Volume 43, Issue 12, December 2002
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Wigner functions for curved spaces. I. On hyperboloids
View Description Hide DescriptionWe propose a Wigner quasiprobability distribution function for Hamiltonian systems in spaces of constant curvature, in this article on hyperboloids, which returns the correct marginals and has the covariance of the Shapiro functions under transformations. To the free systems obeying the Laplace–Beltrami equation on the hyperboloid, we add a conicoscillator potential in the hyperbolic coordinate. As an example, we analyze the onedimensional case on a hyperbola branch, where this conicoscillator is the Pöschl–Teller potential. We present the analytical solutions and plot the computed results. The standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature.

Fixed points of quantum operations
View Description Hide DescriptionQuantum operations frequently occur in quantum measurement theory, quantum probability, quantum computation, and quantum informationtheory. If an operator is invariant under a quantum operation φ, we call a φfixed point. Physically, the φfixed points are the operators that are not disturbed by the action of φ. Our main purpose is to answer the following question. If is a φfixed point, is compatible with the operation elements of φ? We shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes. Our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von Neumann algebras.

Classicality criteria
View Description Hide DescriptionWe present two possible criteria quantifying the degree of classicality of an arbitrary (finite dimensional) dynamical system. The inputs for these criteria are the classical dynamical structure of the system together with the quantum and the classical data providing the two alternative descriptions of its initial time configuration. It is proved that a general quantum system satisfying the criteria up to some extend displays a time evolution consistent with the classical predictions up to some degree and thus it is argued that the criteria provide a suitable measure of classicality. The features of the formalism are illustrated through two simple examples.

Superintegrability with thirdorder integrals in quantum and classical mechanics
View Description Hide DescriptionWe consider here the coexistence of first and thirdorder integrals of motion in twodimensional classical and quantum mechanics. We find explicitly all potentials that admit such integrals, and all their integrals. Quantum superintegrable systems are found that have no classical analog, i.e., the potentials are proportional to so their classical limit is free motion.

Convexity and potential sums for Salpetertype Hamiltonians
View Description Hide DescriptionThe semirelativistic Hamiltonian where is a central potential in is concave in and convex in This fact enables us to obtain complementary energy bounds for the discrete spectrum of By extending the notion of “kinetic potential” we are able to find general energy bounds on the groundstate energy corresponding to potentials with the form In the case of sums of powers and the potential, where the bounds can all be expressed in the semiclassical form “Upper” and “lower” numbers are provided for and for the log potential Some specific examples are discussed, to show the quality of the bounds.

Magnetic translation groups in an ndimensional torus and their representations
View Description Hide DescriptionA charged particle in a uniform magnetic field in a twodimensional torus has a discrete noncommutative translation symmetry instead of a continuous commutative translation symmetry. We study topology and symmetry of a particle in a magnetic field in a torus of arbitrary dimensions. The magnetic translation group (MTG) is defined as a group of translations that leave the gauge field invariant. We show that the MTG in an dimensional torus is isomorphic to a central extension of a cyclic group by with We construct and classify irreducible unitary representations of the MTG in a threetorus and apply the representation theory to three examples. We briefly describe a representation theory for a general torus. The MTG in an torus can be regarded as a generalization of the socalled noncommutative torus.

Theory and application of Fermi pseudopotential in one dimension
View Description Hide DescriptionThe theory of interaction at one point is developed for the onedimensional Schrödinger equation. In analog with the threedimensional case, the resulting interaction is referred to as the Fermi pseudopotential. The dominant feature of this onedimensional problem comes from the fact that the real line becomes disconnected when one point is removed. The general interaction at one point is found to be the sum of three terms, the wellknown deltafunction potential and two Fermi pseudopotentials, one odd under space reflection and the other even. The odd one gives the proper interpretation for the potential, while the even one is unexpected and more interesting. Among the many applications of these Fermi pseudopotentials, the simplest one is described. It consists of a superposition of the deltafunction potential and the even pseudopotential applied to twochannel scattering. This simplest application leads to a model of the quantum memory, an essential component of any quantum computer.

Extension of Bethe ansatz to multiple occupancies for onedimensional SU(4) fermions with δfunction interaction
View Description Hide DescriptionWe consider the problem of consistence between the Bethe ansatz (BA) wave function and the multiparticle (more than two) scattering in onedimensional δfunction interacting SU(4) fermions, which the approach of BA does not explicitly take into account. We find the scattering conditions of three and four particles located at the same position and show that the conditions can be fulfilled by the twoparticle connection conditions of the BA wave function. So the definition of the BA wave function can be exactly extended to those cases with multiple occupancies. The inconsistence between the BA and multiparticle interacting on a same site in the degenerate Hubbard model, which makes the BA fail for the model, is shown to vanish in the limit of small site spacing. A correspondence relation of the BA equation and SU(4) symmetry of the system is also indicated for the fermions. The degeneracy of state with BA eigenenergy is given. Singlet lies in the case when there are equal numbers of particles in each inner component.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Results on the Wess–Zumino consistency condition for arbitrary Lie algebras
View Description Hide DescriptionThe socalled covariant Poincaré lemma on the induced cohomology of the space–time exterior derivative in the cohomology of the gauge part of the BRST differential is extended to cover the case of arbitrary, nonreductive Lie algebras. As a consequence, the general solution of the Wess–Zumino consistency condition with a nontrivial descent can, for arbitrary (super) Lie algebras, be computed in the small algebra of the oneform potentials, the ghosts and their exterior derivatives. For particular Lie algebras that are the semidirect sum of a semisimple Lie subalgebra with an ideal, a theorem by Hochschild and Serre is used to characterize more precisely the cohomology of the gauge part of the BRST differential in the small algebra. In the case of an Abelian ideal, this leads to a complete solution of the Wess–Zumino consistency condition in this space. As an application, the consistent deformations of dimensional Chern–Simons theory based on are rediscussed.

Blackbrane solution for algebra
View Description Hide DescriptionBlack brane solutions for a wide class of intersection rules and Ricciflat “internal” spaces are considered. They are defined up to moduli functions obeying nonlinear differential equations with certain boundary conditions imposed. A new solution with intersections corresponding to the Lie algebra is obtained. The functions and for this solution are polynomials of degree 3 and 4.

Cluster properties in relativistic quantum mechanics of particle systems
View Description Hide DescriptionA general technique is presented for constructing a quantum theory of a finite number of interacting particles satisfying Poincaré invariance, cluster separability, and the spectral condition. Irreducible representations and Clebsch–Gordan coefficients of the Poincaré group are the central elements of the construction. A different realization of the dynamics is obtained for each basis of an irreducible representation of the Poincaré group. Unitary operators that relate the different realizations of the dynamis are constructed. This technique is distinguished from other solutions [S. N. Sokolov, Dokl. Akad. Nauk USSR 233, 575 (1977); F. Coester and W. N. Polyzou, Phys. Rev. D 26, 1348 (1982)] of this problem because it does not depend on the kinematic subgroups of Dirac’s forms [P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949)] of dynamics. Special basis choices lead to kinematic subgroups.

Exactly solvable models of relativistic δsphere interactions in quantum mechanics
View Description Hide DescriptionWe discuss the quantum Hamiltonian describing a δsphere interaction introduced in [J. Math. Phys 30, 2275 (1989)] and formally given by where is the Dirac Hamiltonian and is a real matrix defined by We obtain a series of new results for this model, in particular the resolvent equation, the spectral properties, the nonrelativistic limit and the various quantities related to the scattering theory. These results are generalized to the case of an asymmetric δsphere interaction and a δsphere plus Coulomb interaction, respectively.

Boundary states, extended symmetry algebra, and module structure for certain rational torus models
View Description Hide DescriptionThe massless bosonic field compactified on the circle of rational is reexamined in the presense of boundaries. A particular class of models corresponding to is distinguished by demanding the existence of a consistent set of Neumann boundary states. The boundary states are constructed explicitly for these models and the fusion rules are derived from them. These are the ones prescribed by the Verlinde formula from the matrix of the theory. In addition, the extended symmetry algebra of these theories is constructed, which is responsible for the rationality of these theories. Finally, the chiral space of these models is shown to split into a direct sum of irreducible modules of the extended symmetry algebra.
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 GENERAL RELATIVITY AND GRAVITATION


Scalartensor gravity and conformal continuations
View Description Hide DescriptionGlobal properties of vacuum static, spherically symmetric configurations are studied in a general class of scalartensor theories (STTs) of gravity in various dimensions. The conformal mapping between the Jordan and Einstein frames is used as a tool. Necessary and sufficient conditions are found for the existence of solutions admitting a conformal continuation (CC). The latter means that a singularity in the Einsteinframe manifold maps to a regular surface in the Jordan frame, and the solution is then continued beyond this surface. can be an ordinary regular sphere or a horizon. In the second case, connect two epochs of a KantowskiSachs type cosmology. It is shown that the list of possible types of global causal structure of vacuum space–times in any STT, with any potential function is the same as in general relativity with a cosmological constant. This is even true for conformally continued solutions. A traversable wormhole is shown to be one of the generic structures created as a result of CC. Two explicit examples are presented: the known solution for a conformal field in general relativity, illustrating the emergence of singularities and wormholes due to CC, and a nonsingular threedimensional model with an infinite sequence of CCs.
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 DYNAMICAL SYSTEMS


On the nonlocal equations and nonlocal charges associated with the Harry Dym hierarchy
View Description Hide DescriptionA large class of nonlocal equations and nonlocal charges for the Harry Dym hierarchy is exhibited. They are obtained from nonlocal Casimirs associated with its biHamiltonian structure. The Lax representation for some of these equations is also given.

Infinite symmetries and conservation laws
View Description Hide DescriptionWe will consider partial differential equations of a variational problem whose symmetry group generators contain arbitrary function(s) of one or more independent variables. Unlike the Second Noether Theorem we will be interested in the case of arbitrary functions of not all base variables. We will study the relations between infinite symmetries and local conservation laws. We will demonstrate that infinite symmetries may lead to a finite number of conservation laws through appropriate boundary conditions, or to a set of additional constraints for the function and its derivatives.

Solitonlike solutions of higher order wave equations of the Korteweg–de Vries type
View Description Hide DescriptionIn this work we study second and third order approximations of water wave equations of the Korteweg–de Vries (KdV) type. First we derive analytical expressions for solitary wave solutions for some special sets of parameters of the equations. Remarkably enough, in all these approximations, the form of the solitary wave and its amplitudevelocity dependence are identical to the formula of the onesoliton solution of the KdV. Next we carry out a detailed numerical study of these solutions using a Fourier pseudospectral method combined with a finitedifference scheme, in parameter regions where solitonlike behavior is observed. In these regions, we find solitary waves which are stable and behave like solitons in the sense that they remain virtually unchanged under time evolution and mutual interaction. In general, these solutions sustain small oscillations in the form of radiation waves (trailing the solitary wave) and may still be regarded as stable, provided these radiation waves do not exceed a numerical stability threshold. Instability occurs at high enough wave speeds, when these oscillations exceed the stability threshold already at the outset, and manifests itself as a sudden increase of these oscillations followed by a blowup of the wave after relatively short time intervals.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Driven Newton equations and separable timedependent potentials
View Description Hide DescriptionWe present a class of timedependent potentials in that can be integrated by separation of variables: by embedding them into socalled cofactor pair systems of higher dimension, we are led to a timedependent change of coordinates that allows the time variable to be separated off, leaving the remaining part in separable Stäckel form.
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 FLUIDS


Uniqueness in MHD in divergence form: Right nullvectors and wellposedness
View Description Hide DescriptionMagnetohydrodynamics in divergence form describes a hyperbolic system of covariant and constraintfree equations. It comprises a linear combination of an algebraic constraint and Faraday’s equations. Here, we study the problem of wellposedness, and identify a preferred linear combination in this divergence formulation. The limit of weak magnetic fields shows the slow magnetosonic and Alfvén waves to bifurcate from the contact discontinuity (entropywaves), while the fast magnetosonic wave is a regular perturbation of the hydrodynamical sound speed. These results are further reported as a starting point for characteristic based shock capturing schemes for simulations with ultrarelativistic shocks in magnetized relativistic fluids.
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 STATISTICAL PHYSICS


Fourparticle decay of the Bethe–Salpeter kernel in the hightemperature Ising model
View Description Hide DescriptionIn this article we study the fourparticle decay of the Bethe–Salpeter (BS) kernel for the hightemperature Ising model. We use the hyperplane decoupling method [T. Spencer, Commun. Math. Phys. 44, 143 (1975); R. S. Schor, Nucl. Phys. B 222, 71 (1983)] to prove exponential decay in a set of variables particularly adapted to the methods of Spencer and Zirilli [Commun. Math. Phys. 49, 1 (1976)] for the analysis of scattering and bound states in QFT, transcribed to lattice theories by Auil and Barata [Ann. Henri Poincare 2, 1065 (2001)]. We study arbitrary derivatives of the general point correlation functions with respect to the interpolating variables, and we are able to obtain, in some cases, information about the third derivatives of the BS kernel. As a later consequence, we have twobody asymptotic completeness for the (massive) Euclidean lattice field theory implemented by this model. This allows us to analyze the Ornstein–Zernike behavior of fourpoint functions, related to the specific heat of the model.
