Volume 49, Issue 6, June 2008
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


On the geometry of supersymmetric quantum mechanical systems
View Description Hide DescriptionWe consider some simple examples of supersymmetric quantum mechanical systems and explore their possible geometric interpretation with the help of geometric aspects of real Clifford algebras. This leads to natural extensions of the considered systems to higher dimensions and more complicated potentials.

Visualization and dimensional scaling for some threebody problems in atomic and molecular quantum mechanics
View Description Hide DescriptionThreebody problems in atomic and molecular quantum mechanics, comprising one electron–two nuclei and two electron–one nucleus, are studied from their Schrödinger–Born–Oppenheimer models. The following are main topics of interest in this paper: (1) review of foundational mathematical properties of the multiparticle Schrödinger operator, (2) visualization of (hydrogen molecular ion)like and He (helium)like molecular and atomic states, and (3) spectrum of He obtained by the largedimension scaling limit. The authors begin with topic (1) for the tutorial purpose and devote topics (2) and (3) to new contributions of the analytical, numerical, and visualization nature. Relevant heuristics, graphics, proofs, and calculations are presented.

Sharp thresholds of blowup and global existence for the coupled nonlinear Schrödinger system
View Description Hide DescriptionIn this paper, we establish two new types of invariant sets for the coupled nonlinear Schrödinger system in the Euclidean space and derive two sharp thresholds of blowup and global existence for its solutions. Some analogous results for the nonlinear Schrödinger system posed on the hyperbolic space and on the standard 2sphere are also presented. Our arguments and constructions are improvements of some previous works on this direction. At the end, we give some heuristic analysis about the strong instability of the solitary waves. The relation between the two types of thresholds is a very interesting problem, and we leave it as an open problem for further study.

Purely squeezed states for quantum deformed systems
View Description Hide DescriptionThe generalized purely squeezed states for primary shapeinvariant potentials systems, quantum deformed by different models, are constructed by the ladderoperator method within an algebraic approach based on supersymmetric quantum mechanics. The characteristic properties of these states as well as their quantum statistical properties and squeezing effects for generalized quadrature observables are studied and analyzed in terms of the quantum deformation parameter . An application is given for a quantum deformed Pöschl–Teller potential system, and numerical results are presented and discussed in detail.

Higher entropic uncertainty relations for anticommuting observables
View Description Hide DescriptionUncertainty relations provide one of the most powerful formulations of the quantum mechanical principle of complementarity. Yet, very little is known about such uncertainty relations for more than two measurements. Here, we show that sufficient unbiasedness for a set of binary observables, in the sense of mutual anticommutation, is good enough to obtain maximally strong uncertainty relations in terms of the Shannon entropy. We also prove nearly optimal relations for the collision entropy. This is the first systematic and explicit approach to finding an arbitrary number of measurements for which we obtain maximally strong uncertainty relations. Our results have immediate applications to quantum cryptography.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Interaction of imaginarychargecarrying dyon with particles
View Description Hide DescriptionBy analytic continuation from a gauge field solution, Wu and Yang [Phys. Rev.13, 3233 (1976)] obtained a static and sourceless solution of gauge theory for the group [also for ]. This field configuration resembles a dyon that carries an imaginary charge. We present here a scheme that allows us to derive consistently the equations describing the interaction between the Lorentzgauge field and particles in the classical limit. They look like Wong’s equations in which gauge field components and color charges are complex. The complex charges and complex gauge field components can be understood as auxiliary concepts, while the equations of motion of particles in the outer space as well as kinematical and dynamical characteristics of the motion are real. The obtained equations are applied to investigate the case of particles in the mentioned dyon field configuration. The expressions of total energy and angular momentum, as integrals of motion of particles, are derived. The equations of motion allow planar motions, for which an analytic description of orbits is carried out.

More transition amplitudes on the Riemann sphere
View Description Hide DescriptionWe consider a conformal field theory for bosons on the Riemann sphere. Correlation functions are defined as singular limits of functional integrals. The main result is that these amplitudes define transition amplitudes, that is multilinear Hilbert–Schmidt functionals on a fixed Hilbert space.

Schwinger–Dyson operators as invariant vector fields on a matrix model analog of the group of loops
View Description Hide DescriptionFor a class of large multimatrix models, we identify a group that plays the same role as the group of loops on spacetime does for Yang–Mills theory. is the spectrum of a commutative shuffledeconcatenation Hopf algebra that we associate with correlations. is the exponential of the free Lie algebra. The generating series of correlations is a function on and satisfies quadratic equations in convolution. These factorized Schwinger–Dyson or loop equations involve a collection of Schwinger–Dyson operators, which are shown to be rightinvariant vector fields on , one for each linearly independent primitive of the Hopf algebra. A large class of formal matrix models satisfying these properties are identified, including as special cases, the zero momentum limits of the Gaussian, Chern–Simons, and Yang–Mills field theories. Moreover, the Schwinger–Dyson operators of the continuum Yang–Mills action are shown to be rightinvariant derivations of the shuffledeconcatenation Hopf algebra generated by sources labeled by position and polarization.
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 GENERAL RELATIVITY AND GRAVITATION


The Riemann–Lanczos equations in general relativity and their integrability
View Description Hide DescriptionThe aim of this paper is to examine the RiemannLanczos equations and how they can be made integrable. They consist of a system of linear firstorder partial differential equations that arise in general relativity, whereby the Riemann curvature tensor is generated by an unknown thirdorder tensor potential field called the Lanczos tensor. Our approach is based on the theory of jet bundles, where all field variables and all their partial derivatives of all relevant orders are treated as independent variables alongside the local manifold coordinates on the given spacetime manifold. This approach is adopted in (a) Cartan’s method of exterior differential systems, (b) Vessiot’s dual method using vector field systems, and (c) the Janet–Riquier theory of systems of partial differential equations. All three methods allow for the most general situations under which integrability conditions can be found. They give equivalent results, namely, that involutivity is always achieved at all generic points of the jet manifold after a finite number of prolongations. Two alternative methods that appear in the general relativity literature to find integrability conditions for the Riemann–Lanczos equations generate new partial differential equations for the Lanczos potential that introduce a source term, which is nonlinear in the components of the Riemann tensor. We show that such sources do not occur when either of method (a), (b), or (c) are used.

Massless field perturbations of the spinning metric
View Description Hide DescriptionA single master equation is given describing spin test fields that are gauge and tetradinvariant perturbations of the spinning metricspacetime representing a source with mass , uniformly rotating with angular momentum per unit mass , and uniformly accelerated with acceleration . This equation can be separated into its radial and angular parts. The behavior of the radial functions near the black hole (outer) horizon is studied to examine the influence of on the phenomenon of superradiance, while the angular equation leads to modified spinweighted spheroidal harmonic solutions generalizing those of the Kerr spacetime. Finally the coupling between the spin of the perturbing field and the acceleration parameter is discussed.

Nonimprisonment conditions on spacetime
View Description Hide DescriptionThe nonimprisonment conditions on spacetimes are studied. It is proved that the nonpartial imprisonment property implies the distinction property. Moreover, it is proven that feeble distinction, a property which stays between weak distinction and causality, implies nontotal imprisonment. As a result the nonimprisonment conditions can be included in the causal ladder of spacetimes. Finally, totally imprisoned causal curves are studied in detail, and results concerning the existence and properties of minimal invariant sets are obtained.
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 DYNAMICAL SYSTEMS


Time evolution of the rule 150 cellular automaton activity from a Fibonacci iteration
View Description Hide DescriptionThe rule 150 cellular automaton is a remarkable discrete dynamical system, as it shows spectra if started from a single seed [J. Nagler and J. C. Claussen, Phys. Rev. E71, 067103 (2005)]. Despite its simplicity, a feasible solution for its time behavior is not obvious. Its selfsimilarity does not follow a onestep iteration like other elementary cellular automata. Here it is shown how its time behavior can be solved as a twostep vectorial, or string, iteration, which can be viewed as a generalization of Fibonacci iteration generating the time series from a sequence of vectors of increasing length. This allows us to compute the total activity time series more efficiently than by simulating the whole spatiotemporal process or even by using the closed expression. The results are further extended to the generalization of rule 150 to the twodimensional case and to Bethe lattices and the relation to corresponding integer sequences is discussed.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Invariants from classical field theory
View Description Hide DescriptionWe introduce a method that generates invariant functions from perturbative classical field theories depending on external parameters. By applying our methods to several field theories such as Abelian , Chern–Simons, and twodimensional Yang–Mills theory, we obtain, respectively, the linking number for embedded submanifolds in compact varieties, the Gauss’ and the second Milnor’s invariant for links in , and invariants under areapreserving diffeomorphisms for configurations of immersed planar curves.

Unified formalism for nonautonomous mechanical systems
View Description Hide DescriptionWe present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of regular and nonregular timedependent mechanical systems, which is based on the approach of Skinner and Rusk [“Generalized Hamiltonian dynamics I. Formulation on ,” J. Math. Phys.24, 2589 (1983)]. The dynamical equations of motion and their compatibility and consistency are carefully studied, making clear that all the characteristics of the Lagrangian and the Hamiltonian formalisms are recovered in this formulation. As an example, a semidiscretization of the nonlinear waveequation is studied, proving the applicability of the proposed formalism.

MultiHamiltonian structures for matrix systems
View Description Hide DescriptionFor the rational, elliptic, and trigonometric matrices, we exhibit the links between three “levels” of Poisson spaces: (a) some finitedimensional spaces of matrixvalued meromorphic functions on the complex line, (b) spaces of spectral curves and sheaves supported on them, and (c) symmetric products of a surface. We have, at each level, a linear space of compatible Poissonstructures, and the maps relating the levels are Poisson. This leads in a natural way to the Nijenhuis coordinates for these spaces. At level (b), there are Hamiltonian systems on these spaces which are integrable for each Poissonstructure in the family and which are such that the Lagrangian leaves are the intersections of the symplectic leaves over the Poissonstructures in the family. Specific examples include many of the wellknown integrable systems.

On the holonomy of the Coulomb connection over manifolds with boundary
View Description Hide DescriptionNarasimhan and Ramadas [Commun. Math. Phys.67, 121 (1979)] showed that the restricted holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group when one considers the product principal bundle . Instead of a base manifold, we consider here a base manifold of dimension with a boundary and use Dirichlet boundary conditions on connections as defined by Marini [Commun. Pure Appl. Math.45, 1015 (1992)]. A key step in the method of Narasimhan and Ramadas consisted in showing that the linear space spanned by the curvature form at one specially chosen connection is dense in the holonomyLie algebra with respect to an appropriate Sobolev norm. Our objective is to explore the effect of the presence of a boundary on this construction of the holonomyLie algebra. Fixing appropriate Sobolev norms, it will be shown that the space spanned, linearly, by the curvature form at any one connection is never dense in the holonomyLie algebra. In contrast, the linear space spanned by the curvature form and its first commutators at the flat connection is dense and, in the category, is in fact the entire holonomyLie algebra. The former, negative, theorem is proven for a general principle bundle over , while the latter, positive, theorem is proven only for a product bundle over the closure of a bounded open subset of . Our technique for proving the absence of density consists in showing that the linear space spanned by the curvature form at one point is contained in the kernel of a linear map consisting of a third order differential operator, followed by a restriction operation at the boundary; this mapping is determined by the mean curvature of the boundary.
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 FLUIDS


Nonstandard analysis and jump conditions for converging shock waves
View Description Hide DescriptionNonstandard analysis is an area of modern mathematics that studies abstract number systems containing both infinitesimal and infinite numbers. This article applies nonstandard analysis to derive jump conditions for onedimensional, converging shock waves in a compressible, inviscid, perfect gas. It is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval. Predistributions of the Heaviside function and the Dirac delta measure are introduced to model the flow parameters across a shock wave. The equations of motion expressed in nonconservative form are then applied to derive unambiguous relationships between the jump functions for the flow parameters.
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 STATISTICAL PHYSICS


Domain wall and periodic solutions of a coupled model
View Description Hide DescriptionCoupled triple well onedimensional potentials occur in both condensed matter physics and field theory. Here we provide a set of exact periodic solutions in terms of elliptic functions (domain wall arrays) and obtain single domain wallsolutions in specific limits. Topological, nontopological (e.g., some pulselike solutions), as well as mixed domain walls are obtained. We relate these solutions to structural phase transitions in materials with polarization, shuffle modes, and strain. We calculate the energy and the asymptotic interaction between solitons for various solutions.

Time dependent current in a nonstationary environment: A microscopic approach
View Description Hide DescriptionBased on a microscopic system reservoir model, where the associated bath is not in thermal equilibrium, we simulate the nonstationary Langevin dynamics and obtain the generalized nonstationary fluctuation dissipation relation (FDR) which asymptotically reduces to the traditional form. Our Langevin dynamics incorporates nonMarkovian process also, the origin of which lies in the decaying term of the nonstationary FDR. We then follow the stochastic dynamics of the Langevin particle based on the Fokker–Planck–Smoluchowski description in ratchet potential to obtain the steady and time dependent current in an analytic form. We also examine the influence of initial excitation and subsequent relaxation of bath modes on the transport of the Langevin particle to show that the nonequilibrium nature of the bath leads to both strong nonexponential dynamics as well as nonstationary current.
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 METHODS OF MATHEMATICAL PHYSICS


Scalar products of elementary distributions
View Description Hide DescriptionThe Schrödinger equation with point interaction in one dimension is revisited in a simple framework where the “singular” potential is defined as a symmetric operator in a natural way. The main tool is a scalar product of the elementary distributions constructed after a commutative (and wellordered) field extension of followed by complexification. A contact with the hyperreal numbers that arise in nonstandard analysis is possible but not essential, our extensions of and being obtained by a quite elementary method.
