lie algebraic approach to the constant-mass oscillators.

We construct explicitly groups associated to specific ternary algebras which extend the Lie (super)algebras (called Lie algebras of order three). It turns out that the natural variables which appear in this construction are variables which generate the three-exterior algebra. An explicit matrix representation of a group associated to a peculiar Lie algebra of order three is constructed considering matrices with entry which belong to the three exterior algebra.

Generalized quantum statistics such as para-Bose and para-Fermi statistics are related to the basic classical Lie superalgebras B(0|n) and B_n. We give a quite general definition of 'a generalized quantum statistics associated to a Lie superalgebra G'. This definition is closely related to a certain Z-grading of G. The generalized quantum statistics is determined by a set of root vectors (the creation and annihilation operators of the statistics) and the set of algebraic relations for these operators. Then we give a complete classification of all generalized quantum statistics associated to the Lie superalgebras A_n, B_n, C_n, D_n, G₂, F₄, E₆, E₇, E₈, A(m|n), B(m|n), C(n), D(m|n), G(3), F(4) and D(2; 1; α).

A recently developed method systematically improved the convergence of generic path integrals for transition amplitudes, partition functions, expectation values and energy spectra. This was achieved by analytically constructing a hierarchy of discretized effective actions indexed by a level number p and converging to the continuum limit as 1/N^p. Here we apply the above general method to numerical calculations using Metropolis Monte Carlo simulations of energy expectation values and energy spectra. We analyze and compare the ensuing increase in efficiency of several orders of magnitude.