Nonlocal Structure Research Articles

Nonlocal Structure of Higher Spin Fields

We first review the problem of superluminal propagation or even loss of propagation for higher spin fields minimally coupled to an external electromagnetic field, discovered by Velo and Zwanziger in the 1960's. Then, we show that these fields have a nonlocal structure. This indicates that these fields have internal structure and should not be considered as elementary. The main tool in this investigation is the use of a Peirce decomposition of the algebra of the matrices occuring in the first order wave equations for these fields.

Electronic structure of the layer compounds GaSe and InSe in a tight-binding approach

The three-dimensional band structure of the III-VI layer compounds GaSe and InSe has been investigated in the tight-binding approach. The pseudo-Hamiltonian matrix elements in the ${\mathrm{sp}}^{3}{s}^{*}$ basis are fit in order to reproduce the nonlocal pseudopotential band structure, in the framework of constrained optimization techniques using the conjugate gradient method. The results are in good agreement with the optical and photoemission experimental data. The scaling laws appropriate to the covalent bonding are violated by a fraction of eV only, which suggests that the interlayer interactions are not solely of the van der Waals type.

Integrable Systems and Metrics of Constant Curvature

In this article we present a Lagrangian representation for evolutionary systems with a Hamiltonian structure determined by a differential-geometric Poisson bracket of the first order associated with metrics of constant curvature. Kaup-Boussinesq system has three local Hamiltonian structures and one nonlocal Hamiltonian structure associated with metric of constant curvature. Darboux theorem (reducing Hamiltonian structures to canonical form “d/dx” by differential substitutions and reciprocal transformations) for these Hamiltonian structures is proved.

Notes on q-Electroweak

A gauged SU q (2) theory is characterized by two dual algebras, the first lying close to the Lie algebra of SU(2) while the second introduces new degrees of freedom that may be associated with nonlocality or solitonic structure. The first and second algebras, here called the external and internal algebras respectively, define two sets of fields, also called external and internal. The gauged external fields agree with the Weinberg–Salam model at the level of the doublet representation but differ at the level of the adjoint representation. For example, the g-factor of the charged W-boson differs in the two models. The gauged internal fields remain speculative but are analogous to color fields.

Developments in turbulence research: a review based on the 1999 Programme of the Isaac Newton Institute, Cambridge

Recent research is making progress in framing more precisely the basic dynamical and statistical questions about turbulence and in answering them. It is helping both to define the likely limits to current methods for modelling industrial and environmental turbulent flows, and to suggest new approaches to overcome these limitations. Our selective review is based on the themes and new results that emerged from more than 300 presentations during the Programme held in 1999 at the Isaac Newton Institute, Cambridge, UK, and on research reported elsewhere. A general conclusion is that, although turbulence is not a universal state of nature, there are certain statistical measures and kinematic features of the small-scale flow field that occur in most turbulent flows, while the large-scale eddy motions have qualitative similarities within particular types of turbulence defined by the mean flow, initial or boundary conditions, and in some cases, the range of Reynolds numbers involved. The forced transition to turbulence of laminar flows caused by strong external disturbances was shown to be highly dependent on their amplitude, location, and the type of flow. Global and elliptical instabilities explain much of the three-dimensional and sudden nature of the transition phenomena. A review of experimental results shows how the structure of turbulence, especially in shear flows, continues to change as the Reynolds number of the turbulence increases well above about 104 in ways that current numerical simulations cannot reproduce. Studies of the dynamics of small eddy structures and their mutual interactions indicate that there is a set of characteristic mechanisms in which vortices develop (vortex stretching, roll-up of instability sheets, formation of vortex tubes) and another set in which they break up (through instabilities and self- destructive interactions). Numerical simulations and theoretical arguments suggest that these often occur sequentially in randomly occurring cycles. The factors that determine the overall spectrum of turbulence were reviewed. For a narrow distribution of eddy scales, the form of the spectrum can be defined by characteristic forms of individual eddies. However, if the distribution covers a wide range of scales (as in elongated eddies in the ‘wall’ layer of turbulent boundary layers), they collectively determine the spectra (as assumed in classical theory). Mathematical analyses of the Navier–Stokes and Euler equations applied to eddy structures lead to certain limits being defined regarding the tendencies of the vorticity field to become infinitely large locally. Approximate solutions for eigen modes and Fourier components reveal striking features of the temporal, near-wall structure such as bursting, and of the very elongated, spatial spectra of sheared inhomogeneous turbulence; but other kinds of eddy concepts are needed in less structured parts of the turbulence. Renormalized perturbation methods can now calculate consistently, and in good agreement with experiment, the evolution of second- and third-order spectra of homogeneous and isotropic turbulence. The fact that these calculations do not explicitly include high-order moments and extreme events, suggests that they may play a minor role in the basic dynamics. New methods of approximate numerical simulations of the larger scales of turbulence or ‘very large eddy simulation’ (VLES) based on using statistical models for the smaller scales (as is common in meteorological modelling) enable some turbulent flows with a non-local and non-equilibrium structure, such as impinging or convective flows, to be calculated more efficiently than by using large eddy simulation (LES), and more accurately than by using ‘engineering’ models for statistics at a single point. Generally it is shown that where the turbulence in a fluid volume is changing rapidly and is very inhomogeneous there are flows where even the most complex ‘engineering’ Reynolds stress transport models are only satisfactory with some special adaptation; this may entail the use of transport equations for the third moments or non-universal modelling methods designed explicitly for particular types of flow. LES methods may also need flow-specific corrections for accurate modelling of different types of very high Reynolds number turbulent flow including those near rigid surfaces.This paper is dedicated to the memory of George Batchelor who was the inspiration of so much research in turbulence and who died on 30th March 2000. These results were presented at the last fluid mechanics seminar in DAMTP Cambridge that he attended in November 1999.

Multiparticle Quantum Superposition and Stimulated Entanglement by Parity Selective Amplification of Entangled States

Abstract A multiparticle quantum superposition state has been generated by a novel phase-selective parametric amplifier of an entangled two-photon state. This realization is expected to open a new field of investigations on the persistence of the validity of the standard quantum theory for systems of increasing complexity, in a quasi decoherence-free environment. Because of its nonlocal structure the new system is expected to play a relevant role in the modem endeavor on quantum information and in the basic physics of entanglement. - Pacs: 03.65.Bz, 03.67.-a, 42.50.Ar, 89.70.+C

A Simple Family of Non-Local Poisson Brackets

A dispersionless integrable system with repeated eigenvalues is presented. For N ≥ 3 components the system has no local Hamiltonian structure. Infinitely many simple compatible non-local Hamiltonian structures are given, using a result of Ferapontov.

Shape-changing collisions of coupled bright solitons in birefringent optical fibers

We critically review the recent progress in understanding soliton propagation in birefringent optical fibers. By constructing the most general bright two-soliton solution of the integrable coupled nonlinear Schrödinger equation (Manakov model) we point out that solitons in birefringent fibers can in general change their shape after interaction due to a change in the intensity distribution among the modes even though the total energy is conserved. However, the standard shape-preserving collision (elastic collision) property of the (1 + 1)-dimensional solitons is recovered when restrictions are imposed on some of the soliton parameters. As a consequence the following further properties can be deduced using this shape-changing collision. 1. (i) The exciting possibility of switching of solitons between orthogonally polarized modes of the birefringent fiber exists. 2. (ii) When additional effects due to periodic rotation of birefringence axes are considered, the shape changing collision can be used as a switch to suppress or to enhance the periodic intensity exchange between the orthogonally polarized modes. 3. (iii) For ultra short optical soliton pulse propagation in non-Kerr media, from the governing equation an integrable system of coupled nonlinear Schrödinger equation with cubic-quintic terms is identified. It admits a nonlocal Poisson bracket structure. 4. (iv) If we take into account the higher-order terms in the coupled nonlinear Schrödinger equation then their effect on the shape changing collision of solitons, during optical pulse propagation, can be studied by using a direct perturbational approach.

Decay of solitons in an isotropic collisionless quasineutral plasma with isothermal pressure

Soliton-type solutions of the complete unreduced system of transport equations describing the plane-parallel motions of an isotropic collisionless quasineutral plasma in a magnetic field with constant ion and electron temperatures are studied. The regions of the physical parameters for fast and slow magnetosonic branches, where solitons and generalized solitary waves—nonlocal soliton structures in the form of a soliton “core” with asymptotic behavior at infinity in the form of a periodic low-amplitude wave—exist, are determined. In the range of parameters where solitons are replaced by generalized solitary waves, soliton-like disturbances are subjected to decay whose mechanisms are qualitatively different for slow and fast magnetosonic waves. A specific feature of the decay of such disturbances for fast magnetosonic waves is that the energy of the disturbance decreases primarily as a result of the quasistationary emission of a resonant periodic wave of the same nature. Similar disturbances in the form of a soliton core of a slow magnetosonic generalized solitary wave essentially do not emit resonant modes on the Alfven branch but they lose energy quite rapidly because of continuous emission of a slow magnetosonic wave. Possible types of shocks which are formed by two types of existing soliton solutions (solitons and generalized solitary waves) are examined in the context of such solutions.

The averaging of nonlocal Hamiltonian structures in Whitham′s method

We consider the m‐phase Whitham′s averaging method and propose the procedure of “averaging” nonlocal Hamiltonian structures. The procedure is based on the existence of a sufficient number of local‐commuting integrals of the system and gives the Poisson bracket of Ferapontov type for Whitham′s system. The method can be considered as the generalization of the Dubrovin‐Novikov procedure for the local field‐theoretical brackets.

Dynamics of a Two-Pendulum Model with Impact Interaction and an Elastic Support

This paper examines the dynamic behavior of a double pendulummodel with impact interaction. One of the masses of the two pendulumsmay experience impacts against absolutely rigid container wallssupported by an elastic system forming an inverted pendulum restrainedby a torsional elastic spring. The system equations of motion arewritten in terms of a non-smooth set of coordinates proposed originallyby Zhuravlev. The advantage of non-smooth coordinates is that theyeliminate impact constraints. In terms of the new coordinates, thepotential energy field takes a cell-wise non-local structure, and theimpact events are treated geometrically as a crossing of boundariesbetween the cells. Based on a geometrical treatment of the problem,essential physical system parameters are established. It is found thatunder resonance parametric conditions of the linear normal modes thesystem's response can be either bounded or unbounded, depending on thesystem's parameters. The ability of the system to absorb energy from anexternal source essentially depends on the modal inclination angle,which is related to the principal coordinates.

The Variety of an Assortment

Consumers rank variety of assortment right behind location and price when naming reasons why they patronize their favorite stores. Consumers care about variety because they are more likely to find what they want when going to a store that offers more varied assortments. When tastes are not well formed or are dynamic, perceived variety matters even more because of the desire to become educated about what is available while maintaining flexibility. Variety perception also matters when the variety-seeking motive operates. Retailers care about variety because customers value variety. Therefore, it is important to understand how people perceive the variety contained in an assortment and how these perceptions influence satisfaction and store choice. Remarkably, except for a recent study by Broniarczyk et al. (1998), there has been no research aimed at understanding the variety perception process itself. We offer a general mathematical model of variety based on the complete information structure of an assortment, defined both by the multiattribute structure of the objects and their spatial locations. We impose a psychologically plausible set of restrictions on the general model and obtain a class of simpler estimable models of perceived variety. We utilize the model to develop assortments that vary widely in terms of their information structure and study the influence of three factors on variety perceptions: (a) information structure of each assortment (i.e., the attribute level differences between objects); (b) level of organization of the objects and hence their relative spatial positions; and (c) task orientations, promoting either analytic or holistic processing. We also investigate the influence of variety perception and organization on stated satisfaction and store choice. To summarize our major findings: 1. Information structure has a big impact on variety perceptions, though diminishing returns accompany increases in the number of attributes on which object pairs differ. 2. People are more influenced by local information structure (adjacent objects) than nonlocal information structure. Proximity matters. 3. Organization of the display can either increase or decrease variety perceptions. When people engage in analytic processing, organized displays appear to offer more variety. When processing is holistic, random displays are seen as more varied. 4. Both variety perceptions and organization drive stated satisfaction and store choice. People are more satisfied with and likely to choose stores carrying those assortments that are perceived as offering high variety and that are displayed in an organized rather than random manner. Our work provides a basic framework for thinking about variety. By helping retailers to understand the factors that drive variety perception, it may be possible to design more efficient, lower cost assortments without reducing variety perceptions and the probability of future store visits.

Partitioning using second-order information and stochastic-gain functions

A probability-based partitioning algorithm, PROP, was introduced in [8] that achieved large improvements over traditional "deterministic" iterative-improvement techniques like Fidducia-Mattheyses (FM) and Krishnamurthy's look-ahead (LA) algorithm. While PROP's gain function has a greater futuristic component than PM or LA, it incorporates spatially local information-only information on the removal probabilities of adjacent nets of a cell is used in its gain computation. This prevents a higher-level view of nonlocal structures. Also, giving uniform weights to all nets, results in an inability to differentiate between the futuristic benefit of removing one net from another. This paper investigates for the first time the issues of using nonlocal structural information in gain functions and variable net weights based on the futuristic (stochastic) benefit of moving them from the cutset. The result is a more sophisticated partitioner DEEP-PROP that performs better for circuits with large complexities by incorporating more nonlocal (second order) structural information than PROP. The second-order information is incorporated into cell gains as well as variable net weights-the latter helps to focus future cell moves in the "right" cluster around the currently moved cell and, thus, better utilizes the information that led to its selection as the best move. A lower complexity version, variable weight PROP (VAR-PROP), that also uses dynamically assigned variable net weights, but based on first-order information, has also been developed. Both versions yield significant improvements over PROP on the ACM/SIGDA benchmark suite. DEEP-PROP yields mincut improvements of as much as 39% and an average of 20% for large circuits (10-K to 25-K cells) and an average of 14% over all circuits. DEEP PROP is about a factor of 2.8 times slower than PROP, which is very fast. VAR-PROP, which has a much lower computational complexity than DEEP-PROP, yields for large circuits, maximum and average mincut improvements over PROP of 27% and 18%, respectively, and an average of 12.6% improvement over all circuits. It is only about 14% slower than PROP, For the only very large circuit golem3 in the suite (>100 K cells), the improvements produced by DEEP-PROP and VAR-PROP over PROP are 15.6% and 11.5%, respectively. We also compare DEEP-PROP to FM, PROP and hMetis for a subset of the newer 1SPD98 benchmark circuits, and demonstrate significant improvements over FM and PROP, and comparable mincuts (within 2%) to hMetis, one of the best multilevel partitioners.

Topology Change and Context Dependence

The nonclassical features of quantum mechanicsare reproduced using models constructed with a classicaltheory — general relativity. The inability todefine complete initial data consistently andindependently of future measurements, nonlocality, and thenon-Boolean logical structure are reproduced by theseexamples. The key feature of the models is the role oftopology change. It is the breakdown of causal structure associated with topology change that leads tothe apparently nonclassical behavior. For geons,topology change is required to describe the interactionof particles. It is therefore natural to regard topology change as an essential part of the measurementprocess. This leads to models in which the measurementimposes additional nonredundant boundary conditions. Theinitial state cannot be described independently of the measurement and there is a causalconnection between the measurement and the initialstate.

Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s law

A generalized bidirectional distribution function (BRDF) that relates the specific intensity of the scattered light from a semi-infinite medium to the specific intensity of the incident light is introduced in the framework of coherence theory. This derivation allows us to obtain from first principles several fundamental properties: First, it is established that the generalized BRDF takes the form of a nonlocal relation between the incident and the scattered specific intensities. This nonlocal structure allows us to account naturally for the lateral shift of a beam. Second, the generalized BRDF is the Fourier transform of the correlation function that describes the memory effect. Third, the Helmholtz principle for specific intensities is derived as a theorem from the reciprocity property of the scattering operator for wave fields. This result allows us to prove Kirchhoff’s law.

Relationships between differential substitutions and Hamiltonian structures of the Korteweg-de Vries equation

A new regular method for (a) classification of integrable equations, (b) constructing an infinite set of differential substitutions, and (c) reducing every nonlocal Hamiltonian structure into canonical form is presented. An explanation of the origin of these differential substitutions is given via a relationship between the spectral problem and a Miura transformation. A relationship of these differential substitutions with a generating function of conservation law densities is found. Every Hamiltonian structure of the Korteweg-de Vries equation possesses a transformation to the canonical “ d dx ”- type by a combination of some differential substitutions and reciprocal transformations. Some well-known equations are embedded into a unified chain.

Geometry of the Batalin–Fradkin–Vilkovisky theorem

We describe gauge-fixing at the level of virtual paths in the path integral as a non-symplectic BRST-type of flow on the path phase space. As a consequence a gauge-fixed, non-local symplectic structure arises. Restoring of locality is discussed. A pertinent anti-Lie-bracket and an infinite dimensional group of gauge fermions are introduced. Generalizations to Sp(2)-symmetric BLT-theories are made.

Coset approach to the N = 2 supersymmetric matrix GNLS hierarchies

We discuss a large class of coset constructions of the N=2 sl(n|n-1) affine superalgebra. We select admissible subalgebras, i.e. subalgebras that induce linear chiral/antichiral constraints on the coset supercurrents. We show that all the corresponding coset constructions lead to N=2 matrix GNLS hierarchies. We develop an algorithm to compute the relative Hamiltonians and flows. We spell out completely the case of the N=2 affine sl(3|2), which possesses four admissible subalgebras. The non-local second Hamiltonian structure of the N=2 matrix GNLS hierarchies is obtained via Dirac procedure from the local N=2 sl(n|n-1) affine superalgebra. We observe that to any second Hamiltonian structure with pure bosonic or pure fermionic superfield content there correspond two different N=2 matrix GNLS hierarchies.

High-field transport and electroluminescence in ZnS phosphor layers

A full-band Monte Carlo simulation of the high-field electron transport in the ZnS phosphor layer of an alternating-current thin-film electroluminescent device is performed. The simulation includes a nonlocal empirical pseudopotential band structure for ZnS and the relevant scattering mechanisms for electrons in the first four conduction bands, including band-to-band impact ionization and impact excitation of Mn2+ luminescent centers. The steady-state electron energy distribution in the ZnS layer is computed for phosphor fields from 1 to 2 MV/cm. The simulation reveals a substantial fraction of electrons with energies in excess of the Mn2+ impact excitation threshold. The computed impact excitation yield for carriers transiting the phosphor layer exhibits an approximately linear increase with increasing phosphor field above threshold. The onset of Mn2+ impact excitation coincides with the onset of band-to-band impact ionization of electron-hole pairs which prevents electron runaway at high electric fields.

Bound on nonlocal scale from $g-2$ of muon in a nonlocal W-S model

We consider the nonlocal version of the Weinberg Salam model (following Kleppe et al.) with a finite parameter Λ signifying a fundamental length scale. We calculate the extra contributions to the anomalous magnetic moment of the muon coming from the nonlocal structure in this model. We find that the nonlocal contribution can be comparable to weak contributions and goes to increase theoretical estimates. We use this calculation to determine the limit on the scale of nonlocality. We obtain the result 1/Λ ≲ 3 × 10−16cm, which could be improved when present experimental errors narrow down.