Volume 44, Issue 9, September 2003
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Monotone Riemannian metrics on density matrices with nonmonotone scalar curvature
View Description Hide DescriptionThe theory of monotone Riemannian metrics on the state space of a quantum system was established by Dénes Petz in 1996. In a recent paper he argued that the scalar curvature of a statistically relevant—monotone—metric can be interpreted as an average statistical uncertainty. The present paper contributes to this subject. It is reasonable to expect that states which are more mixed are less distinguishable than those which are less mixed. The manifestation of this behavior could be that for such a metric the scalar curvature has a maximum at the maximally mixed state. We show that not every monotone metric fulfils this expectation, some of them behave in a very different way. A mathematical condition is given for monotone Riemannian metrics to have a local minimum at the maximally mixed state and examples are given for such metrics.

Spontaneous localization of electrons in twodimensional lattices within the adiabatic approximation
View Description Hide DescriptionThe conditions for spontaneous localization of electrons in an isotropic twodimensional electron–phonon lattice are investigated within the zero adiabatic approximation. It is shown that the localization occurs when the electron–phonon coupling takes values within certain finite interval of values At the energy minimum is attained for the delocalized states and at the strong localization on one lattice site takes place. In this paper we introduce an ansatz which, under a variational principle, allows us to describe all three regimes at the same time. The radius of the electron localization, as a function of electron–phonon coupling constant, is evaluated analytically and shown to fit well the numerical data.

On integrable Hamiltonians for higher spin chain
View Description Hide DescriptionIntegrable Hamiltonians for higher spin periodic chains are constructed in terms of the spin generators; explicit examples for spins up to are given. Relations between Hamiltonians for some symmetric and symmetric universal matrices are studied; their properties are investigated. A certain modification of the higher spin periodic chain Hamiltonian is shown to be an integrable symmetric Hamiltonian for an open chain.

Quantum mechanics of damped systems
View Description Hide DescriptionWe show that the quantization of a simple damped system leads to a selfadjoint Hamiltonian with a family of complex generalized eigenvalues. It turns out that they correspond to the poles of energy eigenvectors when continued to the complex energy plane. Therefore, the corresponding generalized eigenvectors may be interpreted as resonant states. We show that resonant states are responsible for the irreversible quantum dynamics of our simple model.

Extended edge states in finite Hall systems
View Description Hide DescriptionWe study edge states of a random Schrödinger operator for an electron submitted to a magnetic field in a finite macroscopic two dimensional system of linear dimensions equal to The direction is periodic and in the direction the electron is confined by two smoothly increasing parallel boundary potentials. We prove that, with large probability, for an energy range in the first spectral gap of the bulk Hamiltonian, the spectrum of the full Hamiltonian consists only on two sets of eigenenergies whose eigenfuntions have average velocities which are strictly positive/negative, uniformly with respect to the size of the system. Our result gives a well defined meaning to the notion of edge states for a finite cylinder with two boundaries, and extends previous studies on systems with only one boundary.

Wigner–Yanase information on quantum state space: The geometric approach
View Description Hide DescriptionIn the search of appropriate Riemannian metrics on quantum state space, the concept of statistical monotonicity, or contraction under coarse graining, has been proposed by Chentsov. The metrics with this property have been classified by Petz. All the elements of this family of geometries can be seen as quantum analogs of Fisher information. Although there exists a number of general theorems shedding light on this subject, many natural questions, also stemming from applications, are still open. In this paper we discuss a particular member of the family, the Wigner–Yanase information. Using a wellknown approach that mimics the classical pullback approach to Fisher information, we are able to give explicit formulas for the geodesic distance, the geodesic path, the sectional and scalar curvatures associated to Wigner–Yanase information. Moreover, we show that this is the only monotone metric for which such an approach is possible.

Quantum fourbody system in dimensions
View Description Hide DescriptionBy the method of generalized spherical harmonic polynomials, the Schrödinger equation for a fourbody system in dimensional space is reduced to the generalized radial equations where only six internal variables are involved. The problem on separating the rotational degrees of freedom from the internal ones for a quantum body system in dimensions is generally discussed.

Global symmetries of timedependent Schrödinger equations
View Description Hide DescriptionSome symmetries of timedependent Schrödinger equations for inverse quadratic, linear, and quadratic potentials have been systematically examined by using a method suitable to the problem. Especially, the symmetry group for the case of the linear potential turns out to be a semidirect product of the with a twodimensional real translation group Here, the time variable transforms as for real constants and satisfying with an accompanying transformation for the space coordinate
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Nonlinear transforms of momenta and Planck scale limit
View Description Hide DescriptionStarting with the generators of the Poincaré group for arbitrary mass (m) and spin (s), a nonunitary transformation is implemented to obtain momenta with an absolute Planck scale limit. In the rest frame (for the transformed energy coincides with the standard one, both being As the latter tends to infinity under Lorentz transformations the former tends to a finite upper limit where is the Planck length and the massdependent nonleading terms vanish exactly for zero rest mass. The invariant is conserved for the transformed momenta. The speed of light continues to be the absolute scale for velocities. We study various aspects of the kinematics in which two absolute scales have been introduced in this specific fashion. The precession of polarization and transformed position operators are among them. A deformation of the Poincaré algebra to the de Sitter one permits the implementation of our transformation in the latter case. A supersymmetric extension of the Poincaré algebra is also studied in this context.

Determination of quantum symmetries for higher systems from the modular matrix
View Description Hide DescriptionWe show that the Ocneanu algebra of quantum symmetries, for an diagram (or for higher Coxeter–Dynkin systems, like the Di Francesco–Zuber system) can be, in most cases, deduced from the structure of the modular matrix in the series. We recover in this way the (known) quantum symmetries of su(2) diagrams and illustrate our method by studying those associated with the three genuine exceptional diagrams of type su(3), namely, and This also provides the shortest way to the determination of twisted partition functions in boundary conformal field theory with defect lines.

“Massive” spin2 field in de Sitter space
View Description Hide DescriptionIn this paper we present a covariant quantization of the “massive” spin2 field on de Sitter (dS) space. By “massive” we mean a field which carries a specific principal series representation of the dS group. The work is in the direct continuation of previous ones concerning the scalar, the spinor, and the vector cases. The quantization procedure, independent of the choice of the coordinate system, is based on the WightmanGärding axiomatic and on analyticity requirements for the twopoint function in the complexified pseudoRiemanian manifold. Such a construction is necessary in view of preparing and comparing with the dS conformal spin2 massless case (dS linear quantum gravity) which will be considered in a forthcoming paper and for which specific quantization methods are needed.

Improved Epstein–Glaser renormalization. II. Lorentz invariant framework
View Description Hide DescriptionThe Epstein–Glaser type subtraction introduced by one of the authors in a previous paper is extended to the Lorentz invariant framework. The advantage of using our subtraction instead of the Epstein and Glaser standard subtraction method is especially important when working in Minkowski space, as then the counterterms necessary to keep Lorentz invariance are simplified. We show how renormalization of primitive diagrams in the Lorentz invariant framework directly relates to causal Riesz distributions. A covariant subtraction rule in momentum space is found, sharply improving upon the BPHZL method for massless theories.

Dynamical structure of irregular constrained systems
View Description Hide DescriptionHamiltonian systems with functionally dependent constraints (irregular systems), for which the standard Dirac procedure is not directly applicable, are discussed. They are classified according to their behavior in the vicinity of the constraint surface into two fundamental types. If the irregular constraints are multilinear (type I), then it is possible to regularize the system so that the Hamiltonian and Lagrangian descriptions are equivalent. When the constraints are power of a linear function (type II), regularization is not always possible and the Hamiltonian and Lagrangian descriptions may be dynamically inequivalent. It is shown that the inequivalence between the two formalisms can occur if the kinetic energy is an indefinite quadratic form in the velocities. It is also shown that a system of type I can evolve in time from a regular configuration into an irregular one, without any catastrophic changes. Irregularities have important consequences in the linearized approximation to nonlinear theories, as well as for the quantization of such systems. The relevance of these problems to Chern–Simons theories in higher dimensions is discussed.
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 GENERAL RELATIVITY AND GRAVITATION


Detecting ill posed boundary conditions in general relativity
View Description Hide DescriptionA persistent challenge in numerical relativity is the correct specification of boundary conditions. In this work we consider a manyparameter family of symmetric hyperbolic initialboundary value formulations for the linearized Einstein equations and analyze its well posedness using the Laplace–Fourier technique. By using this technique ill posed modes can be detected and thus a necessary condition for well posedness is provided. We focus on the following types of boundary conditions:boundary conditions that have been shown to preserve the constraints, boundary conditions that result from setting the ingoing constraint characteristic fields to zero, and boundary conditions that result from considering the projection of Einstein’s equations along the normal to the boundary surface. While we show that in case there are no ill posed modes, our analysis reveals that, unless the parameters in the formulation are chosen with care, there exist ill posed constraint violating modes in the remaining cases.

Dynamical system approach to FRW models in higherorder gravity theories
View Description Hide DescriptionWe study the late time evolution of positively curved FRW models with a scalar field which arises in the conformal frame of the theory. The resulted threedimensional dynamical system has two equilibrium solutions corresponding to a de Sitter space and an ever expanding closed universe. We analyze the structure of the first equilibrium with the methods of the center manifoldtheory and, for the second equilibrium, we apply the normal form theory to obtain a simplified system, which we analyze with special phase plane methods. It is shown that an initially expanding closed FRW space–time avoids recollapse.
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 DYNAMICAL SYSTEMS


Generalized variational principle of Herglotz for several independent variables. First Noethertype theorem
View Description Hide DescriptionThis paper extends the generalized variational principle of Herglotz to one with several independent variables and derives the corresponding generalized Euler–Lagrange equations. The extended principle contains the classical variational principle with several independent variables and the variational principle of Herglotz as special cases. A first Noethertype theorem is proven for the new variational principle, which gives the conserved quantities corresponding to symmetries of the associated functional. This theorem contains the classical first Noether theorem as a special case. As examples for applications we calculate a conserved quantity for the damped nonlinear Klein–Gordon equation and we show that the equations which describe the propagation of electromagnetic fields in a conductive medium can be derived from the generalized variational principle of Herglotz (but not from a classical variational principle).

Two choices of the gauge transformation for the AKNS hierarchy through the constrained KP hierarchy
View Description Hide DescriptionOn the basis of the equivalence between the AKNS hierarchy and the cKP hierarchy with the constraint we point out that there exist two choices to keep the form of the Lax operator when we perform the gauge transformation for the AKNS hierarchy, which results in two classes of functions to trigger the gauge transformation. For the second choice, two theorems for two types of gauge transformation are established. Several new and more general forms of taufunctions for the AKNS hierarchy are obtained by means of gauge transformations of both types. The union of the two choices leads to new forms of τfunctions. We generate the AKNS hierarchy from the “free” Lax operator via a chain of gauge transformations.

TriHamiltonian formulation for certain integrable lattice equations
View Description Hide DescriptionA systematic investigation of integrable differential–difference equations with two independent variables admitting multiHamiltonian structure is presented. Considering the Volterra (VL), Toda (TL), Relativistic Toda (RT), Belov–Chaltikian (BC) and Blaszak–Marciniak both three (BM3) and four (BM4) coupled lattice equations it is shown that they admit a sequence of operators out of which only three are Hamiltonian ones and so they are triHamiltonian systems only. It is observed that the constructed third operator for VL and BC lattice equations is Hamiltonian only if the field variable is periodic with even period.

Spectral curve and Hamiltonian structure of isomonodromic SU(2) Calogero–Gaudin system
View Description Hide DescriptionThis paper presents a new approach to the Hamiltonian structure of isomonodromic deformations of a matrix system of ordinary differential equations (ODEs) on a torus. An isomonodromic analogue of the SU(2) Calogero–Gaudin system is used for a case study of this approach. A clue of this approach is a mapping to a finite number of points on the spectral curve of the isomonodromic Lax equation. The coordinates of these moving points give a new set of Darboux coordinates called the spectral Darboux coordinates. The system of isomonodromic deformations is thereby converted to a nonautonomous Hamiltonian system in the spectral Darboux coordinates. The Hamiltonians turn out to resemble those of a previously known isomonodromic system of a secondorder scalar ODE. The two isomonodromic systems are shown to be linked by a simple relation.

Extended multilinear variable separation approach and multivalued localized excitations for some dimensional integrable systems
View Description Hide DescriptionThe multilinear variable separation approach and the related “universal” formula have been applied to many dimensional nonlinear systems. Starting from the universal formula, abundant dimensional localized excitations have been found. In this paper, the universal formula is extended in two different ways. One is obtained for the modified Nizhnik–Novikov–Veselov equation such that two universal terms can be combined linearly and this type of extension is also valid for the dimensional symmetric sineGordon system. The other is for the dispersive long wave equation, the Broer–Kaup–Kupershmidt system, the higher order Broer–Kaup–Kupershmidt system, and the Burgers system where arbitrary number of variable separated functions can be involved. Because of the existence of the arbitrary functions in both the original universal formula and its extended forms, the multivalued functions can be used to construct a new type of localized excitations, folded solitary waves (FSWs) and foldons. The FSWs and foldons may be “folded” in quite complicated ways and possess quite rich structures and multiplicate interaction properties.
