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The basic ideas of second quantization and Fock space are extended to density operator states, used in treatments of open many-body systems. This can be done for fermions and bosons. While the former only requires the use of a non-orthogonal basis, the latter requires the introduction of a dual set of spaces. In both cases an operator algebra closely resembling the canonical one is developed and used to define the dual sets of bases. We here concentrated on the bosonic case where the unboundedness of the operators requires the definitions of dual spaces to support the pair of bases. Some applications, mainly to non-equilibrium steady states, will be mentioned., To appear in the Proceedings of Symposium Symmetries in Nature in memoriam Marcos Moshinsky. http://www.cicc.unam.mx/activities/2010/SymmetriesInNature/index.html

Locally 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., Comment: 12 pages, 16 figures, Accepted in Proceedings of "Symmetries in Nature", Symposium in Memoriam Marcos Moshinsky

Many 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 fβ noises (β≤0) is fractal because fragments of the series are statistically identical to the original time series. 1/f noise (β = −1) 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.

The 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 under study. 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 e^(-t sqrt(H)), where H is the self-adjoint operator under study. In contrast with the well-studied heat kernel e^(-tH), 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., 10 pages, 3 figures

One of the most important scientific contributions of Professor Marcos Moshinsky has been his study on the harmonic oscillator in quantum theory vis a vis the standard Schrodinger equation with constant mass [1]. However, a simple description of the motion of a particle interacting with an external environment such as happen in compositionally graded alloys consist of replacing the mass by the so‐called effective mass that is in general variable and dependent on position. Therefore, honoring in memoriam Marcos Moshinsky, in this work we consider the position‐dependent mass Schrodinger equations (PDMSE) for the harmonic oscillator potential model as former potential as well as with equi‐spaced spectrum solutions, i.e. harmonic oscillator isospectral partners. To that purpose, the point canonical transformation method to convert a general second order differential equation (DE), of Sturm‐Liouville type, into a Schrodinger‐like standard equation is applied to the PDMSE. In that case, the former potential ass...

Low 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., Comment: 15 pages, with figures

The 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 (≥10−20 s); 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...

The 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., Comment: 13 pages, 3 figures

The group‐theoretical structure of the harmonic oscillator appears in many guises. Originally developed by Marcos Moshinsky among several others for applications in nuclear physics, we point out here that the harmonic oscillator structure appears in aberrations of geometric optics, particularly in their classification by rank, symplectic spin and weight. And further, the finite harmonic oscillator appears again in the nonlinear transformations of finite Hamiltonian systems, when applied to the parallel processing of signals.

We 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 ωaωF = ±4γ2. 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 C2 = {I,exp(iπΛ)}, with Λ 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 v...

We 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.

We 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.

Digital 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.

In this contribution we shall construct a statistical theory of the complex wave interference phenomena that occur when a wave propagates through a disordered waveguide containing a quenched random distribution of scatterers.In studies performed in such systems one has found remarkable statistical regularities, in the sense that the probability distribution of various macroscopic quantities involves a rather small number of relevant physical parameters, while the rest of the microscopic details serves as mere “scaffolding”.We shall review past work in which this feature, which gives rise to universal statistical properties, was captured following a maximum‐entropy approach, as well as later studies in which the existence of a limiting distribution, in the sense of a generalized central‐limit theorem, has been actually demonstrated.We then describe a microscopic potential model that was developed recently, which also gives rise to a limiting macroscopic statistics and to a further generalization of the cen...

We 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.

This 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 Schrodinger equation as an initial value problem and in the analysis of the corresponding transient behavior.

Is it possible to trap a quantum particle in an open geometry? In this work we deal with the boundary value problem of the stationary Schroedinger (or Helmholtz) equation within a waveguide with straight segments and a rectangular bending. The problem can be reduced to a one dimensional matrix Schroedinger equation using two descriptions: oblique modes and conformal coordinates. We use a corner-corrected WKB formalism to find the energies of the one dimensional problem. It is shown that the presence of bound states is an effect due to the boundary alone, with no classical counterpart for this geometry. The conformal description proves to be simpler, as the coupling of transversal modes is not essential in this case., Comment: 16 pages, 10 figures. To appear in the Proceedings of the Symposium "Symmetries in Nature, in memoriam Marcos Moshinsky"

In memory of Marcos Moshinsky, who promoted the algebraic study of the harmonic oscillator, some results recently obtained on an infinite family of deformations of such a system are reviewed. This set, which was introduced by Tremblay, Turbiner, and Winternitz, consists in some Hamiltonians $H_k$ on the plane, depending on a positive real parameter $k$. Two algebraic extensions of $H_k$ are described. The first one, based on the elements of the dihedral group $D_{2k}$ and a Dunkl operator formalism, provides a convenient tool to prove the superintegrability of $H_k$ for odd integer $k$. The second one, employing two pairs of fermionic operators, leads to a supersymmetric extension of $H_k$ of the same kind as the familiar Freedman and Mende super-Calogero model. Some connection between both extensions is also outlined., Comment: 10 pages, no figure; to be published in Proceedings Symmetries in Nature: Symposium in Memoriam of Marcos Moshinsky, Cuernavaca, Mexico, Aug. 9-13, 2010

We 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.

We report results of the ultrasonic investigation of Ba1−xKxBiO3 single crystals for two potassium concentrations x≈0.35 and x≈0.47 in a wide temperature range. The softening of both the transverse c44 and the longitudinal c11 modes have been observed at temperatures between 200 K and 50 K. In the case of Ba0.65K0.35BiO3 a pronounced hysteresis was discovered. We proposed a model in which the softening of the elastic moduli, the hysteresis, and the maximum in the attenuation of sound can be explained by assuming a coupling of the acoustic modes with the anharmonic oscillations of O6 octahedra.