The XI International Conference on Symmetry Methods in Physics (SYMPHYS-11) was held in Prague, the Czech Republic, from 21 to 24 June 2004. It was organized by the Bogoliubov Laboratory of Theoretical Physics, the Doppler Institute, and the Czech Technical University (Prague) in the framework of the Blokhintsev-Votruba programme. A total of about 100 scientists from all over the world took part in the conference.

The conference series was initiated by Professor Yakov A. Smorodinsky (1917-1992), an outstanding theoretical physicist. Professor Smorodinsky organized the first five conferences that were held at the Institute of Physics and Power Engineering in Obninsk from 1986 to 1991. The next three conferences of the series were held at the Joint Institute for Nuclear Research in Dubna in 1993, 1995 and 1997, while the last two took place at the Yerevan State University (Armenia) in 2001 and 2003.

The SYMPHYS-11 conference is devoted to general studies and applications of group theoretical methods in modern physics. It covers fields of research where symmetry-based methods play an important role. The programme included the following topics: symmetries of fundamental interactions; Lie groups, supergroups and nonlinear algebraic structures; symmetries of difference and differential equations; nonlinear systems and quantum chaos; quantum optics and coherent states; periodic and aperiodic structures.

BPS domain wall in massive nonlinear sigma model in harmonic superspace

Masato Arai, Jiri Niederle
Institute of Physics, AS CR, 182 21, Prague 8, Czech Republic
Evgeny Ivanov
Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, 141 980 Moscow region, Russia

BPS wall solutions in four-dimensional massive N = 2 nonlinear sigma models are studied in the off-shell harmonic superspace approach in which N = 2 supersymmetry is manifest. The general nonlinear sigma model can be described by an analytic harmonic potential which is the hyper-Kahler analog of the Kahler potential in N = 1 theory. We examine the massive nonlinear sigma model with multi-center four-dimensional target hyper-Kahler metrics and derive the corresponding BPS equation. We study in some detail two particular cases with the Taub-NUT and double Taub-NUT metrics. The latter embodies, as its two separate limits, both Taub-NUT and Eguchi-Hanson metrics. We find that domain wall solutions exist only in the double Taub-NUT case including its Eguchi-Hanson limit.

PACS: 11.10.-z;11.27.+d;11.30.Pb
Keywords: domain wall, supersymmetry, harmonic superspace
File size: 201 KB

Anomalies and superpotential in N = 1 noncommutative gauge theories

F. Ardalan and N. Sadooghi
Department of Physics, Sharif University of Technology P.O. Box 11365-9161, Tehran, Iran and Institute for Studies in Theoretical Physics and Mathematics (IPM) P.O. Box 19395-5531, Tehran, Iran

The anomaly of various currents in the noncommutative supersymmetric N = 1, U(1) gauge theory are calculated and the effective superpotential obtained.

PACS: 11.15.Bt
Keywords: noncommutative N = 1 supersymmety, effective superpotential
File size: 111 KB

Generalized dimensional reduction of D = 6 (2,0) chiral supergravity

L. Andrianopoli, S. Ferrara
CERN, Theory Division, CH 1211 Geneva 23, Switzerland
M.A. Lledo
Departamento de Fisica Teorica, Universidad de Valencia and IFIC, C/Dr. Moliner, 50, E-46100 Burjassot (Valencia), Spain.

We report on the Scherk-Schwarz reduction of D = 6 (2,0) supergravity coupled to matter and its interpretation as D = 5 gauged N = 2 supergravity of no-scale type.

PACS: 04.65.+e, 04.50.+h, 11.10.Kk
Keywords: Scherk-Schwarz , 6D supergravity, dimensional reduction
File size: 124 KB

Algebraic construction of integrable and super integrable hierarchies

H. Aratyn
Department of Physics, University of Illinois at Chicago 845 W. Taylor St., Chicago, Illinois, 60607-7059
J.F. Gomes and A.H. Zimerman
Instituto de Fisica Teorica - IFT/UNESP Rua Pamplona 145, 01405-900, S~ao Paulo - SP, Brazil

A general construction of integrable hierarchies based on afine Lie algebras is presented. The models are specified according to some algebraic data and their time evolution is obtained from solutions of the zero curvature condition. Such framework provides an unified treatment of relativistic and non relativistic models. The extension to the construction of supersymmetric integrable hierarchies is proposed. An explicit example of N = 2 super mKdV and sinh-Gordon is presented.

PACS: 11.25.Hf, 02.30.Ik
Keywords: integrability, supersymmetric integrable models
File size: 141 KB

On sl(N) and sl(M|N) integrable open spin chains

D. Arnaudon, N. Crampe, A. Doikou, L. Frappat, E. Ragoucy
Laboratoire d'Annecy-le-Vieux de Physique Theorique LAPTH CNRS, UMR 5108, associee a l'Universite de Savoie LAPP, BP 110, F - 74941 Annecy-le-Vieux Cedex, France
J. Avan
Laboratoire de Physique Theorique et Modelisation Universite de Cergy, 5 mail Gay-Lussac, Neuville-sur-Oise F - 95031 Cergy-Pontoise Cedex, France

We study open spin chains based on rational sl(N) and sl(M|N) R-matrices. We classify the solutions of the re ection equations, for both the soliton-preserving and soliton- non-preserving cases. We then write the Bethe equations for these open spin chains.

PACS: 02.20.Uw, 03.65.Fd, 75.10.Pq
Keywords: spin chains, Yangians, quantum groups, Yang-Baxter equation
File size: 154 KB

Non-generic symmetries and surface terms

Dumitru Baleanu
Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, University of Cankaya, Ankara 06530, Turkey and
Institute of Space Sciences, P.O.BOX, MG-23, R 76900, Magurele-Bucharest, Romania

Integrable geometries were obtained by adding a total time derivative involving the components of the angular momentum to a given free Lagrangian. The motion on a sphere and its induced geometries are examined in details.

PACS: 02.40.-ky.
Keywords: Killing-Yano tensors, non-generic symmetries, surface terms
File size: 77 KB

The covering problem related to quasicrystals

Lubomira Balkova, Zuzana Masakova
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Trojanova 13, 120 00 Praha 2, Czech Republic

We study mathematical models of quasicrystalline materials/non-crystallographic solids with long range aperiodic order. A natural generalization of crystallographic lattices are the so-called Meyer sets. They are uniformly discrete, relatively dense point sets with the property of almost lattices. This property ensures that there is only a finite number of local configurations of atoms in the model of the material. The most commonly studied class of Meyer sets arises in the well known cut-and-project scheme. For cut-and-project sets with compact acceptance window we study a finite set of the Meyer property. This task can be transformed into the problem of covering of the difference set by open copies. The cardinality fof the minimal covering is called the Meyer number. We show that f is bounded on the space of convex compact sets. We give estimates on the universal upper bound of the Meyer number of dimension 2 and 3. We determine the values of f for some special types of dimension 2. We further show that f is not bounded if we relax the condition of convexity.

PACS: 61.44.Br, 02.40.Ft, 02.40.-k
Keywords: quasicrystals, cut-and-project sets, Meyer sets, covering problem
File size: 153 KB

Special theory of relativity and conventionality

V.S. Barashenkov
Laboratory of Information Technologies, Joint Institute of Nuclear Research, 141980 Dubna, Russia
E.Kapuscik and D. Wcislo
Department of Physics, University of Lodz, ul. Pomorska 149/153, PL 90236 Lodz, Poland
Department of Theoretical Astrophysics, H.Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, ul. Eliasza-Radzikowskiego 152, PL 31342 Krakow, Poland

The general properties of the clock synchronization in Special Theory of Relativity with different "one way" velocities of light are discussed. It is argued that the customary irreducible element of conventionality of the synchronization problem may be eliminated.

PACS: 03.30.+p
Keywords: special theory of relativity, conventionality
File size: 74 KB

Little group kinematics associated with classical optics

S. Baskal
Department of Physics, Middle East Technical University, 06531 Ankara, Turkey
Y.S. Kim
Department of Physics, University of Maryland, College Park, Maryland 20742, U.S.A.

Little groups are the subgroups of the Poincare group whose transformations leave the four-momentum of a relativistic particle invariant. Massless particle representations can be obtained from their massive counterparts through a contraction procedure. The two by two matrix representations of little groups as well as of other kinematical effects of special relativity, like Wigner rotations are observed to coincide with the matrix formulations of some interesting classical ray optics phenomena. Examples include beam cycles in laser cavities, image focusing in a one-lens-camera, multilens optics and interferometers. Thus, it is argued that optical implementations can be exploited as analogue processors for special relativity.

PACS: 11.30.Cp, 42.79.Bh
Keywords: Wigner's little groups, classical optics
File size: 108 KB

Nonlinear algebraic structures

N. Belalouiy, L. Khodjaz and H. Bennacer
LPMPS, Departement de Physique, Faculte des Sciences, Universite Mentouri Constantine, Constantine, Algeria

We paraquantize the bosonic (resp. the Neuveu Shwarz spinning) string theory. Unlike the Ardalan and Mansouri work, the paraquantum system is so that both the center of mass variables and the excitation modes of the string verify paracommutation relations. We find existence possibilities of parabosonic (resp. paraspinning) strings defined in a noncommutative space-time at space-time dimensions other than D = 26 (resp. D = 10). We investigate then the existence possibilities of the. D = 3, 4, 6 parasuperstring. The two cases, parabose-parafermi (resp. bose-parafermi) superstrings are considered. In the first one, the spectrum is discussed through the partition functions for D = 3, 4, 6. Despite of the parastatistical algebraic structure of the dynamical variables, the combined set of the generators of the symmetries forms the algebra of the Super Symmetric Quantum Mechanic (resp. the ParaSSQM in the sense of Beckers and Debergh).

PACS: 11.30.Cp , 11.25.Hf
Keywords: parastring, Poincar parasuperalgebra, critical dimentions, noncommutativity
File size: 157 KB

Chaining spins from (super)Yang-Mills

S. Bellucci, P.Y. Casteill, J.F. Morales, C. Sochichiu
Laboratori Nazionali di Frascati, Via E. Fermi 40, 00044 Frascati, Italy

We review the spin bit model describing anomalous dimensions of the operators of Super Yang-Mills theory. We concentrate here on the scalar sector. In the limit of large N this model coincides with integrable spin chain while at finite N it has nontrivial chain splitting and joining interaction.

File size: 108 KB

2k-dimensional N = 8 supersymmetric quantum mechanics

S. Bellucci
INFN-Laboratori Nazionali di Frascati, C.P. 13, 00044 Frascati, Italy
S. Krivonos, A. Shcherbakov
Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia
A. Nersessian
Yerevan State University and Yerevan Physics Institute, Yerevan, Armenia, Artsakh State University, Stepanakert, Nagorny Karabakh, Armenia

We demonstrate that two-dimensional N = 8 supersymmetric quantum mechanics which inherits the most interesting properties of N = 2; d = 4 SYM can be constructed if the reduction to one dimension is performed in terms of the basic object - N = 2; d = 4 vector multiplet. In such a reduction only complex scalar fields from the N = 2; d = 4 vector multiplet become physical bosons in d = 1, while the rest of the bosonic components are reduced to auxiliary fields thus giving rise to (2, 8, 6) supermultiplet in d = 1. We construct the most general action for this supermultiplet with all possible FI terms included and explicitly demonstrate that the action possesses duality symmetry extended to the fermionic sector of theory. To deal with the second-class constraints presented in the system, we introduce the Dirac brackets for the canonical variables and find supercharges and Hamiltonian which form the N = 8 super Poincare algebra with central charges. Finally, we explicitly present the generalization of the two-dimensional N = 8 SQM to the 2k-dimensional case with a special Kahler geometry in the target space.

PACS: 11.25.Hf, 11.30.Pb
Keywords: supersymmetry, quantum mechanics, special Kahler geometry
File size: 158 KB

The representation theory of the Heisenberg group and beyond

Alastair Brodlie
School of Mathematics, University of Leeds

In this paper we present some recent and new developments in the theory of p- mechanics. p-Mechanics is a consistent physical theory which contains both classical and quantum mechanics. The Heisenberg group and its representation theory is the basis of p-mechanics. We give a summary of recent results on p-mechanical observables, states and canonical transformations. In doing so we exhibit relations between the quantum and classical image of these objects. We also present some new work on the Kepler/Coulomb problem. This involves constructing a new Hilbert space which represents the dynamics of the Kepler/Coulomb problem in a simple form.

PACS: 03.65.Ca, 03.65.Db.
Keywords: Heisenberg group, quantum mechanics, coherent states, p-mechanics, canonical transformations, Kepler/Coulomb problem
File size: 184 KB

Algebraic solutions for Schrödinger equations with time-varying potentials and time-dependent boundary conditions

B.L. Burrows
Staffordshire University Stafford UK
M. Cohen
The Hebrew University Jerusalem Israel

Lie algebraic methods, which have been used widely for stationary states of quantum mechanical systems are extended here to treat time–dependent problems. Difficulties may arise at points where the potential is discontinuous or has discontinuous derivatives and from certain imposed boundary conditions. The simplicity and elegance of the usual algebraic methods can be retained for such problems by redefining the domain of the operators using techniques developed by Lighthill to introduce generalized functions. We treat a model double–well subject to time–varying external fields as well as prob­lems with time–dependent boundary conditions.

PACS: 81.R.15
Keywords: algebraic solutions, time–dependent
File size: 104 KB

Exact plasma equilibria from symmetries and transformations of MHD and CGL equilibrium equations

Alexei F. Cheviakov
Department of Mathematics, Queen's University at Kingston, ON, Canada K7L 3N6

Exact isotropic and anisotropic plasma equilibria are constructed as solutions to nonlinear 3D Magnetohydrodynamic (MHD) and anisotropic Chew-Goldberger-Low (CGL) plasma equilibrium equations, using the representation of equilibrium equations in coordinates connected with magnetic surfaces. Infinite-dimensional symmetries of MHD and CGL equilibrium equations used in this construction are discussed from the prospective of Lie group analysis. The infinite-parameter set of transformations between MHD and CGL equilibrium systems is employed to produce families of anisotropic (CGL) equilibria from particular isotropic (MHD) ones. Solutions produced with the presented method are generally fully 3D solutions with no geometrical symmetries; they have different topologies and physical properties, and can serve as models of astrophysical phenomena.

PACS: 52.30.Cv, 05.45.-a, 02.30.Jr, 02.90.+p.
Keywords: MHD, plasma equilibria, Lie group of symmetries, exact solutions
File size: 333 KB

Stationary axially symmetric gravitation fields in Einstein theory

E. Chubaryan and H. Abazyan
Department of Physics, Yerevan State University 1 Alex Manoogian St., 375049 Yerevan, Armenia

In many recent astrophysical applications of the theory of dense matter it is necessary to investigate the properties of rapidly rotating compact objects within general relativity theory. The reason of this development is the hope that changes in the internal structure of the dense matter, e.g. during phase transitions, could have observable consequences for the dynamics of the rotational behavior of these objects. Particular examples are the observations of glitches and postglitch relaxation in pulsars, which are discussed as signals for superfluidity in nuclear matter and the suggestion that the braking index is remarkably enhanced when a quark matter core occurs in the center of a pulsar during its spin–down evolution. Further constraints for the nuclear equation of state come from the observation of quasi–periodic brightness oscillations (QPO’s) in low–mass–X–ray binaries, which entail mass and radius limits for rapidly rotating neutron stars.

File size: 191 KB

On the self-similarities of the rhombic Penrose tilings

Nicolae Cotfas
Faculty of Physics, University of Bucharest PO Box 76 - 54, Postal Office 76, Bucharest, Romania

We prove that the original Penrose tilings of the plane admit an infinite number of independent scaling factors and an infinite number of in ation centers. Our results are based on the definition of these tilings in terms of strip projection method proposed by Katz and Duneau shortly after the discovery of quasicrystals.

PACS: 61.44.Br
Keywords: Penrose tiling, self-similarity, scaling factor, in ation center, quasicrystal
File size: 96 KB

Extension of Moyal-deformed hierarchies of soliton equations

Aristophanes Dimakis
Department of Financial and Management Engineering University of the Aegean, 31 Fostini Str., GR–82100 Chios, Greece
Folkert Müller-Hoissen
Max-Planck-Institut für Strömungsforschung Bunsenstrasse 10, D–37073 Göttingen, Germany

Moyal–deformed hierarchies of soliton equations can be extended to larger hierarchies by including additional evolution equations with respect to the deformation parameters. A general framework is presented in which the extension is universally determined and which applies to several deformed hierarchies, including the noncommutative KP, modified KP, and Toda lattice hierarchy. We prove a Birkhoff factorization relation for the extended ncKP and ncmKP hierarchies. Also reductions of the latter hierarchies are briefly discussed. Furthermore, some results concerning the extended ncKP hierarchy are recalled from previous work.

PACS: 02.30.Ik,05.45.Yv
Keywords: deformation, factorization, hierarchy, KP, Moyal, soliton, star–product, Toda lattice
File size: 286 KB

Characters of D = 4 conformal supersymmetry

V.K. Dobrev
Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria

We give character formulae for the positive energy unitary irreducible representations of the N-extended D = 4 conformal superalgebras su(2; 2=N). Using these we also derive decompositions of long superfields as they descend to the unitarity threshold. These results are also applicable to irreps of the complex Lie superalgebras sl(4=N).

PACS: 11.30.Pb,11.25.Hf
Keywords: supersymmetry, conformal, characters
File size: 211 KB

Non-Hermitian Hamiltonians and supersymmetric quantum mechanics

Quentin Duret and Francois Gieres
Institut de Physique Nucleaire de Lyon, Universite Lyon 1, Bat. Paul Dirac 4 rue Enrico Fermi, F-69622 Villeurbanne Cedex

We review non-Hermitian Hamiltonians following Mostafazadeh, while expanding on the underlying mathematical details. To conclude, we shortly summarize pseudo-supersymmetric quantum mechanics.

PACS: 03.65.Ca, 03.65.Fd
Keywords: Non-Hermitian operators with real spectrum, PT-symmetry, supersymmetric quantum mechanics
File size: 210 KB

Solutions of the Camassa-Holm hierarchy in 2+1 dimensions

P.G. Estevez
Area de Fisica Teorica. Universidad de Salamanca, Salamanca 37008, Spain
J. Prada
Departamento de Matematicas. Universidad de Salamanca, Salamanca 37008, Spain

We consider solutions of a generalization of the Camassa-Holm hierarchy to (2+1) dimensions that include, in particular, the well-known multipeakons solutions for the celebrated Camassa-Holm equation.

PACS: 02.30.Jr, 02.30.Ik
Keywords: Camassa-Holm, peakons, shallow-water
File size: 116 KB

On supersymmetric Q-balls

Andrzej M. Frydryszak
Institute of Theoretical Physics, University of Wroclaw, pl. M. Borna 9, 50-204 Wroclaw, Poland
Michael A. Knyazev
Department of Informative Technologies and Robototechnics, Belarussian National Technical University, Scarina Av. 65, 220013 Minsk, Belarus

We present some facts related to the charged nontopological solutions of nonlinear field equations known as Q-balls. Using simplified field equations from the bosonic sector of the supersymmetric model we discuss an approximate solution with the spherical symmetry.

PACS: 11.30.Pb, 05.45.Yu
Keywords: solitons, supersymmetry, nontopological solutions
File size: 108 KB

Space groups for aperiodic crystals

Jean-Pierre Gazeau and Avi Elkharrat
Laboratoire de Physique Theorique de la Matiere Condensee, Boite 7020, Universite Paris 7-Denis Diderot, 75251 Paris Cedex 05
Christiane Frougny
Laboratoire d'Informatique Algorithmique: Fondements et Applications, UMR 7089 CNRS, Boite 7014, Universite Paris 7-Denis Diderot, 75251 Paris Cedex 05, and Universite Paris 8
Jean-Louis Verger-Gaugry
Institut Fourier, UMR 5582 CNRS, Universite Grenoble I, BP 74, 38402 Saint-Martin d'Heres

We report on the existence of symmetry plane-groups for quasiperiodic point-sets named beta-lattices. Like lattices are vector superpositions of integers, beta-lattices are vector superpositions of beta-integers. When beta > 1 is a quadratic Pisot-Vijayaraghavan (PV) algebraic unit, the set of beta-integers can be equipped with an abelian group structure and an internal multiplicative law. When beta is equal to special values, we show that these arithmetic and algebraic structures lead to freely generated symmetry plane- groups for beta-lattices. These plane-groups are based on repetitions of discrete adapted rotations and translations we shall refer to as beta-rotations and beta-translations. Hence beta-lattices, endowed with beta-rotations and beta-translations, can be viewed like lattices. We also show that, at large distances, beta-lattices and their symmetries behave asymptotically like lattices and lattice symmetries respectively.

PACS: 02.20.-a,61.44.-n
Keywords: beta-lattices, Pisot numbers, quasicrystals, tilings, plane groups
File size: 278 KB

Surfaces in su(N) algebra via sigma models on Minkowski space

A.M. Grundland
Centre de Recherches Mathematiques, Universite de Montreal C. P. 6128, Succ. Centre-ville, Montreal, (QC) H3C 3J7, Canada, Universite du Quebec, Trois-Rivieres CP500 (QC) G9A 5H7, Canada
L. Snobl
Centre de Recherches Mathematiques, Universite de Montreal C. P. 6128, Succ. Centre-ville, Montreal, (QC) H3C 3J7, Canada, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University Brehova 7, 115 19 Prague 1, Czech Republic

We review our recent results concerning two-dimensional smooth orientable surfaces immersed in su(N) Lie algebras. These are derived from the sigma model defined on Minkowski space. The structural equations of such surfaces expressed in terms of any regular solution of the model are found. This is carried out using a moving frame adapted to the surface. A procedure for construction of such surfaces is proposed and illustrated by several examples obtained from the model in dim = 1.

PACS: 02.40.Hw, 02.20.Qs
Keywords: sigma models, structural equations of surfaces, integrable systems, Lie algebras
File size: 340 KB

Quantum (anti)de Sitter algebras and generalizations of the kappa-Minkowski space

Francisco J. Herranz, Angel Ballesteros
Departamento de Fisica, Universidad de Burgos Avda. Cantabria s.n., 09006 Burgos, Spain
N. Rossano Bruno
Dipartimento di Fisica, Universita di Roma Tre and INFN Sez. Roma Tre Via Vasca Navale 84, 00146 Roma, Italy

We present two different quantum deformations for the (anti)de Sitter algebras and groups. The former is a non-standard (triangular) deformation of SO(4; 2) realized as the conformal group of the (3+1)D Minkowskian spacetime, while the latter is a standard (quasitriangular) deformation of both SO(2; 2) and SO(3; 1) expressed as the kinematical groups of the (2+1)D anti-de Sitter and de Sitter spacetimes, respectively. The Hopf structure of the quantum algebra and a study of the dual quantum group are presented for each deformation. These results enable us to propose new non-commutative spacetimes that can be interpreted as generalizations of the kappa-Minkowski space, either by considering a variable deformation parameter (depending on the boost coordinates) in the conformal deformation, or by introducing an explicit curvature/cosmological constant in the kinematical one; kappa-Minkowski turns out to be the common first-order structure for all of these quantum spaces. Some properties provided by these deformations, such as dimensions of the deformation parameter (related with the Planck length), space isotropy, deformed boost transformations, etc., are also commented.

PACS: 02.20.Uw, 11.30.-j, 04.60.-m
Keywords: quantum algebras, deformation, Minkowski, anti-de Sitter, Poincare, noncommutative spacetime
File size: 176 KB

Quasi-exact solvability of the Dirac equations

Choon-Lin Ho
Department of Physics, Tamkang University, Tamsui 25137, Taiwan, R.O.C.

We present a general procedure for determining quasi–exact solvability of the Dirac and the Pauli equation with an underlying sl(2) symmetry. This procedure makes full use of the close connection between quasi–exactly solvable systems and supersymmetry.

PACS: 03.65.-w, 03.65.Pm
Keywords: quasi–exact solvability, Dirac equations
File size: 113 KB

A product formula and combinatorial field theory

A. Horzela, P. Blasiak
H. Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, ul. Eliasza-Radzikowskiego 152, PL 31342 Krakow, Poland
Laboratoire de Physique Theorique des Liquides, Universite Pierre et Marie Curie, Tour 24 - 2e et., 4 Pl.Jussieu, F 75252 Paris Cedex 05, France

G. E. H. Duchamp
Universite de Rouen, LIFAR, F 76821 Mont-Saint Aignan Cedex, France
K. A. Penson and A. I. Solomon
Laboratoire de Physique Theorique des Liquides, Universite Pierre et Marie Curie, Tour 24 - 2e et., 4 Pl.Jussieu, F 75252 Paris Cedex 05, France
The Open University, Physics and Astronomy Department, Milton Keynes MK7 6AA, United Kingdom

We treat the problem of normally ordering expressions involving the standard boson operators a, a+ where [a, a+] = 1. We show that a simple product formula for formal power series - essentially an extension of the Taylor expansion - leads to a double exponential formula which enables a powerful graphical description of the generating functions of the combinatorial sequences associated with such functions - in essence, a combinatorial field theory. We apply these techniques to some examples related to specific physical Hamiltonians.

PACS: 03.65.Fd, 05.30.Jp,
Keywords: boson normal ordering, combinatorics
File size: 422 KB

Symmetries and graded contractions of the Pauli graded sl(3, C)

Jiri Hrivnak, Petr Novotny
Department of Physics, Faculty of Nuclear sciences and Physical Engineering, Czech Technical University, Brehova 7, 115 19 Prague 1, Czech Republic

Presented results were achieved in collaboration with Miloslav Havlicek, Jiri Patera and Jiri Tolar. We consider the Pauli grading of the Lie algebra sl(3, C) and use a concept of graded contractions to construct non-isomorphic Lie algebras of dimension 8, while preserving the Pauli grading. We show how the symmetry group of a grading simplifies the solution of contraction equations and identification of results. We give examples of resulting Lie algebras. Complete results will be published elsewhere.

PACS: 02.20.Sv
Keywords: Lie algebra, grading
File size: 172 KB

Recoupling theory of many-body quantum theory

William P. Joyce
Department of Physics and Astronomy, University of Canterbury Private Bag 4800, New Zealand

In this paper we sketch the foundations of recoupling theory. Introduction of an indistinguishability principle leads to Pauli Exclusion and confinement. We discuss its application to SU(3) colour.

PACS: 02.10.Ws 12.38.Aw
Keywords: recoupling, monoidal category, Pauli exclusion confinement, quarks
File size: 138 KB

On a group-theoretical approach to the periodic table of chemical elements

Maurice R. Kibler
Institut de Physique Nucleaire de Lyon, IN2P3-CNRS et Universite Claude Bernard 43 Bd du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France

This paper is concerned with the application of the group SO(4; 2) SU(2) to the periodic table of chemical elements. It is shown how the Madelung rule of the atomic shell model can be used for setting up a periodic table that can be further rationalized via the group SO(4; 2) SU(2) and some of its subgroups. Qualitative results are obtained from the table and the general lines of a programme for a quantitative approach to the properties of chemical elements are developed on the basis of the group SO(4; 2) SU(2).

PACS: 03.65.Fd, 31.15.Hz
Keywords: hydrogen-like atom, harmonic oscillator, invariance and non-invariance groups, Lie algebra under constraints, Madelung rule, periodic table
File size: 300 KB

Quantum entanglement and dynamical symmetries

Alexander A. Klyachko and Alexander S. Shumovsky
Faculty of Science, Bilkent University, Bilkent, Ankara, 06800, Turkey

Definition of maximum entanglement in terms of a novel variational principle for quantum uctuations and its corollaries are discussed.

PACS: 03.65.Ud, 03.67.Mn
Keywords: entanglement, quantum fluctuations, quantum measurements
File size: 133 KB

SU(1,1) algebra and interacting families of Calogero particles

Marijan Milekovic
Physics Department, Faculty of Science, Bijenicka c. 32, 10002 Zagreb, Croatia
Stjepan Meljanac, Andjelo Samsarov, Marko Stojic
Rudjer Boskovic Institute, Bijenicka c. 54, 10002 Zagreb, Croatia

A one-dimensional model with interacting families of Calogero-type particles is studied. It includes harmonic, two-body and three-body interactions among particles. We find the exact eigenenergies corresponding to a class of the exact eigenstates of the model. We emphasize the universal SU(1; 1) structure of the model. We show how SU(1; 1) generators for the whole system are composed of SU(1; 1) generators of arbitrary subsystems. By imposing the conditions for the absence of the three-body interaction, we find certain relations between the coupling constants.

PACS: 03.65.Fd; 03.65.Sq
Keywords: multispecies Calogero model, SU(1; 1) symmetry, interacting families
File size: 117 KB

Newton-Wigner postulates and commutativity of position operators

R.M. Mir-Kasimov
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research 141980, Dubna, Russia and
Department of Mathematics, Izmir Institute of Tehnology 35430, Urla/Izmir, Turkey

Commutativity of the position operator components is one of the Newton-Wigner postulates for the localized states tacitly included in the list. Omitting it gives additional possibilities to use the Poincare group representations for the analysis of the concept of the relativistic quantum localized states.

PACS: 02.20.Uw, 02.20.-a
Keywords: localization, noncommutative, relativistic
File size: 160 KB

Method of categorical extension of Cayley-Klein groups

S.S. Moskaliuk
Bogolyubov Institute for Theoretical Physics, 14b Metrolohichna st., Kyiv, Ukraine

The method of categorical extension of the Cayley-Klein groups is developed. The method uses the Cayley-Klein spaces, as objects of the Cayley-Klein category, endowed with all possible linear relations or bilinear forms as morphisms.

PACS: 02.20.Qs
Keywords: group theory, category theory
File size: 140 KB

Non-trivial extension of the Poincare algebra for antisymmetric gauge fields

G. Moultaka
Laboratoire de Physique Mathematique et Theorique, CNRS UMR 5825, Universite Montpellier II, Place E. Bataillon, 34095 Montpellier, France
M. Rausch de Traubenberg and A. Tanasak
Laboratoire de Physique Theorique, CNRS UMR 7085, Universite Louis Pasteur 3 rue de l'Universite, 67084 Strasbourg, France

We investigate a non-trivial extension of the D-dimensional Poincare algebra. Matrix representations are obtained. The bosonic multiplets contain antisymmetric tensor fields. It turns out that this symmetry acts in a natural geometric way on these p-forms. Some field theoretical aspects of this symmetry are studied and invariant Lagrangians are explicitly given.

PACS: 03.50.Kk, 03.65.Fd,11.10.Kk,11.30.Ly
Keywords: algebraic methods, extension of the Poincare algebra, p-forms, field theory
File size: 159 KB

On conservation laws for the potential Zabolotskaya-Khokhlov equation

V. Rosenhaus
Department of Mathematics and Statistics, California State University, Chico, CA 95929, USA

We study local conservation laws for the potential Zabolotskaya-Khokhlov equation in three-dimensional case. We analyze an infinite Lie point symmetry group of the equation, and generate a finite number of conserved quantities corresponding to infinite symmetries through appropriate boundary conditions.

PACS: 11.30.-j, 02.20.Tw
Keywords: infinite symmetries, conservation laws
File size: 104 KB

Some features about toric quaternionic Kahler geometry in D = 4

O.P. Santillan
Bogoliubov Laboratory of Theoretical Physics, JINR 141 980 Dubna, Moscow Reg., Russia
A.G. Zorin
Faculty of Physics, MSU, Vorobjovy Gory, Moscow, 119899, Russia

We explain some features about toric self dual structures and toric quaternionic Kahler manifolds in four dimensions. Applications are outlined.

Keywords: Toda equation, Einstein-Weyl structures
File size: 100 KB

Lorentz-like formulation of Galilean field theories

Esdras Santos
Department of Physics, University of Alberta, Edmonton, Alberta, Canada, T6G 2J1
Faqir Khanna
Department of Physics, University of Alberta, Edmonton, Alberta, Canada, T6G 2J1, TRIUMF, 4004, Wesbrook Mall, Vancouver, British Columbia, Canada, V6T 2A3
Marc de Montigny
Faculte Saint-Jean, University of Alberta, Edmonton, Alberta, Canada, T6C 4G9
Department of Physics, University of Alberta, Edmonton, Alberta, Canada, T6G 2J1

We construct non-relativistic Lagrangian field models by enforcing Galilean covariance with a (4; 1) Minkowski manifold followed by a projection onto the (3; 1) Newtonian space- time. We discuss scalar, Fermi and gauge fields, as well as interactions between some of these fields. The Galilean covariant formalism provides an elegant construction of the Lagrangians which describe the electric and magnetic limits of Galilean electromagnetism. As further examples of scalar fields, we discuss various models of fluids and super fluids. Then, we turn to linear wave equations, and consider the Dirac Lagrangian which allows one to retrieve the Levy-Leblond wave equations. We examine the situation where the Fermi field interacts with an abelian gauge field. Finally, we study the Bhabha equations for spins 0 and 1.

PACS: 03.50.De; 03.65.Pm; 47.10.+g
Keywords: Galilean invariance, wave equations, electromagnetism
File size: 159 KB

Saturation and non-linear effects in diffractive processes

O.V. Selyugin and J.R. Cudell
Institut de Physique, Bat. B5a, Universite de Liege 4000 Liege, Belgique

Through a direct implementation of the saturation regime resulting from the unitarity limit in the impact parameter representation, we explore various possibilities for the energy dependence of hadronic scattering. We show that it is possible to obtain a good description of the scattering amplitude from a hard pomeron provided one includes non-linear effects

PACS: 62.20
Keywords: diffraction, unitarity, non-linear equations, total cross sections
File size: 151 KB

The evolution of solutions of plane ideal plasticity

S.I. Senashov
Siberian State Aero-Cosmic University, Krasnoyarsk, 660014 Russia
A. Yakhno
CUCEI, Universidad de Guadalajara, Guadalajara, Mexico

It's well known, that the symmetries of a system of differential equations allow transforming its solutions to solutions of this system. Using this property, from two known solutions of the theory of plasticity: the solution of Nadai for circular cavity stressed by normal and shear pressure, and Prandtl's solution for a block compressed between perfectly rough plates, there were constructed new analytical exact solutions of the system of two-dimensional ideal plasticity equations.

PACS: 46.35+z; 02.20-a
Keywords: plasticity, symmetry analysis, Lie group and algebra methods, exact solutions of differential equations
File size: 93 KB

Partition functions and graphs: A combinatorial approach

A. I. Solomon
The Open University, Physics and Astronomy Department Milton Keynes MK7 6AA, United Kingdom
Laboratoire de Physique Theorique des Liquides, Universite Pierre et Marie Curie Tour 24 - 2e et., 4 Pl.Jussieu, F 75252 Paris Cedex 05, France

P. Blasiak
H.Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences ul. Eliasza-Radzikowskiego 152, PL 31342 Krakow, Poland
Laboratoire de Physique Theorique des Liquides, Universite Pierre et Marie Curie Tour 24 - 2e et., 4 Pl.Jussieu, F 75252 Paris Cedex 05, France

G. E. H. Duchamp,
Universite de Rouen, LIFAR, F 76821 Mont-Saint Aignan Cedex, France
A. Horzela
H.Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences ul. Eliasza-Radzikowskiego 152, PL 31342 Krakow, Poland
K. A. Penson
Laboratoire de Physique Theorique des Liquides, Universite Pierre & Marie Curie Tour 24 - 2e et., 4 Pl.Jussieu, F 75252 Paris Cedex 05, France

Although symmetry methods and analysis are a necessary ingredient in every physicist's toolkit, rather less use has been made of combinatorial methods. One exception is in the realm of Statistical Physics, where the calculation of the partition function, for example, is essentially a combinatorial problem. In this talk we shall show that one approach is via the normal ordering of the second quantized operators appearing in the partition function. This in turn leads to a combinatorial graphical description, giving essentially Feynman-type graphs associated with the theory. We illustrate this methodology by the explicit calculation of two model examples, the free boson gas and a super uid boson model. We show how the calculation of partition functions can be facilitated by knowledge of the combinatorics of the boson normal ordering problem; this naturally gives rise to the Bell numbers of combinatorics. The associated graphical representation of these numbers gives a perturbation expansion in terms of a sequence of graphs analogous to zero-dimensional Feynman diagrams.

PACS: 03.65.Fd, 05.30.Jp,
Keywords: boson normal ordering, combinatorics
File size: 211 KB

On systems of diffusion equations

C. Sophocleous
Department of Mathematics and Statistics, University of Cyprus, CY 1678 Nicosia, Cyprus
R.J. Wiltshire
The Division of Mathematics and Statistics, The University of Glamorgan, Pontypridd CF37 1DL, Great Britain

We consider systems of diffusion equations that have considerable interest in Soil Science and Mathematical Biology. We construct non-local symmetries, known as potential symmetries. Furthermore we present linearizing mappings.

PACS: 02.30.Jr; 02.20.Tw
Keywords: diffusion equations, potential symmetries, linearizing mappings
File size: 126 KB

Deformed solitons: The case of two coupled scalar fields

A. de Souza Dutra
UNESP-Campus de Guaratinguet´a-DFQ, Av. Dr. Ariberto Pereira da Cunha, 333 C.P. 205, 12516-410 Guaratinguet´a SP Brasil

In this work, we present a general procedure, which is able to generate new exact solitonic models in 1+1 dimensions, from a known one, consisting of two coupled scalar fields. An interesting consequence of the method, is that of the appearing of nontrivial extensions, where the deformed systems presents other BPS solitons than that appearing in the original model. Finally we take a particular example, in order to check the above mentioned features.

PACS: 11.27.+d, 11.30.Er
Keywords: solitons, deformations
File size: 108 KB

A division algebra classification of generalized supersymmetries

Francesco Toppan
CBPF, CCP, Rua Dr. Xavier Sigaud 150, cep 22290–180, Rio de Janeiro, Brazil

Generalized supersymmetries admitting bosonic tensorial central charges are classi- fied in accordance with their division algebra properties. Division algebra consistent constraints lead (in the complex and quaternionic cases) to the classes of hermitian and holomorphic generalized supersymmetries. Applications to the analytic continuation of the M–algebra to the Euclidean and the systematic investigation of certain classes of models in generic space–times are briefly mentioned.

PACS: 11.30.Pb
Keywords: supersymmetry, M–theory
File size: 225 KB

Finiteness of generalized Chern-Simons thoeries

Institute for Theoretical and Experimental Physics, 117259, Moscow, Russia
Institute for Theoretical Physics, 03143, Kyiv, Ukraine and
Physics Department, T.Shevchenko Kyiv State University, 01003, Kyiv, Ukraine

We study the perturbation theory for the example of a topological Batalin-Vilkovisky theory and show that it is free from the UV divergences. In fact, the property of finiteness can be generalized for some class of theories, which we describe here. 3-dimensional Chern-Simons and 2-dimensional topological Yang-Mills theories belong to this class.

PACS: 00.00.xy
Keywords: BV formalism, topological theory, Chern-Simons, BF
File size: 98 KB

Similarity solutions of an equation describing ice sheet dynamics

R.J. Wiltshire
The Division of Mathematics and Statistics, The University of Glamorgan Pontypridd CF37 1DL, Great Britain

This paper focus's upon the derivation of the similarity solutions of a free boundary problem arising in glaciology. With reference to shallow ice sheet ow we present a potential symmetry analysis of the second order non-linear degenerate parabolic equation that describe non-Newtonian ice sheet dynamics in the isothermal case. A full classical and also a non-classical symmetry analysis is presented. A particular example of a similarity solution to a problem formulated with Cauchy boundary conditions is described. This demonstrates the existence of a free moving boundary and also an accumulation-ablation function with realistic physical properties.

PACS: 02.30.Jr,02.30.Gp,83.10.Bb,47.50+d
Keywords: non-linear degenerate equations, ice ow dynamics, potential symmetries
File size: 142 KB

Space-group approach to the wavefunction of a Cooper pair. Application to unconventional superconductors

V.G. Yarzhemsky
Institute of General and Inorganic Chemistry of RAS

Zero-total momentum two-electron wavefunction in crystal space-groups are constructed making use of induced representation method and projection operator technique. Theory is applied to analysis of nodal structure of Cooper pairs in unconventional supercondectors. Theoretical results are compared with experimental data.

PACS: 03.65.Fd,74.70.Tx,74.72.-h
Keywords: Space groups, superconducting order parameter, strongly correlated electronic systems, unconventional superconductors
File size: 154 KB

PT-symmetry, ghosts, supersymmetry and Klein-Gordon equation

Miloslav Znojil
Nuclear Physics Institute, 250 68 Rez, Czech Republic

Parallels between the concepts of symmetry, supersymmetry and (recently introduced) PT -symmetry are reviewed and discussed, with particular emphasis on the new insight in quantum mechanics which is rendered possible by their combined use.

PACS: 03.65.Ge
Keywords: supersymmetric quantum mechanics, parity times time reversal symmetry, non-Hermitian Hamiltonians with real spectra, pseudo-metrics in Hilbert space, factorization method, Klein-Gordon equation
File size: 122 KB