Volume 47, Issue 4, April 2006
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


The WignerWeyl transformation and the quantum path integral
View Description Hide DescriptionWe show that the space of quantum states for a spinless particle possesses a (trivial) fiber bundle structure where is the classical phase space. This geometric point of view allows us to define a “quantum path integral” that connects quantum observables with their classical counterparts. We show that this path integral reduces in fact to the WignerWeyl transformation.

Reduced Gutzwiller formula with symmetry: Case of a finite group
View Description Hide DescriptionWe consider a classical Hamiltonian on , invariant by a finite group of symmetry , whose Weyl quantization is a selfadjoint operator on . If is an irreducible character of , we investigate the spectrum of its restriction to the symmetry subspace of coming from the decomposition of PeterWeyl. We give reduced semiclassical asymptotics of a regularized spectral density describing the spectrum of near a noncritical energy . If is compact, assuming that periodic orbits are nondegenerate in , we get a reduced Gutzwiller trace formula which makes periodic orbits of the reduced space appear. The method is based upon the use of coherent states, whose propagation was given in the work of Combescure and Robert.

Reverse inequalities in deformed SegalBargmann analysis
View Description Hide DescriptionWe prove reverse hypercontractivity inequalities as well as reverse logSobolev inequalities in the context of a space of holomorphic functions, which is called the deformed SegalBargmann space and arises in the works of Wigner, Rosenblum, and Marron. To achieve this we define deformations of energy and entropy. Our principle results generalize earlier works of Carlen and Sontz. We also show that the semigroup of this theory is bounded, and we conjecture that it is contractive and, even more strongly, that it is hypercontractive.

Phaseintegral calculation of the phase of the regular Coulomb wave function
View Description Hide DescriptionThe phase of the regular Coulomb wave function is calculated by means of the phaseintegral approximation of arbitrary order with a convenient choice of the base function. The result agrees with the most accurate asymptotic expansion of the exact expression, , for the phase, truncated at an arbitrary order of approximation. It is seldom the case that the phaseintegral expression for a physical quantity can be obtained in an arbitrary order of the approximation, and it is remarkable that in the present case this expression is the same as that obtained from the most accurate asymptotic formula for the quantity in question.

Motion of a charged particle in a periodic AharonovBohm potential
View Description Hide DescriptionIn this paper we introduce a periodic AharonovBohm potential, produced by two parallel infinite thin layers. The motion of a quantum charged particle without spin, under the action of this AharonovBohm potential is examined. Finally we introduce a periodic AharonovBohm potential, produced by a system of parallel solenoids and the motion of the charged particle in this field is also examined.

New multiplicativity results for qubit maps
View Description Hide DescriptionLet be a tracepreserving, positivitypreserving (but not necessarily completely positive) linear map on the algebra of complex matrices, and let be any finitedimensional completely positive map. For and , we prove that the maximal norm of the product map is the product of the maximal norms of and . Restricting to the class of completely positive maps, this settles the multiplicativity question for all qubit channels in the range of values .

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


Superconformal symmetry in the interacting theory of (2,0) tensor multiplets and selfdual strings
View Description Hide DescriptionWe investigate the concept of superconformal symmetry in six dimensions, applied to the interacting theory of (2,0) tensor multiplets and selfdual strings. The action of a superconformal transformation on the superspace coordinates is found, both from a sixdimensional perspective and by using a superspace with eight bosonic and four fermionic dimensions. The transformation laws for all fields in the theory are derived, as well as general expressions for the transformation of onshell superfields. Superconformal invariance is shown for the interaction of a selfdual string with a background consisting of onshell tensor multiplet fields, and we also find an interesting relationship between the requirements of superconformal invariance and those of a local fermionic symmetry. Finally, we try to construct a superspace analogue of the Poincaré dual to the string worldsheet and consider its properties under superconformal transformations.

Friedel theorem for two dimensional relativistic spin systems
View Description Hide DescriptionThe Friedel sum rule is generalized to relativistic systems of spin particles in two dimensions. The change in energy due to the presence of an impurity is studied. The relation of the sum rule with the relativistic Levinson theorem is presented. Density oscillations in such systems are discussed. Since the Friedel theorem has been of major importance in understanding the impurity scattering in materials, the present results may be useful to explain some phenomena in two dimensional fermionic many body systems.

 GENERAL RELATIVITY AND GRAVITATION


The Cauchy problem for the wave equation in the Schwarzschild geometry
View Description Hide DescriptionThe Cauchy problem is considered for the scalar wave equation in the Schwarzschild geometry. We derive an integral spectral representation for the solution and prove pointwise decay in time.

Spacetime matter inflation
View Description Hide DescriptionWe study a model of powerlaw inflationary inflation using the spacetimematter theory of gravity for a fivedimensional canonical metric that describes an apparent vacuum. In this approach the expansion is governed by a single scalar (neutral) quantum field. In particular, we study the case where the power of expansion of the universe is . This kind of model is more successful than others in accounting for galaxy formation.

A Morsetheoretical analysis of gravitational lensing by a KerrNewman black hole
View Description Hide DescriptionConsider, in the domain of outer communication of a KerrNewman black hole, a point (observation event) and a timelike curve (worldline of light source). Assume that (i) has no past endpoint, (ii) does not intersect the caustic of the past light cone of , and (iii) goes neither to the horizon nor to infinity in the past. We prove that then for infinitely many positive integers there is a pastpointing lightlike geodesic of (Morse) index from to , hence an observer at sees infinitely many images of . Moreover, we demonstrate that all lightlike geodesics from an event to a timelike curve in are confined to a certain spherical shell. Our characterization of this spherical shell shows that in the KerrNewman spacetime the occurrence of infinitely many images is intimately related to the occurrence of centrifugalplusCoriolis force reversal.

Gravitational magnetic monopoles and MajumdarPapapetrou stars
View Description Hide DescriptionDuring the 1990s a large amount of work was dedicated to studying general relativity coupled to nonAbelian YangMills type theories. Several remarkable results were accomplished. In particular, it was shown that the magnetic monopole, a solution of the YangMillsHiggs equations can indeed be coupled to gravitation. For a low Higgs mass it was found that there are regular monopole solutions, and that for a sufficiently massive monopole the system develops an extremal magnetic ReissnerNordström quasihorizon with all the matter fields laying inside the horizon. These latter solutions, called quasiblack holes, although nonsingular, are arbitrarily close to having a horizon, and for an external observer it becomes increasingly difficult to distinguish these from a true black hole as a critical solution is approached. However, at precisely the critical value the quasiblack hole turns into a degenerate spacetime. On the other hand, for a high Higgs mass, a sufficiently massive monopole develops also a quasiblack hole, but at a critical value it turns into an extremal true horizon, now with matter fields showing up outside. One can also put a small Schwarzschild black hole inside the magnetic monopole, the configuration being an example of a nonAbelian black hole. Surprisingly, MajumdarPapapetrou systems, Abelian systems constructed from extremal dust (pressureless matter with equal charge and energy densities), also show a resembling behavior. Previously, we have reported that one can find MajumdarPapapetrou solutions which are everywhere nonsingular, but can be arbitrarily close of being a black hole, displaying the same quasiblackhole behavior found in the gravitational magnetic monopole solutions. With the aim of better understanding the similarities between gravitational magnetic monopoles and MajumdarPapapetrou systems, here we study a particular system, namely a system composed of two extremal electrically charged spherical shells (or stars, generically) in the EinsteinMaxwellMajumdarPapapetrou theory. We first review the gravitational properties of the magnetic monopoles, and then compare with the gravitational properties of the double extremal electric shell system. These quasiblackhole solutions can help in the understanding of true black holes, and can give some insight into the nature of the entropy of black holes in the form of entanglement.

Decoupling of the general scalar field mode and the solution space for Bianchi type I and V cosmologies coupled to perfect fluid sources
View Description Hide DescriptionThe scalar field degree of freedom in Einstein’s plus matter field equations is decoupled for Bianchi type I and V general cosmological models. The source, apart from the minimally coupled scalar field with arbitrary potential , is provided by a perfect fluid obeying a general equation of state. The resulting ODE is, by an appropriate choice of final time gauge affiliated to the scalar field, reduced to first order, and then the system is completely integrated for arbitrary choices of the potential and the equation of state.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Isochronous and partially isochronous Hamiltonian systems are not rare
View Description Hide DescriptionA technique is provided that allows to associate to a Hamiltonian another, modified, Hamiltonian, which reduces to the original one when the parameter vanishes, and for features an open, hence fully dimensional, region in its phase space where all its solutions are isochronous, i.e., completely periodic with the same period. The class of Hamiltonians to which this technique is applicable is large: it includes for instance the Hamiltonian characterizing the classical manybody problem with potentials that are translationinvariant but otherwise completely arbitrary, which is largely used in this paper to illustrate these findings. We also discuss variants of this technique that yield partially isochronous Hamiltonians, which also feature a region in their phase space where allsolutions are isochronous, that region having however a bit less than full dimensionality (for instance codimension one or two) in phase space.

Lowfrequency currents induced in adjacent spherical cells
View Description Hide DescriptionThe currents induced inside cells by external electric fields in the frequency range 50–60 Hz are studied analytically by accounting for thin cell membranes with transverse conductivity that is small compared to the conductivity of the saline fluid. A general perturbation scheme is formulated and applied to two adjacent spherical cells of equal radii by using a reflection principle and solving a nonlinear difference equation. The presence of the second cell is found to cause a no more than 10% increase to the current induced in an isolated spherical cell.

On spatial and material covariant balance laws in elasticity
View Description Hide DescriptionThis paper presents some developments related to the idea of covariance in elasticity. The geometric point of view in continuum mechanics is briefly reviewed. Building on this, regarding the reference configuration and the ambient space as Riemannian manifolds with their own metrics, a Lagrangian field theory of elastic bodies with evolving reference configurations is developed. It is shown that even in this general setting, the EulerLagrange equations resulting from horizontal (referential) variations are equivalent to those resulting from vertical (spatial) variations. The classical GreenNaghdiRivilin theorem is revisited and a material version of it is discussed. It is shown that energy balance, in general, cannot be invariant under isometries of the reference configuration, which in this case is identified with a subset of . Transformation properties of balance of energy under rigid translations and rotations of the reference configuration is obtained. The spatial covariant theory of elasticity is also revisited. The transformation of balance of energy under an arbitrary diffeomorphism of the reference configuration is obtained and it is shown that some nonstandard terms appear in the transformed balance of energy. Then conditions under which energy balance is materially covariant are obtained. It is seen that material covariance of energy balance is equivalent to conservation of mass, isotropy, material DoyleEricksen formula and an extra condition that we call configurational inviscidity. In the last part of the paper, the connection between Noether’s theorem and covariance is investigated. It is shown that the DoyleEricksen formula can be obtained as a consequence of spatial covariance of Lagrangian density. Similarly, it is shown that the material DoyleEricksen formula can be obtained from material covariance of Lagrangian density.

Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a twodimensional manifold
View Description Hide DescriptionIn this paper we prove that the twodimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the Poisson algebra of the integrals of motion. There are six general fundamental classes of superintegrable systems. Analytic formulas for the involved integrals are calculated in all the cases. All the known superintegrable systems are classified as special cases of these six general classes.

Hamiltonian theory of constrained impulsive motion
View Description Hide DescriptionThis paper considers systems subject to nonholonomic constraints which are not uniform on the whole configuration manifold. When the constraints change, the system undergoes a transition in order to comply with the new imposed conditions. Building on previous work on the Hamiltonian theory of impact, we tackle the problem of mathematically describing the classes of transitions that can occur. We propose a comprehensive formulation of the transition principle that encompasses the various impulsive regimes of Hamiltonian systems. Our formulation is based on the partial symplectic formalism, which provides a suitable framework for the dynamics of nonholonomic systems. We pay special attention to mechanical systems and illustrate the results with several examples.

 FLUIDS


Flow acoustics and linearized equations for ideal barotropic fluid flows
View Description Hide DescriptionIn this paper we analyze the acoustic flow in periodic structures for general timedependent problems. Mathematically, the acoustic problem reduces to analysis of the linearized Helmholtz equations of an ideal barotropic fluid in the neighbourhood of the stationary background flow. We show that in the linear approximation stationary flows are generally unstable. Explicit unbounded solutions are determined and, in some cases, expressions for the general solution of the acoustic problem are stated.

 STATISTICAL PHYSICS


On metastable regimes in stochastic Lamb system
View Description Hide DescriptionWe consider the long time behavior of the coupled Hamilton system of onedimensional string and nonlinear oscillator, in contact with a heat bath modeled by the white noise. For any temperature the system converges to a statistical equilibrium described by the Boltzmann equilibrium measure. The convergence is caused by radiation provided by the nonlinear coupling. If the oscillator potential has more than one well and the temperature is small, the relaxation time is large, and the system goes through a sequence of metastable states located near local minima of the potential. When both, the temperature and the radiation rate are small, the metastable states are distributions among the minima of the potential.
