Volume 46, Issue 7, July 2005
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Hybrid states of two and three level atoms
View Description Hide DescriptionWe calculate atomphoton resonances in the WignerWeisskopf model, admitting two photons and choosing a particular coupling function. We also present a rough description of the set of resonances in a model for a threelevel atom coupled to a scalarphoton field. We give a general picture of matterfield resonances that these results fit into.

Principal bundle structure of quantum adiabatic dynamics with a Berry phase which does not commute with the dynamical phase
View Description Hide DescriptionA geometric model is proposed to describe the Berry phase phenomenon when the geometric phase does not commute with the dynamical phase. The structure used is a principal composite bundle in which the adiabatic transport appears as a horizontal lift. The formulation is applied to a simple quantum dynamical system controlled by two lasers.

Canonical quantization of SU(3) Skyrme model in a general representation
View Description Hide DescriptionA complete canonical quantization of the SU(3) Skyrme model performed in the collective coordinate formalism in general irreducible representations. In the case of SU(3) the model differs qualitatively in different representations. The WessZuminoWitten term vanishes in all selfadjoint representations in the collective coordinate method for separation of space and time variables. The canonical quantization generates representation dependent quantum mass corrections, which can stabilize the soliton solution. The standard symmetry breaking mass, which in general leads to representation mixing, degenerates to the SU(2) form in all selfadjoint representations.

Changes in nonlinear potential scattering theory in electron gases brought about by reducing dimensionality
View Description Hide DescriptionRecent work has shown the essential equivalence of stopping power, forceforce correlation function, and phaseshift analysis for nonlinear potential scattering in a threedimensional electron gas. In the present study, we first demonstrate that the above situation is markedly different when the scattering occurs from a localized potential in a twodimensional (2D) electron gas. Only to second order in the potential do the three methods referred to above precisely agree. However, all these methods can still be applied in 2D, some fully nonlinear evaluation proving possible. The onedimensional case is also discussed, albeit more briefly. Scattering from a twocenter modeling of the localized potential is also calculated, but now only in the Born approximation, due to the added complication of a noncentral potential.

Spectral and localization properties for the onedimensional Bernoulli discrete Dirac operator
View Description Hide DescriptionAn onedimensional (1D) Dirac tightbinding model is considered and it is shown that its nonrelativistic limit is the 1D discrete Schrödinger model. For random Bernoulli potentials taking two values (without correlations), for typical realizations and for all values of the mass, it is shown that its spectrum is pure point, whereas the zero mass case presents dynamical delocalization for specific values of the energy. The massive case presents dynamical localization (excluding some particular values of the energy). Finally, for general potentials the dynamical moments for distinct masses are compared, especially the massless and massive Bernoulli cases.

component deformed charge coherent states and their nonclassical properties
View Description Hide Descriptioncomponent deformed chargecoherent states are constructed, their (over)completeness proved and their generation explored. The deformed chargecoherent states and the even (odd) deformed chargecoherent states are the two special cases of them as becomes 1 and 2, respectively. A algebra realization of the generators is given in terms of them. Their nonclassical properties are studied and it is shown that for , they exhibit twomode antibunching, but neither squeezing, nor one or twomode squeezing.

Stargenfunctions, generally parametrized systems and a causal formulation of phase space quantum mechanics
View Description Hide DescriptionWe address the deformation quantization of generally parametrized systems displaying a natural time variable. The purpose of this exercise is twofold: first, to illustrate through a pedagogical example the potential of quantum phase space methods in the context of constrained systems and particularly of generally covariant systems. Second, to show that a causal representation for quantum phase space quasidistributions can be easily achieved through general parametrization. This result is succinctly discussed.

Quantum phase space for the onedimensional hydrogen atom on the hyperbola
View Description Hide DescriptionWe obtain the solution of the onedimensional hydrogen atom on the hyperbola by transforming its Schrödinger equation into the modified PöschlTeller equation. We write explicitly the wave functions both in configuration and momentum space, and check the contraction limit of the system to the flat space model. Finally, we find the closed form of the Wigner function for the states of this system.
 Top

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


NonAbelian conversion and quantization of nonscalar secondclass constraints
View Description Hide DescriptionWe propose a general method for the deformation quantization of any secondclass constrained system on a symplectic manifold. The constraints determining an arbitrary constraint surface are in general defined only locally and can be components of a section of a nontrivial vector bundle over the phasespace manifold. The covariance of the construction with respect to the change of the constraint basis is provided by introducing a connection in the “constraint bundle,” which becomes a key ingredient of the conversion procedure for the nonscalar constraints. Unlike in the case of scalar secondclass constraints, no Abelian conversion is possible in general. Within the BRST framework, a systematic procedure is worked out for converting nonscalar secondclass constraints into nonAbelian firstclass ones. The BRSTextended system is quantized, yielding an explicitly covariant quantization of the original system. An important feature of secondclass systems with nonscalar constraints is that the appropriately generalized Dirac bracket satisfies the Jacobi identity only on the constraint surface. At the quantum level, this results in a weakly associative starproduct on the phase space.

Local superfield Lagrangian BRST quantization
View Description Hide DescriptionA local formulation of superfield Lagrangian quantization in nonAbelian hypergauges is proposed on the basis of an extension of general reducible gauge theories to special superfield models with a Grassmann parameter . We solve the problem of describing the quantum action and the gauge algebra of an stagereducible superfield model in terms of a BRST charge for a formal dynamical system with firstclass constraints of stage reducibility. Starting from local functions of the quantum and gaugefixing actions, with an essential use of Darboux coordinates on the antisymplectic manifold, we construct a superfield generatingfunctionals of Green’s functions, including the effective action. We present two superfield forms of BRST transformations, considered as shifts along vector fields defined by Hamiltonianlike systems constructed in terms of the quantum and gaugefixing actions and an arbitrary local boson function, as well as in terms of corresponding fermion functionals, through Poisson brackets with opposite Grassmann parities. The gauge independence of the Smatrix is proved. The Ward identities are derived. Connection is established with the BV method, the multilevel BatalinTyutin formalism, as well as with the superfield quantization scheme of Lavrov, Moshin, and Reshetnyak, extended to the case of general coordinates.

On a classification of irreducible almost commutative geometries. III
View Description Hide DescriptionWe extend a classification of irreducible, almost commutative geometries whose spectral action is dynamically nondegenerate to internal algebras that have four simple summands.

SeibergWitten equations in : Lie symmetries, particular solutions, integrability
View Description Hide DescriptionIt is shown that the 11parameter automorphism group of Lie algebra of fourdimensional Euclidean group is a maximal Lie symmetries group of the SeibergWitten equations in . Particular explicit solutions which are invariant under SO(3) subgroups of the maximal Lie symmetries group are constructed. It is established that SeibergWitten equations do not possess the Painlevé property. Nevertheless, SO(3) invariant solutions obtained are turned to admit a characteristic singularity structure.

A generalization of ConnesKreimer Hopf algebra
View Description Hide Description“Bonsai” Hopf algebras, introduced here, are generalizations of ConnesKreimer Hopf algebras, which are motivated by Feynman diagrams and renormalization. We show that we can find operad structure on the set of bonsais. We introduce a differential on these bonsai Hopf algebras, which is inspired by the tree differential. The cohomologies of these are computed here, and the relationship of this differential with the appending operation ∗ of ConnesKreimer Hopf algebras is investigated.

Bosonic colorflavor transformation for the special unitary group
View Description Hide DescriptionWe extend Zirnbauer’s colorflavor transformation in the bosonic sector to the color group . Because the flavor group is noncompact, the algebraic method by which the original colorflavor transformation was derived leads to a useful result only for . Using the character expansion method, we obtain a different form of the transformation in the extended range . This result can also be used for the color group . The integrals to which the transformation can be applied are of relevance for the recently proposed bosoninduced lattice gauge theory.

Integration of massive states as contractions of nonlinear models
View Description Hide DescriptionWe consider the contraction of some nonlinear models which appear in effective supergravitytheories. In particular we consider the contractions of maximally symmetric spaces corresponding to and theories, as they appear in certain low energy effective supergravity actions with mass deformations. The contraction procedure is shown to describe the integrating out of massive modes in the presence of interactions, as it happens in many supergravity models after spontaneous supersymmetry breaking.

Near threshold expansion of Feynman diagrams
View Description Hide DescriptionThe near threshold expansion of Feynman diagrams is derived from their configuration space representation, by performing all integrations. The general scalar Feynman diagram is considered, with an arbitrary number of external momenta, an arbitrary number of internal lines and an arbitrary number of loops, in dimensions and all masses may be different. The expansions are considered both below and above threshold. Rules, giving real and imaginary part, are derived. Unitarity of a sunset diagram with internal lines is checked in a direct way by showing that its imaginary part is equal to the phase space integral of particles.
 Top

 GENERAL RELATIVITY AND GRAVITATION


Laplacian in the hyperbolic space and linearization stability of the Einstein equation for RobertsonWalker models
View Description Hide DescriptionWe prove that some operators related to the rough Laplacian in the hyperbolic space give isomorphisms between Sobolev spaces of 1forms. By using these results we prove that the Einstein equation of the hyperbolic RobertsonWalker cosmological model is linearization stable. We also study the linearization stability for RobertsonWalker models, , with compact, complete, having either constant negative or zero curvature.

Birkhoff’s theorem in Lovelock gravity
View Description Hide DescriptionWe show that the solutions of the Lovelock equations with spherical, planar, or hyperbolic symmetry are locally isometric to the corresponding static Lovelock black hole. As a consequence, these solutions are locally static: they admit an additional Killing vector that can either be spacelike or timelike, depending on the position. This result also holds in the presence of an abelian gauge field, in which case the solutions are locally isometric to a charged static black hole.
 Top

 DYNAMICAL SYSTEMS


The Kepler problem with anisotropic perturbations
View Description Hide DescriptionWe study a twobody problem given by the sum of the Newtonian potential and an anisotropic perturbation that is a homogeneous function of degree , . For , the sets of initial conditions leading to collisions∕ejections and the one leading to escapes∕captures have positive measure. For and , the flow on the zeroenergy manifold is chaotic. For , a case we prove integrable, the infinity manifold of the zeroenergy level has heteroclinic connections with the collision manifold.

Boundedness, invariant algebraic surfaces and global dynamics for a spectral model of largescale atmospheric circulation
View Description Hide DescriptionWe consider a threedimensional quadratic system S in with six parameters which appears in geophysical fluid dynamics (atmospheric blocking). In this paper we start its systematic study from the point of view of dynamical systems. First, we reduce the number of its parameters from six to three. Thus, we must study a threedimensional quadratic system with three parameters, which recalls us the famous Lorenz63 system. Traditionally, system S has been studied by considering two subcases, called the conservative and the dissipative case, as the parameter responsible for dissipation is zero or not. In the conservative case, we reduce system S to systems without parameters. Among these there are two interesting systems: one is homeomorphic to the simple pendulum, and the other is a perturbation of it. In the latter system the saddle point corresponding to topographic instability is connected to two homoclinic orbits to it. In the dissipative case we prove that all trajectories of system S enter in an ellipsoid for any values of the parameters. We characterize their invariant algebraic surfaces of degree 2, and for those systems having such invariant algebraic surfaces we describe their global phase portraits.
