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Dec 2003

Volume 44, Issue 12, pp. 5461-6232

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Sharp reconstruction of unsharp quantum observables

Roberto Beneduci and Giuseppe Nisticò

J. Math. Phys. 44, 5461 (2003); http://dx.doi.org/10.1063/1.1623615 (13 pages) | Cited 4 times

Online Publication Date: 18 November 2003

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A well defined procedure exists which allows us to “reconstruct” a sharp, i.e., standard, quantum observable A starting from a given commutative unsharp observable F. In this work we prove that the outcomes of measurements of F can be consistently interpreted as the result of a stochastic diffusion of outcomes of its sharp reconstruction A. Furthermore, for every sharp observable B, such that F is unsharp realization of B, we explicitly construct a real mapping g such that A = g(B). © 2003 American Institute of Physics.
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03.65.Ta Foundations of quantum mechanics; measurement theory

Lüders theorem for coherent-state POVMs

Stefan Weigert and Paul Busch

J. Math. Phys. 44, 5474 (2003); http://dx.doi.org/10.1063/1.1623001 (13 pages)

Online Publication Date: 18 November 2003

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Lüders’ 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 operator-valued measurements (or POVMs) associated with coherent states on Lie groups. In general, operators turn out to be invariant under the Lüders map if their P- and Q-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, H3. The Lüders map is also shown to implement the anti-normal ordering of creation and annihilation operators of a particle. © 2003 American Institute of Physics.
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03.65.Fd Algebraic methods
02.20.Qs General properties, structure, and representation of Lie groups
02.20.Uw Quantum groups
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Casimir force between surfaces close to each other

H. Ahmedov and I. H. Duru

J. Math. Phys. 44, 5487 (2003); http://dx.doi.org/10.1063/1.1624471 (17 pages) | Cited 2 times

Online Publication Date: 18 November 2003

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Casimir 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. © 2003 American Institute of Physics.
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12.20.Ds Specific calculations
02.30.Gp Special functions

On the nonrelativistic Lee model

Ali Ayan and O. Teoman Turgut

J. Math. Phys. 44, 5504 (2003); http://dx.doi.org/10.1063/1.1624093 (13 pages) | Cited 2 times

Online Publication Date: 18 November 2003

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In 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, hep-th 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. © 2003 American Institute of Physics.
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11.10.Gh Renormalization
11.10.St Bound and unstable states; Bethe-Salpeter equations

Boundary conformal fields and Tomita–Takesaki theory

K. C. Hannabuss and M. Semplice

J. Math. Phys. 44, 5517 (2003); http://dx.doi.org/10.1063/1.1625872 (13 pages)

Online Publication Date: 18 November 2003

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Motivated 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 state-operator correspondence. © 2003 American Institute of Physics.
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11.25.Hf Conformal field theory, algebraic structures
02.20.Sv Lie algebras of Lie groups

Stochastic Wess–Zumino–Novikov–Witten model on the torus

Rémi Léandre

J. Math. Phys. 44, 5530 (2003); http://dx.doi.org/10.1063/1.1614870 (39 pages) | Cited 2 times

Online Publication Date: 18 November 2003

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We define the Brownian motion on a torus group. We define the stochastic integral of a one-form over each canonical cycle of the torus and the stochastic integral on a two-form 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. © 2003 American Institute of Physics.
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11.25.Hf Conformal field theory, algebraic structures
05.40.Jc Brownian motion
02.50.Ey Stochastic processes

Charge density and electric charge in quantum electrodynamics

G. Morchio and F. Strocchi

J. Math. Phys. 44, 5569 (2003); http://dx.doi.org/10.1063/1.1623928 (19 pages) | Cited 2 times

Online Publication Date: 18 November 2003

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The 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 vacuum polarization effects 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 well-defined correlation functions with local fields. © 2003 American Institute of Physics.
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12.20.Ds Specific calculations
11.15.Tk Other nonperturbative techniques
11.10.Nx Noncommutative field theory

A geometric renormalization group in discrete quantum space–time

Manfred Requardt

J. Math. Phys. 44, 5588 (2003); http://dx.doi.org/10.1063/1.1619579 (28 pages) | Cited 4 times

Online Publication Date: 18 November 2003

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We 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 geometric renormalization group on these (random) networks leading to a hierarchy of increasingly coarse-grained 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. © 2003 American Institute of Physics.
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04.60.Pp Loop quantum gravity, quantum geometry, spin foams
02.20.-a Group theory
11.10.Gh Renormalization
02.10.Ox Combinatorics; graph theory
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Existence of the self-graviting Chern–Simons vortices

Dongho Chae and Kwangseok Choe

J. Math. Phys. 44, 5616 (2003); http://dx.doi.org/10.1063/1.1625871 (21 pages)

Online Publication Date: 18 November 2003

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We prove existence of multivortex solutions of the self-dual 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.© 2003 American Institute of Physics.
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04.20.Cv Fundamental problems and general formalism
11.15.Bt General properties of perturbation theory
02.30.Rz Integral equations

General solutions of Einstein’s spherically symmetric gravitational equations with junction conditions

A. Das, A. DeBenedictis, and N. Tariq

J. Math. Phys. 44, 5637 (2003); http://dx.doi.org/10.1063/1.1621056 (19 pages) | Cited 3 times

Online Publication Date: 18 November 2003

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Einstein’s spherically symmetric interior gravitational equations are investigated. Following Synge’s procedure, the most general solution of the equations is furnished in case T11 and T44 are prescribed. The existence of a total mass function, M(r,t), is rigorously proved. Under suitable restrictions on the total mass function, the Schwarzschild mass M(r,t) = m, implicitly defines the boundary of the spherical body as r = B(t). 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 T-domains, and D-dimensional generalizations. © 2003 American Institute of Physics.
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04.20.Jb Exact solutions
04.40.Nr Einstein-Maxwell spacetimes, spacetimes with fluids, radiation or classical fields
95.30.Lz Hydrodynamics
95.30.Sf Relativity and gravitation
97.60.Lf Black holes

Double structures and double symmetries for the general symplectic gravity models

Ya-Jun Gao

J. Math. Phys. 44, 5656 (2003); http://dx.doi.org/10.1063/1.1624092 (8 pages) | Cited 4 times

Online Publication Date: 18 November 2003

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By using the so-called double-complex 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 double-complex (n+1)×(n+1) matrix Ernst-like potential for any non-negative integer n can be constructed and the associated motion equations can be extended into a double-complex matrix Ernst-like form. Then double symmetry symplectic groups Sp(2(n+1), R(J)) of the theories are given and verified that their actions can be realized concisely by double-complex 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 n = 0 and n = 1 correspond, respectively, to the pure Einstein gravity and the Einstein–Maxwell-dilaton–axion theories. Moreover, as an application, for each n = 0,1,2,…, an infinite chain of double-solutions of the general symplectic gravity model is obtained, which shows that the double-complex method is more effective. Some of the results in this paper cannot be obtained by the usual (nondouble) scheme.© 2003 American Institute of Physics.
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04.20.Gz Spacetime topology, causal structure, spinor structure
04.20.Ex Initial value problem, existence and uniqueness of solutions
02.10.Yn Matrix theory
02.30.Uu Integral transforms

Universes encircling five-dimensional black holes

Sanjeev S. Seahra and Paul S. Wesson

J. Math. Phys. 44, 5664 (2003); http://dx.doi.org/10.1063/1.1623617 (17 pages) | Cited 7 times

Online Publication Date: 18 November 2003

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We clarify the status of two known solutions to the five-dimensional vacuum Einstein field equations derived by Liu, Mashhoon, and Wesson (LMW) and Fukui, Seahra, and Wesson (FSW), respectively. Both 5-metrics explicitly embed four-dimensional Friedman–Lemaître–Robertson–Walker cosmologies with a wide range of characteristics. We show that both metrics are also equivalent to five-dimensional 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. © 2003 American Institute of Physics.
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04.70.Bw Classical black holes
04.20.Jb Exact solutions
04.20.Gz Spacetime topology, causal structure, spinor structure
02.40.-k Geometry, differential geometry, and topology

The second variation of a null geodesic

Guihua Tian and Zhao Zheng

J. Math. Phys. 44, 5681 (2003); http://dx.doi.org/10.1063/1.1623931 (7 pages) | Cited 1 time

Online Publication Date: 18 November 2003

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Confined 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 β2 is constant on the null geodesic. © 2003 American Institute of Physics.
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02.40.Hw Classical differential geometry
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Hamiltonian equations in math3

Ahmet Ay, Metin Gürses, and Kostyantyn Zheltukhin

J. Math. Phys. 44, 5688 (2003); http://dx.doi.org/10.1063/1.1619204 (18 pages) | Cited 13 times

Online Publication Date: 18 November 2003

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The Hamiltonian formulation of N = 3 systems is considered in general. The most general solution of the Jacobi equation in math3 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 bi-Hamiltonian systems are also discussed. Hamiltonian structures, the classification of irregular points and the corresponding reduced first order differential equations of several examples are given. © 2003 American Institute of Physics.
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45.30.+s General linear dynamical systems
02.30.Hq Ordinary differential equations

Lax pair and super-Yangian symmetry of the nonlinear super-Schrödinger equation

V. Caudrelier and E. Ragoucy

J. Math. Phys. 44, 5706 (2003); http://dx.doi.org/10.1063/1.1625078 (27 pages) | Cited 2 times

Online Publication Date: 18 November 2003

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We 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 super-Zamolodchikov–Faddeev (ZF) algebra. Then we prove that the hierarchy associated to this model admits a super-Yangian Y(gl(M∣N)) symmetry. We exhibit the corresponding (classical and quantum) Lax pairs. Finally, we construct explicitly the super-Yangian 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 well-bred operators introduced recently. © 2003 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
11.30.Pb Supersymmetry
02.30.Hq Ordinary differential equations
02.10.-v Logic, set theory, and algebra
03.65.Fd Algebraic methods

Integrability characteristics of two-dimensional generalizations of NLS type equations

S. Roy Choudhury

J. Math. Phys. 44, 5733 (2003); http://dx.doi.org/10.1063/1.1623929 (18 pages) | Cited 2 times

Online Publication Date: 18 November 2003

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A recent procedure based on truncated Painlevé expansions is used to derive Lax Pairs, Darboux transformations, and various soliton solutions for integrable (2+1) 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 (2+1) Korteweg–de Vries type equations, the Nizhnik–Novikov–Veselov equation, and the (2+1) Broer–Kaup system. © 2003 American Institute of Physics.
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02.30.Jr Partial differential equations

N = 4 characters in Gepner models, orbits and elliptic genera

Daniel B. Grünberg

J. Math. Phys. 44, 5751 (2003); http://dx.doi.org/10.1063/1.1624470 (42 pages)

Online Publication Date: 18 November 2003

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We review the properties of characters of the N = 4 superconformal algebra in the context of a nonlinear sigma model on K3, 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 K3 elliptic genus using three different Gepner models (16, 24, and 43 theories), detailing the orbits and verifying that their coefficients Fi are given by elementary modular functions. We also reveal the orbits for the 1322, 144, and 1242 theories. We derive relations for cubes of theta functions and study the function (1/η) math(−1)n(6n+1)k q(6n+1)2/24 for k = 1,2,3,4. © 2003 American Institute of Physics.
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11.10.Lm Nonlinear or nonlocal theories and models
11.30.Pb Supersymmetry
02.40.Pc General topology

A new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling

Fukui Guo and Yufeng Zhang

J. Math. Phys. 44, 5793 (2003); http://dx.doi.org/10.1063/1.1623000 (11 pages) | Cited 97 times

Online Publication Date: 18 November 2003

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A type of new interesting loop algebra mathM (M = 1,2,…) with a simple commutation operation just like that in the loop algebra 1 is constructed. With the help of the loop algebra mathM, a new multicomponent integrable system, M-AKNS-KN hierarchy, is worked out. As reduction cases, the M-AKNS hierarchy and M-KN hierarchy are engendered, respectively. In addition, the system 1-AKNS-KN, which is a reduced case of the M-AKNS-KN hierarchy above, is a unified expressing integrable model of the AKNS hierarchy and the KN hierarchy. Obviously, the M-AKNS-KN hierarchy is again a united expressing integrable model of the multicomponent AKNS hierarchy (M-AKNS) and the multicomponent KN hierarchy(M-KN). This article provides a simple method for obtaining multicomponent integrable hierarchies of soliton equations. Finally, we work out an integrable coupling of the M-AKNS-KN hierarchy.© 2003 American Institute of Physics.
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05.45.Yv Solitons
02.10.-v Logic, set theory, and algebra

Kolmogorov entropy of global attractor for dissipative lattice dynamical systems

Qiuli Jia, Shengfan Zhou, and Fuqi Yin

J. Math. Phys. 44, 5804 (2003); http://dx.doi.org/10.1063/1.1626269 (7 pages) | Cited 4 times

Online Publication Date: 18 November 2003

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We consider Kolmogorov’s ε-entropy 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 ε-entropy of the global attractor. © 2003 American Institute of Physics.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
05.70.Ce Thermodynamic functions and equations of state
02.30.Hq Ordinary differential equations

Superintegrable systems in Darboux spaces

E. G. Kalnins, J. M. Kress, W. Miller, and P. Winternitz

J. Math. Phys. 44, 5811 (2003); http://dx.doi.org/10.1063/1.1619580 (38 pages) | Cited 47 times

Online Publication Date: 18 November 2003

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Almost 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 two-dimensional spaces of nonconstant curvature. It turns out that all of these potentials are equivalent to superintegrable potentials in complex Euclidean 2-space or on the complex 2-sphere, via “coupling constant metamorphosis” (or equivalently, via Stäckel multiplier transformations). We present a table of the results. © 2003 American Institute of Physics.
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02.30.Jr Partial differential equations

Semiclassical nonlinear Schrödinger on the half line

Spyridon Kamvissis

J. Math. Phys. 44, 5849 (2003); http://dx.doi.org/10.1063/1.1624091 (20 pages) | Cited 7 times

Online Publication Date: 18 November 2003

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We are studying the semiclassical limit of the 1+1 dimensional integrable nonlinear Schrödinger equation with defocusing cubic nonlinearity on the half line. Our analysis relies on the recent theory of Fokas et al., which reduces boundary value problems for soliton equations to Riemann–Hilbert factorization problems. We employ the method of nonlinear steepest descent to asymptotically deform the given Riemann–Hilbert problem to an explicilty solvable one. © 2003 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
03.65.Sq Semiclassical theories and applications

Virasoro structure and localized excitations of the LKR system

S. Y. Lou, C. Rogers, and W. K. Schief

J. Math. Phys. 44, 5869 (2003); http://dx.doi.org/10.1063/1.1625077 (19 pages) | Cited 11 times

Online Publication Date: 18 November 2003

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A symmetry analysis is conducted for a master 2+1-dimensional soliton system. The classical symmetries are shown to constitute an infinite dimensional Kac–Moody–Virasoro algebra. Finite symmetry group transformations are then used to construct localized excitations of the system. © 2003 American Institute of Physics.
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05.45.Yv Solitons
02.10.Ud Linear algebra
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Noncommutative geometry framework and the Feynman’s proof of Maxwell equations

A. Boulahoual and M. B. Sedra

J. Math. Phys. 44, 5888 (2003); http://dx.doi.org/10.1063/1.1625891 (14 pages) | Cited 4 times

Online Publication Date: 18 November 2003

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The main focus of the present work is to study the Feynman’s proof of the Maxwell equations using the NC geometry framework. To accomplish this task, we consider two kinds of noncommutativity formulations going along the same lines as Feynman’s approach. This allows us to go beyond the standard case and discover nontrivial results. In fact, while the first formulation gives rise to the static Maxwell equations, the second formulation is based on the following assumption m[xj,mathk] = δjk+imθjkf. The results extracted from the second formulation are more significant since they are associated to a nontrivial θ-extension of the Bianchi-set of Maxwell equations. We find divθ B = ηθ and (∂Bs/∂t)+ϵkjs(∂Ej/∂xk) = A1(d2f/dt2)+A2(df/dt)+A3, where ηθ, A1, A2, and A3 are local functions depending on the NC θ-parameter. The novelty of this proof in the NC space is revealed notably at the level of the corrections brought to the previous Maxwell equations. These corrections correspond essentially to the possibility of existence of magnetic charge sources that we can associate to the magnetic monopole since divθ B = ηθ is not vanishing in general.© 2003 American Institute of Physics.
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03.50.De Classical electromagnetism, Maxwell equations
02.40.Hw Classical differential geometry

Hilbert space structure in classical mechanics. I

E. Deotto, E. Gozzi, and D. Mauro

J. Math. Phys. 44, 5902 (2003); http://dx.doi.org/10.1063/1.1623333 (35 pages) | Cited 7 times

Online Publication Date: 18 November 2003

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In this paper we study the Hilbert space structure underlying the Koopman–von Neumann (KvN) operatorial formulation of classical mechanics. KvN limited themselves to study the Hilbert space of zero-forms that are the square integrable functions on phase space. They proved that in this Hilbert space the evolution is unitary for every system. In this paper we extend the KvN Hilbert space to higher forms which are basically functions of the phase space points and the differentials on phase space. We prove that if we equip this space with a positive definite scalar product the evolution can turn out to be nonunitary for some systems. Vice versa, if we insist in having a unitary evolution for every system then the scalar product cannot be positive definite. Identifying the one-forms with the Jacobi fields we provide a physical explanation of these phenomena. We also prove that the unitary/nonunitary character of the evolution is invariant under canonical transformations. © 2003 American Institute of Physics.
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45.20.D- Newtonian mechanics
45.05.+x General theory of classical mechanics of discrete systems
02.30.Uu Integral transforms
02.10.Ud Linear algebra

Hilbert space structure in classical mechanics. II

E. Deotto, E. Gozzi, and D. Mauro

J. Math. Phys. 44, 5937 (2003); http://dx.doi.org/10.1063/1.1623334 (21 pages) | Cited 5 times

Online Publication Date: 18 November 2003

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In this paper we analyze two different functional formulations of classical mechanics. In the first one the Jacobi fields are represented by bosonic variables and belong to the vector (or its dual) representation of the symplectic group. In the second formulation the Jacobi fields are given as condensates of Grassmannian variables belonging to the spinor representation of the metaplectic group. For both formulations we shall show that, differently from what happens in the case presented in paper I, it is possible to endow the associated Hilbert space with a positive definite scalar product and to describe the dynamics via a Hermitian Hamiltonian. The drawback of this formulation is that higher forms do not appear automatically and that the description of chaotic systems may need a further extension of the Hilbert space. © 2003 American Institute of Physics.
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05.45.Pq Numerical simulations of chaotic systems
45.05.+x General theory of classical mechanics of discrete systems
02.20.-a Group theory
02.10.Ud Linear algebra
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