Volume 44, Issue 12, December 2003
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Sharp reconstruction of unsharp quantum observables
View Description Hide DescriptionA well defined procedure exists which allows us to “reconstruct” a sharp, i.e., standard, quantum observable starting from a given commutative unsharp observable In this work we prove that the outcomes of measurements of can be consistently interpreted as the result of a stochastic diffusion of outcomes of its sharp reconstruction Furthermore, for every sharp observable such that is unsharp realization of we explicitly construct a real mapping such that

Lüders theorem for coherentstate POVMs
View Description Hide DescriptionLüders’ theorem states that two observables commute if measuring one of them does not disturb the measurement outcomes of the other. We study measurements which are described by continuous positive operatorvalued measurements (or POVMs) associated with coherent states on Lie groups. In general, operators turn out to be invariant under the Lüders map if their  and symbols coincide. For a spin corresponding to SU(2), the identity is shown to be the only operator with this property. For a particle, a countable family of linearly independent operators is identified which are invariant under the Lüders map generated by the coherent states of the Heisenberg–Weyl group, The Lüders map is also shown to implement the antinormal ordering of creation and annihilation operators of a particle.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Casimir force between surfaces close to each other
View Description Hide DescriptionCasimir interactions (due to the massless scalar field fluctuations) of two surfaces which are close to each other are studied. After a brief general presentation of the technique, explicit calculations are performed for specific geometries.

On the nonrelativistic Lee model
View Description Hide DescriptionIn this work we present two rigorous results on the nonrelativistic Lee model following a method proposed by Rajeev in an unpublished article (S. G. Rajeev, hepth 9902025). Thus this short paper should be considered as a commentary on Rajeev. In the unpublished paper of Rajeev, the renormalization of the Hamiltonian is accomplished at the level of resolvents. We first establish that the renormalized resolvent of the interacting Hamiltonian indeed defines a unique closed densely defined operator acting on the free Fock space of bosons. Next we give a justification in the mean field approximation that the ground state energy is bounded from below and the system has a good thermodynamic limit by elaborating along the original arguments of Rajeev. Our arguments in two dimensions do not yield better bounds, but this could be due to the inadequacy of the method used. In both cases though the ground state energy is not significantly altered to give a nontrivial ground state energy per particle.

Boundary conformal fields and Tomita–Takesaki theory
View Description Hide DescriptionMotivated by formal similarities between the continuum limit of the Ising model and the Unruh effect, this paper connects the notion of an Ishibashi state in boundary conformal field theory with the Tomita–Takesaki theory for operator algebras. A geometrical approach to the definition of Ishibashi states is presented, and it is shown that, when normalizable, the Ishibashi states are cyclic separating states, justifying the operator state corespondence. When the states are not normalizable Tomita–Takesaki theory offers an alternative approach based on left Hilbert algebras, making possible extensions of our construction and the stateoperator correspondence.

Stochastic Wess–Zumino–Novikov–Witten model on the torus
View Description Hide DescriptionWe define the Brownian motion on a torus group. We define the stochastic integral of a oneform over each canonical cycle of the torus and the stochastic integral on a twoform over the torus. We cannot apply martingale theory in order to define these stochastic integrals. We define a stochastic cohomology in the Chen–Souriau sense of the torus group, which allows us to define the stochastic Wess–Zumino term on the torus group. We show that it is related to the stochastic holonomy over a stochastic line bundle on the loop group.

Charge density and electric charge in quantum electrodynamics
View Description Hide DescriptionThe convergence of integrals over charge densities is discussed in relation with the problem of electric charge and (nonlocal) charged states in quantum electrodynamics. Delicate points like the domain dependence of local charges as quadratic forms and the class of time smearing ensuring strong convergence of integrals of charge densities are analyzed and shown to be crucial in QED, also for the control of vacuumpolarizationeffects leading to time dependence of the charge (Swieca phenomenon). The possibility of constructing physical charged states in the Feynman–Gupta–Bleuler gauge as limits of local state vectors is discussed, compatibly with the vanishing of the Gauss charge on local states. A modification of the Dirac exponential factor which yields the physical Coulomb fields from the Feynman–Gupta–Bleuler fields is shown to remove the infrared divergence of scalar products of local and physical charged states, allowing for a construction of physical charged fields with welldefined correlation functions with local fields.

A geometric renormalization group in discrete quantum space–time
View Description Hide DescriptionWe model quantum space–time on the Planck scale as dynamical networks of elementary relations or time dependent random graphs, the time dependence being an effect of the underlying dynamical network laws. We formulate a kind of geometricrenormalization group on these (random) networks leading to a hierarchy of increasingly coarsegrained networks of overlapping lumps. We provide arguments that this process may generate a fixed limit phase, representing our continuous space–time on a mesoscopic or macroscopic scale, provided that the underlying discrete geometry is critical in a specific sense (geometric long range order). Our point of view is corroborated by a series of analytic and numerical results, which allow us to keep track of the geometric changes, taking place on the various scales of the resolution of space–time. Of particular conceptual importance are the notions of dimension of such random systems on the various scales and the notion of geometric criticality.
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 GENERAL RELATIVITY AND GRAVITATION


Existence of the selfgraviting Chern–Simons vortices
View Description Hide DescriptionWe prove existence of multivortex solutions of the selfdual Einstein–Chern–Simons–Higgs system, proposed by Clément [Phys. Rev. D 54, 1844–1847 (1996)]. We consider both the topological and the nontopological boundary conditions for open, conformally flat manifolds. For nontopological boundary conditions we use perturbation argument from a solution of the Liouville equation combined with the implicit function theorem. Using this argument we have existence for arbitrary positive number for the gravitational constant. For topological boundary condition we construct solutions for small gravitational constant by using the super/subsolution method. For sufficiently large gravitational constant we have a nonexistence result for the radially symmetric topological solutions. We also obtain the decay estimates near infinity for both of the topological and the nontopological solutions.

General solutions of Einstein’s spherically symmetric gravitational equations with junction conditions
View Description Hide DescriptionEinstein’s spherically symmetric interior gravitational equations are investigated. Following Synge’s procedure, the most general solution of the equations is furnished in case and are prescribed. The existence of a total mass function, is rigorously proved. Under suitable restrictions on the total mass function, the Schwarzschild mass implicitly defines the boundary of the spherical body as Both Synge’s junction conditions as well as the continuity of the second fundamental form are examined and solved in a general manner. The weak energy conditions for an arbitrary boost are also considered. The most general solution of the spherically symmetric anisotropic fluid model satisfying both junction conditions is furnished. In the final section, various exotic solutions are explored using the developed scheme including gravitational instantons, interior domains, and dimensional generalizations.

Double structures and double symmetries for the general symplectic gravity models
View Description Hide DescriptionBy using the socalled doublecomplex function method, a doubleness symmetry for each member of the class of stationary axisymmetric general symplectic gravity models is found and exploited so that some doublecomplex matrix Ernstlike potential for any nonnegative integer can be constructed and the associated motion equations can be extended into a doublecomplex matrix Ernstlike form. Then double symmetry symplectic groups of the theories are given and verified that their actions can be realized concisely by doublecomplex matrix form generalizations of the fractional linear transformation on the Ernst potential. These results demonstrate that the theories under consideration possess more and richer symmetry structures. The special cases and correspond, respectively, to the pure Einstein gravity and the Einstein–Maxwelldilaton–axion theories. Moreover, as an application, for each an infinite chain of doublesolutions of the general symplectic gravity model is obtained, which shows that the doublecomplex method is more effective. Some of the results in this paper cannot be obtained by the usual (nondouble) scheme.

Universes encircling fivedimensional black holes
View Description Hide DescriptionWe clarify the status of two known solutions to the fivedimensional vacuum Einstein field equations derived by Liu, Mashhoon, and Wesson (LMW) and Fukui, Seahra, and Wesson (FSW), respectively. Both 5metrics explicitly embed fourdimensional Friedman–Lemaı̂tre–Robertson–Walker cosmologies with a wide range of characteristics. We show that both metrics are also equivalent to fivedimensional topological black hole (TBH) solutions, which is demonstrated by finding explicit coordinate transformations from the TBH to LMW and FSW line elements. We argue that the equivalence is a direct consequence of Birkhoff’s theorem generalized to five dimensions. Finally, for a special choice of parameters we plot constant coordinate surfaces of the LMW patch in a Penrose–Carter diagram. This shows that the LMW coordinates are regular across the black and/or white hole horizons.

The second variation of a null geodesic
View Description Hide DescriptionConfined to the second derivative of the variation of a null geodesic, the proper acceleration of the timelike curves obtained from the variation goes infinity as they approach the null geodesic except that the variation vector is a generalized Jacobi field on the null geodesic and the second variation is constant on the null geodesic.
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 DYNAMICAL SYSTEMS


Hamiltonian equations in
View Description Hide DescriptionThe Hamiltonian formulation of systems is considered in general. The most general solution of the Jacobi equation in is proposed. The form of the solution is shown to be valid also in the neighborhood of some irregular points. Compatible Poisson structures and corresponding biHamiltonian systems are also discussed. Hamiltonian structures, the classification of irregular points and the corresponding reduced first order differential equations of several examples are given.

Lax pair and superYangian symmetry of the nonlinear superSchrödinger equation
View Description Hide DescriptionWe consider a version of the nonlinear Schrödinger equation with M bosons and N fermions. We first solve the classical and quantum versions of this equation, using a superZamolodchikov–Faddeev (ZF) algebra. Then we prove that the hierarchy associated to this model admits a superYangian symmetry. We exhibit the corresponding (classical and quantum) Lax pairs. Finally, we construct explicitly the superYangian generators, in terms of the canonical fields on the one hand, and in terms of the ZF algebra generators on the other hand. The latter construction uses the wellbred operators introduced recently.

Integrability characteristics of twodimensional generalizations of NLS type equations
View Description Hide DescriptionA recent procedure based on truncated Painlevé expansions is used to derive Lax Pairs, Darboux transformations, and various soliton solutions for integrable generalizations of NLS type equations. In particular, diverse classes of solutions are found analogous to the dromion, instanton, lump, and ring soliton solutions derived recently for Korteweg–de Vries type equations, the Nizhnik–Novikov–Veselov equation, and the Broer–Kaup system.

characters in Gepner models, orbits and elliptic genera
View Description Hide DescriptionWe review the properties of characters of the superconformal algebra in the context of a nonlinear sigma model on how they are used to span the orbits, and how the orbits produce topological invariants like the elliptic genus. We derive the same expression for the elliptic genus using three different Gepner models and theories), detailing the orbits and verifying that their coefficients are given by elementary modular functions. We also reveal the orbits for the and theories. We derive relations for cubes of theta functions and study the function for

A new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling
View Description Hide DescriptionA type of new interesting loop algebra with a simple commutation operation just like that in the loop algebra is constructed. With the help of the loop algebra a new multicomponent integrable system, MAKNSKN hierarchy, is worked out. As reduction cases, the MAKNS hierarchy and MKN hierarchy are engendered, respectively. In addition, the system 1AKNSKN, which is a reduced case of the MAKNSKN hierarchy above, is a unified expressing integrable model of the AKNS hierarchy and the KN hierarchy. Obviously, the MAKNSKN hierarchy is again a united expressing integrable model of the multicomponent AKNS hierarchy (MAKNS) and the multicomponent KN hierarchy(MKN). This article provides a simple method for obtaining multicomponent integrable hierarchies of soliton equations. Finally, we work out an integrable coupling of the MAKNSKN hierarchy.

Kolmogorov entropy of global attractor for dissipative lattice dynamical systems
View Description Hide DescriptionWe consider Kolmogorov’s εentropy of the global attractor for first and second order dissipative lattice dynamical systems. By using the element decomposition and the covering property of a polyhedron by balls of radii ε in the finite dimensional space, we obtain an estimate of the upper bound for Kolmogorov’s εentropy of the global attractor.

Superintegrable systems in Darboux spaces
View Description Hide DescriptionAlmost all research on superintegrable potentials concerns spaces of constant curvature. In this paper we find by exhaustive calculation, all superintegrable potentials in the four Darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta, in addition to the Hamiltonian. These are twodimensional spaces of nonconstant curvature. It turns out that all of these potentials are equivalent to superintegrable potentials in complex Euclidean 2space or on the complex 2sphere, via “coupling constant metamorphosis” (or equivalently, via Stäckel multiplier transformations). We present a table of the results.
