Participants

The following provisional list displays the names of those who have already shown their interest to take part in S-3:

Víctor Aldaya
Instituto de Astrofísica de Andalucía (Spain)
Talk: Symmetries from the Solution Manifold

Abstract: In this talk, extended symmetries (non-point symmetries) of a physical system are used to characterize the corresponding solution manifold by means of Noether invariants. This is the starting point to the correct quantization in non-linear cases, where the success of Canonical Quantization is not guaranteed. The use of the Poincaré-Cartan form permits finding both the symplectic structure on the solution manifold, through the Hamilton-Jacobi transformation, and the required symmetries, realized as Hamiltonian vector fields, associated with functions on the solution manifold (thus constituting an inverse of the Noether Theorem) lifted back to the evolution space through the inverse of this Hamilton-Jacobi mapping.
We present simple non-trivial examples where the symmetries are found in a perturbative way.
Juan José Álvarez
Universidad de Valladolid (Spain)
Óscar Arratia
Universidad de Valladolid (Spain)
Engin Aslar
Ankara University (Turkey)
Ángel Ballesteros
Universidad de Burgos (Spain)
Arno Bohm
The University of Texas at Austin (USA)
Talk: Time asymmetric quantum mechanics

Abstract: The meaning of time asymmetry in quantum physics is discussed. On the basis of a mathematical theorem, the Stone-von Neumann theorem, the solutions of the dynamical equations, the Schrödinger equation for states or the Heisenberg equation for observables are given by a unitary group. Dirac kets require the concept of a RHS (rigged Hilbert space) of Schwartz functions; for this kind of RHS a mathematical theorem also leads to time symmetric group evolution. Scattering theory suggests to distinguish mathematically between states (defined by a preparation apparatus) and observables (defined by a registration apparatus (detector)). If one requires that scattering resonances of width $\Gamma$ and exponentially decaying states of lifetime $\tau =\frac{\hbar}{\Gamma}$ should be the same physical entities (for which there is sufficient evidence) one is led to a pair of RHS's of Hardy functions and connected with it, to a semigroup time evolution $t_0 \le ^T t < \infty$, with the puzzling result that there is a quantum mechanical beginning of time, just like the big bang time for the universe, when it was a quantum system. The decay of quasi-stable particles is used to illustrate this quantum mechanical time asymmetry. From the analysis of these processes, we show that the properties of rigged Hilbert spaces of Hardy functions are suitable for a formulation of time asymmetry in quantum mechanics.
Luis J. Boya
Universidad de Zaragoza (Spain)
Juan Antonio Calzada
Universidad de Valladolid (Spain)
Rutwig Campoamor-Stursberg
UCM (Spain)
José F. Cariñena
Universidad de Zaragoza (Spain)
Talk: A generalized approach to integrability by quadratures

Abstract: The classical result of Lie on integrability by quadratures will be reviewed and some generalizations will be proposed. After a short review of the classical Lie theorem it will be shown that if we are able to construct in an iterative way a nested sequence of subalgebras L_i of vector fields including the dynamical vector field and after some steps the Lie subalgebra is Abelian, then the dynamics is integrable by k-1 quadratures. The theory will be illustrated with examples and an extension of the theorem where the Lie algebras are replaced by some distributions will also be presented.
Enrico Celeghini
Università di Firenze (Italy)
José María Cerveró
Universidad de Salamanca (Spain)
Dogukan Cevik
Ankara University (Turkey)
José Adolfo de Azcárraga
Universitat de València (Spain)
Marina De La Torre Mayado
Universidad de Salamanca (Spain)
Erik Diaz-Bautista
CINVESTAV-IPN (México)
David J. Fernández
CINVESTAV (México)
Talk: Penning trap in a rotating magnetic field: coherent states approach

Abstract: The quantum problem of a non-relativistic charged particle subject to the electric and magnetic fields created by an ideal Penning trap plus a magnetic field rotating around the symmetry axis is addressed. The transition to the rotating frame is performed in order to eliminate the initial time-dependence in the Hamiltonian; then a coherent states approach for the resulting time-independent problem is implemented. It is determined the parameters domain for the particle to be in the trap regime. Several physical quantities for the system being in a coherent state are also evaluated.
Federico Finkel Morgenstern
Universidad Complutense de Madrid (Spain)
Manuel Gadella
Universidad de Valladolid (Spain)
Pilar G. Estévez
Universidad de Salamanca (Spain)
Talk: 1+1 spectral problems arising from the Manakov-Santini system

Abstract: This talk deals with the spectral problem of the Manakov Santini system. The point Lie symmetries of the Lax pair have been identified. Several similarity reductions arise from these symmetries. An important benefit of our procedure is that the study of the Lax pair instead of the partial differential equations yields the reductions of the eigenfunctions and also the spectral parameter. Therefore, we have obtained five interesting spectral problems in 1+1 dimensions.
David Gómez-Ullate
Universidad Complutense (Spain)
Talk: Exceptional orthogonal polynomials

Abstract: Exceptional orthogonal polynomials are extensions of classical orthogonal polynomials in the sense that they are also eigenfunctions of a Sturm-Liouville problem although the degree sequence has a finite number of gaps (missing degrees). Equivalently, in mathematical physics they appear as eigenfunctions of rational extensions of exactly solvable potentials. It has been conjectured that all exceptional orthogonal polynomials can be obtained through Darboux-Crum transformations of their classical counterparts, and this conjecture has been proved in the Hermite case using the connection between trivial monodromy potentials and Darboux transformations. In this talk we will review some of these recent results and we will mention current open problems.
Fernando Gómez-Cubillo
Universidad de Valladolid (Spain)
Toño González
Universidad de Valladolid (Spain)
Miguel Ángel González León
Universidad de Salamanca (Spain)
Talk: On the planar Demkov wave functions

Abstract: The quantum spectral problem of diatomic molecular ions, in the Born-Oppenheimer approximation, is separable using spheroidal coordinates obtained by rotating the two-dimensional elliptic coordinates about the focal axis \cite{Wilson} \cite{Ponomarev}. Separability allows to split the Schr\"odinger equation in a system of ODEs consisting in two different Generalized Spheroidal equations for the three dimensional case, and the Razavy equation plus the Whittaker Hill equation in the two dimensional one. Demkov has searched for eigen-wave functions of the Hamiltonian corresponding to the energy levels of a hydrogenoid atom. This procedure lead, for certain values of the nuclear charges, to the construction of ``finite" wave functions, the so-called Demkov wave functions. We have already explained this behaviour in terms of the Quasi-Exact Solvability of the underlying equation, the Confluent Heun equation. We present here a complete analysis of the analogous planar Demkov wave functions.
Artemio González-López
Universidad Complutense de Madrid (Spain)
Nikolai Gromov
Syktyvkar University (Russian Federation)
Talk: Natural Limits of Electroweak Model and Contraction of its Gauge Group

Abstract: Energy is the natural and most importantt parameter in particle physics. The Electroweak Model is a gauge theory based on the group $SU(2)\times U(1)$, acting in the boson, lepton and quark sectors, which are describes by vectors of $C_2$ with different physical interpretations in different sectors. The contracted group $SU(2;j)$ and its fundamental representation space $C_2(j)$ are obtained by the consistent rescaling of $SU(2)$ and $C_2$
\[ z'(j) = \left(\begin{array}{c} j z_1' \[1.ex] z_2' \end{array}\right) = \left(\begin{array}{cc} \alpha & j \beta \[1.ex] -j \bar \beta & \bar \alpha \end{array}\right) \left(\begin{array}{c} j z_1 \[1.ex] z_2 \end{array}\right)= u(j) z(j) \]
when $j\to o$. Transformation rules of the boson felds as well as of the left lepton and quark fields are as follows: $W_\mu^\pm \to j W_\mu^\pm$, $Z_\mu \to Z_\mu$, $A_\mu \to A_\mu$, $\nu_l \to j \nu_l$, $e_l \to e_l$, $u_l \to j u_l$, $d_l \to d_l$.
The right lepton and quark fields are $SU(2)$-singlets, i.e. scalars, and therefore are not transformed. The parameter $j$ is connected with the energy $s$ in center-of-mass system $j^2(s) = g \sqrt{s}/m_W$, where $m_W$ is $W$-boson mass and $g$ is constant. So contraction $j \to 0$ corresponds to zero energy limit of the Electroweak Model.
Julio Guerrero
Universidad de Murcia (Spain)
Faruk Gungor
Istambul Technical University (Turkey)
Talk: Kac-Moody-Virasoro symmetries of variable coefficient nonlinear evolution equations in 2+1 dimensions

Abstract: In this talk I will consider variable coefficient generalizations of two well-known physical models: Kadomtsev-Petviashvili (KP) and Zabolotskaya-Khoklov (or dispersionless KP) equation and discuss how the existence of Kac-Moody-Virasoro algebras as their point symmetry algebras, when combined with equivalence transformations, can serve as an additional test for integrability.
Francisco José Herranz
Universidad de Burgos (Spain)
Véronique Hussin
Université de Montréal (Canada)
Talk: Invariant solutions of the supersymmetric ℂP^N-1 sigma model

Abstract: Constant curvature surfaces are constructed from the finite action solutions of the supersymmetric ℂP^N-1 sigma model. They are shown to be related to invariant solutions of the model. In the case of holomorphic solutions, we get an unicity theorem. Some new results are obtained for the case of non-holomorphic solutions.
Alberto Ibort
Universidad Carlos III de Madrid (Spain)
Talk: On the structure of Schwinger's measurement algebra: groups, groupoids and 2-groupoids

Abstract: The structure of Schwinger's measurement algebra will be reviewed. We would point out that it carries the structure of a 2-groupoid and the basic notions of groups, gropoids and 2-groupoids will be discussed as well as some elementary examples.
José Manuel Izquierdo
Universidad de Valladolid (Spain)
Şengül Kuru
Ankara University (Turkey)
María A. Lledó
IFIC (Universitat de València-CSIC) (Spain)
Carlos López
Universidad de Alcalá de Henares (Spain)
Talk: Vacuum

Abstract: Vacuum as a Lorentz invariant fluid (distribution of density of particles in momentum space), equivalently a relativistic ether, is the subject of this talk. In QM, the vacuum of virtual particles will be considered in a theory of hidden variables, not as a scientific proposal (beacuse of the uncertainty principle), but as a (metaphysical) analogy between some QM phenomena and classical Brownian motion. In GR, there is wide agreement that the Cosmological constant, a property of spacetime vacuum, is the source of dark energy. Although the existence of exotic particles is the main research line in the dark matter issue, quintessence theories, grounded in some new cosmological field, have the advantage of unifying dark matter and dark energy in a single phenomenon. Vacuum inhomogeneity at a cosmological scale is suggested as a candidate for quintessence; a brief qualitative analysis of a specific model will be presented.
Juan M. Mateos Guilarte
Universidad de Salamanca (Spain)
Talk: Scalar field fluctuations distorted by two pairs of δ-δ' interactions

Abstract: Scattering solutions by a double $\delta-\delta^\prime$ will be used as one-particle solutions of an scalar field quantum field theory in a $(1+1)$-dimensional Minkowskian space-time. Following the trend of mimicking the conducting plates used in a Casimir experiment set-up by means of point $\delta$ interactions I will discuss the quantum vacuum interaction between plates relying on the Lipmann-Schwinger $T$-matrix.
Vicente Said Morales Salgado
CINVESTAV-IPN (México)
Talk: Supersymmetric partners of the truncated harmonic oscillator

Abstract: Supersymmetry transformations of first and second order are used to generate Hamiltonians with known spectra departing from the harmonic oscillator with an infinite potential barrier. Certain systems obtained in a straightforward way through said procedure possess differential ladder operators of both types, third and fourth order. Since systems with this kind of operators are linked with the Painleve IV and Painleve V equations respectively, several solutions of these non-linear second-order differential equations will be simply found, along with a chain of Backlund transformations connecting such solutions.
Miguel Carlos Muñoz
Universitat Politècnica de Catalunya (Spain)
Javier Negro
Universidad de Valladolid (Spain)
Luis Miguel Nieto
Universidad de Valladolid (Spain)
José Fernando Pascual
Universidad de Valladolid (Spain)
Mikhail Plyushchay
Universidad de Santiago de Chile (Chile)
Talk: Exotic supersymmetry of reflectionless systems, and solitons

Abstract: Finite-gap and reflectionless quantum systems are characterized by exotic supersymmetry. We review the peculiarities of such supersymmetric structure appearing in reflectionless systems associated with the soliton and multi-kink-antikink solutions of the KdV and mKdV equations. We also discuss the transmutations of supersymmetry related to the soliton scattering.
Orlando Ragnisco
Università di Roma 3 (Italy)
Talk: Some new superintegrable hamiltonian systems, solvable by factorization method

Abstract: Following a recent paper by Kuru and Negro, we apply the "spectrum generating algebra" method (in the classical case) and the factorisation method (in the quantum case) to solve a pair of new hamiltonian systems, Taub-Nut and Darboux III, which are superintegrable deformations on curved spaces of the Kepler-Coulomb and Harmonic Oscillator.
Miguel Ángel Rodríguez
Universidad Complutense de Madrid (Spain)
Talk: Invariant transformations in Euclidean hyperkahler structures

Abstract: We present a detailed construction of the invariance algebra of the quaternionic structure in Euclidean spaces. Related results, as maximal subgroups of simple Lie groups and the s list of holonomy groups, will be also discussed. Berger's list of holonomy groups, will be also discussed.
Yvan Saint-Aubin
Université de Montréal (Canada)
Talk: The role of indencomposable representations in statistical physics

Abstract: The transfer matrix is a common tool in statistical physics and shares many properties with the Hamiltonian in quantum physics. However, contrarily to Hamiltonians, transfer matrices do not need to be Hermitian. They do not even need to have real eigenvalues. This distinction is of physical importance and efforts have been given to understand the representation theory of algebras arising in lattice models like, for example, the families of Temperley-Lieb algebras (TL algebras).
One telltale signature of this distinction is the existence of Jordan blocks in transfer matrices and, more generally, of indecomposable representations. I shall give a physical example of these Jordan blocks and report on the efforts to classify all indecomposable representations of the TL algebras and dilute TL algebras.
Mariano Santander
Universidad de Valladolid (Spain)
Piergiulio Tempesta
Univesidad Complutense de Madrid (Spain)
Talk: Haantjes Manifolds and Integrable Systems

Abstract: A general theory of integrable systems is proposed, based on the theory of Haantjes manifolds. We introduce the notion of symplectic-Haantjes manifold (or !H manifold), as the natural setting where the notion of integrability can be formulated. We propose an approach to the separation of variables for classical systems, related to the geometry of Haantjes manifolds. A special class of coordinates, called Darboux-Haantjes coordinates, will be constructed from the Haantjes structure associated with an integrable systems. They enable the additive separation of variables of the Hamilton-Jacobi equation. We also present an application of our approach to the study of some finite-dimensional integrable models, as the Hénon-Heiles systems and a stationary reduction of the KdV hierarchy.
Jaromir Tosiek
Technical University of Lodz (Poland)
Talk: The WKB approximation in deformation quantization

Abstract: An adaptation of the WKB approximation to the deformation quantization is presented. A relationship between the phase $\sigma(\vec{r})$ of a wave function $\exp \left(\frac{i}{\hbar} \sigma(\vec{r}) \right)$ and a respective Wigner function is derived. Formulas for a Wigner function of a wave function being a product of functions and a superposition of functions are proposed. An example of the semiclassical approximation in deformation quantization is analysed.
Miguel Ángel Velasco
CIEMAT (Spain)
Sujeev Wickramasekara
Grinnell College ( USA)
Kurt Bernardo Wolf
Universidad Nacional Autónoma de México (México)
Talk: Position and momentum in the monochromatic Maxwell fish-eye

Abstract: In geometric optics the Maxwell fish-eye is a medium where light rays follow circles, while in scalar wave optics this medium can only 'trap' fields of certain discrete frequencies. In the monochromatic case characterized by a positive integer ℓ, there are 2ℓ+1 independent fields. We identify two bases of functions that serve as wavefields of definite position and definite momentum. Their construction uses the stereographic projection of the sphere, and the identification is corroborated in the ℓ → ∞ limit to a homogeneous Helmholtz medium.

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