SYMMETRIES IN NATURE: SYMPOSIUM IN MEMORIAM MARCOS MOSHINSKY

Character expansions in physics
View Description Hide DescriptionExpanding products of invariant functions of a group element as a series in the basis of characters of the irreducible representations of a group is widely used in many areas of physics and related fields. In this contribution a formula to generate such expansions and its various applications are briefly reviewed.

Fidelity decay of the two‐level bosonic embedded ensembles of random matrices
View Description Hide DescriptionWe study the fidelity decay of the k‐body embedded ensembles of random matrices for bosons distributed over two single‐particle states. Fidelity is defined in terms of a reference Hamiltonian, which is a purely diagonal matrix consisting of a fixed one‐body term and includes the diagonal of the perturbing k‐body embedded ensemble matrix, and the perturbed Hamiltonian which includes the residual off‐diagonal elements of the k‐body interaction. This choice mimics the typical mean‐field basis used in many calculations. We study separately the cases and 3. We compute the ensemble‐averaged fidelity decay as well as the fidelity of typical members with respect to an initial random state. Average fidelity displays a revival at the Heisenberg time, and a freeze in the fidelity decay, during which periodic revivals of period are observed. We obtain the relevant scaling properties with respect to the number of bosons and the strength of the perturbation. For certain members of the ensemble, we find that the period of the revivals during the freeze of fidelity occurs at fractional times of These fractional periodic revivals are related to the dominance of specific k‐body terms in the perturbation.

Extended spin symmetry and the standard model
View Description Hide DescriptionWe review unification ideas and explain the spin‐extended model in this context. Its consideration is also motivated by the standard‐model puzzles. With the aim of constructing a common description of discrete degrees of freedom, as spin and gauge quantum numbers, the model departs from q‐bits and generalized Hilbert spaces. Physical requirements reduce the space to one that is represented by matrices. The classification of the representations is performed through Clifford algebras, with its generators associated with Lorentz and scalar symmetries. We study a reduced space with up to two spinor elements within a matrix direct product. At given dimension, the demand that Lorentz symmetry be maintained, determines the scalar symmetries, which connect to vector‐and‐chiral gauge‐interacting fields; we review the standard‐model information in each dimension. We obtain fermions and bosons, with matter fields in the fundamental representation, radiation fields in the adjoint, and scalar particles with the Higgs quantum numbers. We relate the fields’ representation in such spaces to the quantum‐field‐theory one, and the Lagrangian. The model provides a coupling‐constant definition.

Algebraic cluster model with tetrahedral symmetry
View Description Hide DescriptionWe propose an algebraic treatment of a four‐body system in terms of a U(10) spectrum generating algebra. The formalism for the case of four identical objects is developed in detail. This includes a discussion of the permutation symmetry, a study of special solutions which are shown to correspond to the harmonic oscillator, the deformed oscillator and the spherical top with tetrahedral symmetry.

Symmetry adapted states and the quantum phase transition in the Dicke model
View Description Hide DescriptionWe study the phase transition properties of the ground state of the Dicke model by means of a variational procedure together with the catastrophe formalism. The proposed variational state is the tensorial product of the SU(2) and Weyl coherent states, and the expectation value of the Dicke Hamiltonian with respect this state is calculated. The stability properties of the energy surface exhibit a phase transition from the normal to the super‐radiant behavior of the two‐level atoms, when At the same time, the analytic form of the ground state in terms of the minimum critical points of the system is obtained. The Dicke Hamiltonian is invariant under transformations of the point group with Λ̂ the excitation number operator. To restore this parity symmetry, we separate the variational proposed states into orthogonal even and odd components, which give rise to analytic expressions for the ground and first excited states. The fidelity of these projected variational states is verified by studying their overlap with the exact quantum results.

Umbral orthogonal polynomials
View Description Hide DescriptionWe present an umbral operator version of the classical orthogonal polynomials. We obtain three families which are the umbral counterpart of the Jacobi, Laguerre and Hermite polynomials in the classical case.

Building and destroying symmetry in 1‐D elastic systems
View Description Hide DescriptionLocally periodic rods, which show approximate invariance with respect to translations, are constructed by joining N unit cells. The spectrum then shows a band spectrum. We then break the local periodicity by including one or more defects in the system. When the defects follow a certain definite prescription, an analog of the Wannier‐Stark ladders is gotten; when the defects are random, an elastic rod showing Anderson localization is obtained. In all cases experimental values match the theoretical predictions.

Scale invariance as a symmetry in physical and biological systems: listening to photons, bubbles and heartbeats
View Description Hide DescriptionMany dynamical systems from different areas of knowledge can be studied within the theoretical framework of time series , where the system can be considered as a black box, that only needs to be “listened” to. In this framework, non‐correlated series (white noise) and strongly correlated series (brownian noise or periodic series) constitute two extremes. Certain dynamical systems auto‐organize in a critical state that is characterized by 1/f or flicker noise. The family of noises is fractal because fragments of the series are statistically identical to the original time series. 1/f noise is critical because it maximizes important complexity‐related quantities as memory, information content, efficiency and fractality. 1/f noise has been observed in classical systems, but also in quantum systems, and could possibly offer a unifying bridge of understanding between the macroscopic and the quantum world. In the present article, we will discuss some examples from both worlds.

Doorway states and billiards
View Description Hide DescriptionWhenever a distinct state is immersed in a sea of complicated and dense states, the strength of the distinct state, which we refer to as a doorway, is distributed in their neighboring states. We analyze this mechanism for 2‐D billiards with different geometries. One of them is symmetric and integrable, another is symmetric but chaotic, and the third has a capricious form. The fact that the doorway‐state mechanism is valid for such highly diverse cases, proves that it is robust.

Diffraction in time in tunneling phenomena
View Description Hide DescriptionThis paper reviews exact analytical results on diffraction in time for tunneling of a particle through an arbitrary potential of finite range. It emphasizes the essential role played by the formalism of resonant states to derive the analytical solution to the time‐dependent Schrödinger equation as an initial value problem and in the analysis of the corresponding transient behavior.

New dimensions of the periodic system: superheavy, superneutronic, superstrange, antimatter nuclei
View Description Hide DescriptionThe possibilities for the extension of the periodic system into the islands of superheavy (SH) elements, to and beyond the neutron drip line and to the sectors of strangeness and antimatter are discussed. The multi‐nucleon transfer processes in low‐energy damped collisions of heavy actinide nuclei may help us to fill the gap between the nuclei produced in the “hot” fusion reactions and the continent of known nuclei. In these reactions we may also investigate the “island of stability”. In many such collisions the lifetime of the composite giant system consisting of two touching nuclei turns out to be rather long sufficient for observing line structure in spontaneous positron emission from super‐strong electric fields (vacuum decay), a fundamental QED process not observed yet experimentally. At the neutron‐rich sector near the drip line islands and extended ridges of quasistable nuclei are predicted by HF calculations. Such nuclei, as well as very long living superheavy nuclei may be provided in double atomic bomb explosions. A tremendously rich scenario of new nuclear structure emerges with new magic numbers in the strangeness domain. Various production mechanisms are discussed for these objects and for antinuclei in high energy heavy‐ion collisions.

The linear canonical transform as a propagation model in digital holography
View Description Hide DescriptionDigital holographic systems are a class of two step, opto‐numerical, three‐dimensional imaging techniques which permit nanometric resolution of in‐vivo objects. The linear canonical transform (LCT) describes the paraxial propagation of a scalar wave field through any system comprising of lenses and free space. There are a great many different recording architectures used in holography. The interaction of these with the limitations of the recording device is poorly understood. Use of the LCT as a propagation model provides a unified description of the optical systems, facilitating direct comparison. We demonstrate how to use phase space diagrams to compare different digital holographic recording set‐ups, analyze the effects of the digital camera, and compare the sampling requirements of different numerical reconstruction algorithms. The results presented in this paper will allow digital holography practitioners to select an optical system to maximize the quality of their reconstructed image using a priori knowledge of the camera and object.

Solvable models and hidden symmetries in QCD
View Description Hide DescriptionWe show that QCD Hamiltonians at low energy exhibit an SU(2) structure, when only few orbital levels are considered. In case many orbital levels are taken into account we also find a semi‐analytic solution for the energy levels of the dominant part of the QCD Hamiltonian. The findings are important to propose the structure of phenomenological models.

Seniority in quantum many‐body systems
View Description Hide DescriptionThe use of the seniority quantum number in many‐body systems is reviewed. A brief summary is given of its introduction by Racah in the context of atomic spectroscopy. Several extensions of Racah’s original idea are discussed: seniority for identical nucleons in a single‐j shell, its extension to the case of many, non‐degenerate j shells and to systems with neutrons and protons. To illustrate its usefulness to this day, a recent application of seniority is presented in Bose‐Einstein condensates of atoms with spin.

Quantum vacuum energy in graphs and billiards
View Description Hide DescriptionThe vacuum (Casimir) energy in quantum field theory is a problem relevant both to new nanotechnology devices and to dark energy in cosmology. The crucial question is the dependence of the energy on the system geometry. Despite much progress since the first prediction of the Casimir effect in 1948 and its subsequent experimental verification in simple geometries, even the sign of the force in nontrivial situations is still a matter of controversy. Mathematically, vacuum energy fits squarely into the spectral theory of second‐order self‐adjoint elliptic linear differential operators. Specifically one promising approach is based on the small‐t asymptotics of the cylinder kernel where H is the self‐adjoint operator under study. In contrast with the well‐studied heat kernel the cylinder kernel depends in a non‐local way on the geometry of the problem. We discuss some results by the Louisiana‐Oklahoma‐Texas collaboration on vacuum energy in model systems, including quantum graphs and two‐dimensional cavities. The results may shed light on general questions, including the relationship between vacuum energy and periodic or closed classical orbits, and the contribution to vacuum energy of boundaries, edges, and corners.

Multipole analysis in cosmic topology
View Description Hide DescriptionLow multipole amplitudes in the Cosmic Microwave Background CMB radiation can be explained by selection rules from the underlying multiply‐connected homotopy. We apply a multipole analysis to the harmonic bases and introduce point symmetry. We give explicit results for two cubic 3‐spherical manifolds and lowest polynomial degrees, and derive three new spherical 3‐manifolds.

Two interacting electrons in a magnetic field: comparison of semiclassical, quantum, and variational solutions
View Description Hide DescriptionThe quantum mechanical many‐body problem is rarely analytically solvable. One notable exception is the case of two electrons interacting via a Coulomb potential in a uniform magnetic field. The motion is confined to a two‐dimensional plane, which is commonly the case in nanodevices. We compare the exact solution with the semiclassical energy spectrum and study the time‐dependent dynamics of the system using the time‐dependent variational principle.

Symmetry projection of the rovibrational functions of methane
View Description Hide DescriptionIn this work we propose a symmetry projection approach to build a rovibrational basis for methane. In our method, symmetry adapted functions are obtained by simultaneous diagonalization of a set of commuting operators, whose representation is given in terms of direct products of Wigner’s D functions and vibrational matrix representations provided by a local scheme. The proposed approach is general and permits to obtain in a systematic fashion an orthonormal set of symmetry‐projected functions, with good total angular momentum, and carrying the irreducible representations of the molecular symmetry group.

Integrability test for discrete equations via generalized symmetries
View Description Hide DescriptionIn this article we present some integrability conditions for partial difference equations obtained using the formal symmetries approach. We apply them to find integrable partial difference equations contained in a class of equations obtained by the multiple scale analysis of the general multilinear dispersive difference equation defined on the square.