Volume 51, Issue 4, April 2010
Index of content:
- Quantum Mechanics (General and Nonrelativistic)
51(2010); http://dx.doi.org/10.1063/1.3339676View Description Hide Description
It is shown that polynomials associated with solutions of certain conditionally exactly solvable potentials obtained via unbroken as well as broken supersymmetry belong to the category of exceptional orthogonal polynomials. Some properties of such polynomials, e.g., recurrence relation, ladder operators, differential equations, etc., have been obtained.
51(2010); http://dx.doi.org/10.1063/1.3299294View Description Hide Description
We examine the problem of using local operations and classical communication (LOCC) to distinguish a known pure state from an unknown (possibly mixed) state, bounding the error probability from above and below. We study the asymptotic rate of detecting multiple copies of the pure state and show that, if the overlap of the two states is great enough, then they can be distinguished asymptotically as well with LOCC as with global measurements; otherwise, the maximal Schmidt coefficient of the pure state is sufficient to determine the asymptotic error rate.
An alternative construction of the positive inner product for pseudo-Hermitian Hamiltonians: Examples51(2010); http://dx.doi.org/10.1063/1.3373551View Description Hide Description
In this paper, we build on our earlier proposal for the construction of a positive inner product for pseudo-Hermitian Hamiltonians and present examples to clarify the procedure. We focus on two detailed calculations where the method is used, namely, a simple (generalized matrix) pseudo-Hermitian Hamiltonian, which can be diagonalized, and a second system where the Hamiltonian cannot be diagonalized, but can be described as a perturbation of the harmonic oscillator. When the quantum mechanical system cannot be diagonalized exactly, our construction can be carried out perturbatively and we develop the general formalism for such a perturbative calculation systematically (for real eigenvalues).
51(2010); http://dx.doi.org/10.1063/1.3372725View Description Hide Description
The theory of Schrödinger bridges for diffusion processes is extended to classical and quantum discrete-time Markovian evolutions. The solution of the path-space maximum entropy problems is obtained from the a priori model in both cases via a suitable multiplicative functional transformation. In the quantum case, nonequilibrium time reversal of quantum channels is discussed and space-time harmonic processes are introduced.
- Quantum Information and Computation
51(2010); http://dx.doi.org/10.1063/1.3309418View Description Hide Description
Recently, Hastings [“A counterexample to additivity of minimum output entropy,” Nat. Phys.5, 255 (2009); e-print arXiv:0809.3972v3] proved the existence of random unitary channels, which violate the additivity conjecture. In this paper, we use Hastings’ method to derive new bounds for the entanglement of random subspaces of bipartite systems. As an application we use these bounds to prove the existence of nonunital channels, which violate additivity of minimal output entropy.
51(2010); http://dx.doi.org/10.1063/1.3364793View Description Hide Description
The positive partial transpose test is one of the main criteria for detecting entanglement, and the set of states with positive partial transpose is considered as an approximation of the set of separable states. However, we do not know to what extent this criterion, as well as the approximation, is efficient. In this paper, we show that the positive partial transpose test gives no bound on the distance of a density matrix from separable states. More precisely, we prove that, as the dimension of the space tends to infinity, the maximum trace distance of a positive partial transpose state from separable states tends to 1. Using similar techniques, we show that the same result holds for other well-known separability criteria such as reduction criterion, majorization criterion, and symmetric extension criterion. We also bring in evidence that the sets of positive partial transpose states and separable states have totally different shapes.
51(2010); http://dx.doi.org/10.1063/1.3374022View Description Hide Description
We report on a new computer study of the existence of equiangular lines in complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite dimensions and are the underlying mathematical objects defining symmetric informationally complete measurements in quantum theory. We provide numerical solutions in all dimensions and, moreover, a putatively complete list of Weyl–Heisenberg covariant solutions for . A symmetry analysis of this list leads to new algebraic solutions in dimensions , 35, and 48, which are given together with algebraic solutions for , and 19.
- Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)
51(2010); http://dx.doi.org/10.1063/1.3321531View Description Hide Description
Galilean conformal algebra (GCA) is an Inönü–Wigner (IW) contraction of a conformal algebra, while Newton–Hooke string algebra is an IW contraction of an Anti-de Sitter (AdS) algebra, which is the isometry of an AdS space. It is shown that the GCA is a boundary realization of the Newton–Hooke string algebra in the bulk AdS. The string lies along the direction transverse to the boundary, and the worldsheet is . The one-dimensional conformal symmetry so(2,1) and rotational symmetry contained in the GCA are realized as the symmetry on the string worldsheet and rotational symmetry in the space transverse to the in , respectively. It follows from this correspondence that 32 supersymmetric GCAs can be derived as IW contractions of superconformal algebras,, , and . We also derive less supersymmetric GCAs from , , , and .
- General Relativity and Gravitation
51(2010); http://dx.doi.org/10.1063/1.3352935View Description Hide Description
A theory of a new gravitational interaction is described. This theory follows naturally from a new Lagrangian formulation of Maxwell’stheory for photons and electrons (and positrons) whose associated Euler Lagrange equations imply the conventional Maxwell equations, but which possesses new bosonic spinor degrees of freedom that may be associated with a new type of fundamental gravitational interaction. The precise character of this gravitational interaction with a photon vector potential is explicitly defined in terms of a local U(1)-invariant Lagrangian in Eq. (86). However, in Sec. ???, in order to parallel the well known Friedmann model in cosmology, a phenomenological description of the new gravitational interaction coupled to Newton–Einstein gravity that is sourced by an ideal fluid is discussed. To lay the foundation for a description of the new gravitational interaction, our new formulation of Maxwell’stheory must first be described. It is cast on the real, eight-dimensional pseudo-Euclidean vector space defined by the split octonion algebra, regarded as a vector space over and denoted as . (Here denotes real four-dimensional Minkowski space-time and denotes its dual; resembles the phase space of a single relativistic particle.) The new gravitational interaction is carried by a field that defines an algebraically distinguished element of the split octonion algebra, namely, the multiplicative unit element. We call this interaction the “unit” interaction and more descriptively refer to it as “second gravity.”
51(2010); http://dx.doi.org/10.1063/1.3352557View Description Hide Description
We discuss exterior differential systems (EDSs) for the vacuum gravitational field. These EDSs are derived by varying the Hilbert–Einstein Lagrangian, given most elegantly as a Cartan 4-form calibrating 4-spaces embedded in ten flat dimensions. In particular, we thus formulate with tetrad equations the Regge–Teitelboim (RT) dynamics “à la string;” it arises when variation of the 4-spaces gives the Euler–Lagrange equations of a multicontact field theory. We calculate the Cartan character table of this EDS, showing the field equations to be well posed with no gauge freedom. The Hilbert Lagrangian as usually varied over just the intrinsic curvature structure of a 4-space yields only a subset of this dynamics, viz., solutions satisfying additional conditions constraining them to be Ricci flat. In the static spherically symmetric case, we present a new tetrad embedding in flat six dimensions, which allows reduction of the RT field equations to a quadrature; the Schwarzschild metric is a special case. As has previously been noted, there may be a classical correspondence of the RT theory with the hidden dimensions of brane theory, and perhaps this extended general relativistic dynamics holds in extreme circumstances where it can be interpreted as including a sort of dark or bulk energy even though no term with a cosmological constant is included in the Lagrangian. As a multicontact system, canonical quantization should be straightforward.
51(2010); http://dx.doi.org/10.1063/1.3387251View Description Hide Description
We study the quantum dynamics of a Klein–Gordon field on de Sitter space based on the Euclidean vacuum. We prove time evolution is not unitarily implementable. We also consider a Klein–Gordon field perturbed by a local potential . In this case we prove that the deviation from the dynamics is unitarily implementable.
- Dynamical Systems
51(2010); http://dx.doi.org/10.1063/1.3371032View Description Hide Description
We investigate the integrable -dimensional Manakov–Santini equation from the Lax–Sato form. Several particular two- and three-component reductions are considered so that the Manakov–Santini equation can be reduced to systems of hydrodynamic type. Then one can construct infinitely many exact solutions of the equation by the hodograph method.
51(2010); http://dx.doi.org/10.1063/1.3371010View Description Hide Description
Invariant manifolds provide geometric structures for understanding dynamical behavior of nonlinear systems. However, these nonlinear systems are often subject to random fluctuations or noises. It is thus desirable to quantify the impact of noises on the invariant manifolds. When the noise intensity is small, this impact is estimated via asymptotic analysis in the context of Liapunov–Perron formulation. Namely, the random invariant manifold is represented as a perturbation of the deterministic invariant manifold, with a well-defined bound for the deviation.
51(2010); http://dx.doi.org/10.1063/1.3385777View Description Hide Description
This paper deals with the following equation, which is proposed by Z. J. Qiao [J. Math. Phys.48, 082701 (2007)] and Qiao and Liu [Chaos, Solitons Fractals41, 587 (2009)]. By adopting the phase analysis method of planar dynamical systems and the theory of the singular traveling wave systems to the traveling wavesolutions of the equation, it is shown that for different , the equation may have infinitely many solitary wavesolutions, periodic wavesolutions, kink/antikink wavesolutions, cusped solitary wavesolutions, and breaking loop solutions. We discuss in a detail the cases of , and parametric representations of all possible bounded traveling wavesolutions are given in the different -parameter regions.
51(2010); http://dx.doi.org/10.1063/1.3387343View Description Hide Description
We show how to apply to Hamiltonian differential systems recent results for studying the periodic orbits of a differential system using the averaging theory. We have chosen two classical integrable Hamiltonian systems, one with the Hooke potential and the other with the Kepler potential, and we study the periodic orbits which bifurcate from the periodic orbits of these integrable systems, first perturbing the Hooke Hamiltonian with a nonautonomous potential, and second perturbing the Kepler problem with an autonomous potential.
51(2010); http://dx.doi.org/10.1063/1.3393506View Description Hide Description
The dynamics of point vortices in a fluid is described by the Helmholtz–Kirchhoff (HK) equations which lead to a completely integrable Hamiltonian system for or 3 but chaotic dynamics for . Here we consider a generalization of the HK equations to describe the dynamics of magnetic vortices within a collective-coordinate approximation. In particular, we analyze in detail the dynamics of a system of three magnetic vortices by a suitable generalization of the solution for three point vortices in an ordinary fluid obtained by Gröbli more than a century ago. The significance of our results for the dynamics of ferromagnetic elements is briefly discussed.
- Classical Mechanics and Classical Fields
A new proof of the higher-order superintegrability of a noncentral oscillator with inversely quadratic nonlinearities51(2010); http://dx.doi.org/10.1063/1.3374665View Description Hide Description
The superintegrability of a rational harmonic oscillator (noncentral harmonic oscillator with rational ratio of frequencies) with nonlinear “centrifugal” terms is studied. In the first part, the system is directly studied in the Euclidean plane; the existence of higher-order superintegrability (integrals of motion of higher order than 2 in the momenta) is proved by introducing a deformation in the quadratic complex equation of the linear system. The constants of motion of the nonlinear system are explicitly obtained. In the second part, the inverse problem is analyzed in the general case of degrees of freedom; starting with a general Hamiltonian and introducing appropriate conditions for obtaining superintegrability, the particular “centrifugal” nonlinearities are obtained.
51(2010); http://dx.doi.org/10.1063/1.3345125View Description Hide Description
In this paper, we consider the inverse problem of central configurations of the -body problem. For a given , let be the admissible set of masses by For a given , let be the permutational admissible set about by Here, is called a super central configuration if there exists such that is nonempty. For any in the planar four-body problem, is not a super central configuration as an immediate consequence of a theorem proved by MacMillan and Bartky [“Permanent configurations in the problem of four bodies,” Trans. Am. Math. Soc.34, 838 (1932)]. The main discovery in this paper is the existence of super central configurations in the collinear three-body problem. We proved that for any in the collinear three-body problem and any , has at most one element and the detailed classification of is provided.
51(2010); http://dx.doi.org/10.1063/1.3352555View Description Hide Description
Stochastic solutions are obtained for the Maxwell–Vlasovequation in the approximation where magnetic field fluctuations are neglected and the electrostatic potential is used to compute the electric field. This is a reasonable approximation for plasmas in a strong external magnetic field. Both Fourier and configuration space solutions are constructed.
A new lubrication theory to derive far-field axial pressure difference due to force singularities in cylindrical or annular vessels51(2010); http://dx.doi.org/10.1063/1.3321580View Description Hide Description
In simulations of creeping flow, the effect of suspended particles on a fluid is often represented by force singularities responsible for singular Stokesian solutions, which are infinite at the sphere center and decay far from the particle. In this article, we consider such singular fields centered at a point inside a cylindrical or annular conduit containing highly viscous medium. Across an unbounded infinite domain, these singular flows cannot produce a finite pressure difference as they decay to zero far from the center. However, in the presence of bounding cylindrical surfaces, the reflected flow from the walls creates a finite pressure difference between the far fields across the force singularity along the axial direction. To quantify the effect of the reflected flow, we present a new lubricationanalysis, which, on the one hand, identifies the specific singular fields capable of producing axial pressure difference and, on the other hand, provides explicit expressions for the far-field pressure. Though we specifically focus on cylindrical or annular geometry, the outlined approach can also be extended to other confinements. Thus, the general formulation can be used in larger context to quantify the effect of small particles on wall-bounded fluid medium.