Volume 52, Issue 3, March 2011
Index of content:
 ARTICLES


Quantum Mechanics (General and Nonrelativistic)

Intrinsic structure of state space of a quantum system
View Description Hide DescriptionQuantum states play a fundamental role in quantum physics; so it is necessary to study intrinsic structure of quantum states. In this paper, we study topological structure and measurable structure of state space of a quantum system, and find that almost all physical important properties on quantum states coincide.

Noncommutative oscillators from a Hopf algebra twist deformation. A first principles derivation
View Description Hide DescriptionNoncommutative oscillators are firstquantized through an abelian Drinfel'd twist deformation of a Hopf algebra and its action on a module. Several important and subtle issues making the quantization possible are solved. The spectrum of the singleparticle Hamiltonians is computed. The multiparticle Hamiltonians are fixed, unambiguously, by the Hopf algebra coproduct. The symmetry under particle exchange is guaranteed. In d = 2 dimensions the rotational invariance is preserved, while in d = 3 the so(3) rotational invariance is broken down to an so(2) invariance.

On the Cauchy problem for Gross–Pitaevskii hierarchies
View Description Hide DescriptionThe purpose of this paper is to investigate the Cauchy problem for the Gross–Pitaevskii infinite linear hierarchy of equations on n ⩾ 1. We prove local existence and uniqueness of solutions in certain Sobolevtype spaces of sequences of marginal density operators with α > n/2. In particular, we give a clear discussion of all cases α > n/2, which covers the local wellposedness problem for the Gross–Pitaevskii hierarchy in this situation.

Information entropy of conditionally exactly solvable potentials
View Description Hide DescriptionWe evaluate Shannon entropy for the position and momentum eigenstates of some conditionally exactly solvable potentials which are isospectral to harmonic oscillator and whose solutions are given in terms of exceptional orthogonal polynomials. The Bialynicki–Birula–Mycielski inequality has also been tested for a number of states.

A nonAbelian SO(8) monopole as generalization of DiracYang monopoles for a 9dimensional space
View Description Hide DescriptionWe establish an explicit form of a nonAbelian SO(8) monopole in a 9dimensional space and show that it is indeed a direct generalization of Dirac and Yang monopoles. Using the generalized Hurwitz transformation, we have found a connection between a 16dimensional harmonic oscillator and a 9dimensional hydrogenlike atom in the field of the SO(8) monopole (MICZKepler problem). Using the built connection the group of dynamical symmetry of the 9dimensional MICZKepler problem is found as SO(10, 2).

Transition representations of quantum evolution with application to scattering resonances
View Description Hide DescriptionA Lyapunov operator is a selfadjoint quantum observable whose expectation value varies monotonically as time increases and may serve as a marker for the flow of time in a quantum system. In this paper it is shown that the existence of a certain type of Lyapunov operator leads to representations of the quantum dynamics, termed transition representations, in which an evolving quantum state ψ(t) is decomposed into a sum ψ(t) = ψ^{ b }(t) + ψ^{ f }(t) of a backward asymptotic component and a forward asymptotic component such that the evolution process is represented as a transition from ψ^{ b }(t) to ψ^{ f }(t). When applied to the evolution of scattering resonances, such transition representations separate the process of decay of a scattering resonance from the evolution of outgoing waves corresponding to the probability “released” by the resonance and carried away to spatial infinity. This separation property clearly exhibits the spatialprobability distribution profile of a resonance. Moreover, it leads to the definition of exact resonance states as elements of the physical Hilbert space corresponding to the scattering problem. These resonance states evolve naturally according to a semigroup law of evolution.

Exact diagonalization of 1D interacting spinless Fermions
View Description Hide DescriptionWe acquire a method of constructing an infinite set of exact eigenfunctions of 1D interacting spinless Fermionic systems. Creation and annihilation operators for the interacting system are found and thereby the manybody Hamiltonian is diagonalized. The formalism is applied to several examples. One example is the theory of Jack polynomials. For the Calogero–Moser–Sutherland Hamiltonian a direct proof is given that the asymptotic Bethe ansatz is correct.

Quantum Information and Computation

On quantum network coding
View Description Hide DescriptionWe study the problem of errorfree multiple unicast over directed acyclic networks in a quantum setting. We provide a new informationtheoretic proof of the known result that network coding does not achieve a larger quantum information flow than what can be achieved by routing for twopair communication on the butterfly network. We then consider a kpair multiple unicast problem and for all k ⩾ 2 we show that there exists a family of networks where quantum network coding achieves ktimes larger quantum information flow than what can be achieved by routing. Finally, we specify a graphtheoretic sufficient condition for the quantum information flow of any multiple unicast problem to be bounded by the capacity of any sparsest multicut of the network.

Decoherence for positive semigroups on M _{2}()
View Description Hide DescriptionWe extend Blanchard and Olkiewicz's definition of decoherence to open quantum systems whose dynamics are described by semigroups of positive (and not necessarily completely positive) operators on . In particular, in the case , we completely characterize the decomposition of in the sum of a decoherencefree part and of a space on which the semigroup vanishes with time.

Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

Canonical quantization of lattice Higgs–Maxwell–Chern–Simons fields: Osterwalder–Schrader positivity
View Description Hide DescriptionA Euclidean representation is given for a canonically quantized relativistic Maxwell–Chern–Simons field on a lattice, which approximates a complex measure on a space of distributions. Using a pathspace formula for the nonselfadjoint Hamiltonian, the relation between Euclidean Osterwalder–Schrader positivity, the Krein metric, and Gauss’ law is examined.

Laplacians in polar matrix coordinates and radial fermionization in higher dimensions
View Description Hide DescriptionWe consider the quantum mechanical Hamiltonian of two, space indexed, Hermitian matrices. By introducing matrix valued polar coordinates, we obtain the form of the Laplacian acting on invariant states. For potentials depending only on the eigenvalues of the radial matrix, we establish that the radially invariant sector is equivalent to a system of noninteracting 2 + 1 dimensional fermions and obtain its density description. For a larger number of matrices, the presence of a repulsive radial intereigenvalue potential is identified.

Toward an axiomatic formulation of noncommutative quantum field theory
View Description Hide DescriptionWe propose new Wightman functions as vacuum expectation values of products of field operators in the noncommutative space–time. These Wightman functions involve the ⋆product among the fields, compatible with the twisted Poincaré symmetry of the noncommutative quantum field theory (NC QFT). In the case of only space–space noncommutativity (θ_{0i } = 0), we prove the CPT theorem using the noncommutative form of the Wightman functions. We also show that the spinstatistics theorem, demonstrated for the simplest case of a scalar field, holds in NC QFT within this formalism.

General Relativity and Gravitation

Angular momentum at null infinity in five dimensions
View Description Hide DescriptionIn this paper, using the Bondi coordinates, we discuss the angular momentum at null infinity in five dimensions and address the Poincare covariance of the Bondi mass and angular momentum. We also show the angular momentum loss/gain law due to gravitational waves. In four dimensions, the angular momentum at null infinity has the supertranslational ambiguity and then it is known that we cannot construct welldefined angular momentum there. On the other hand, we would stress that we can define angular momentum at null infinity without any ambiguity in higher dimensions. This is because of the nonexistence of supertranslations in higher dimensions.

The motion of a body in Newtonian theories
View Description Hide DescriptionA theorem due to Bob Geroch and Pong Soo Jang [J. Math. Phys.16(1), 65 (1975)] provides the sense in which the geodesic principle has the status of a theorem in general relativity (GR). Here we show that a similar theorem holds in the context of geometrized Newtonian gravitation (often called Newton–Cartan theory). It follows that in Newtonian gravitation, as in GR, inertial motion can be derived from other central principles of the theory.

Dynamical Systems

Sufficient conditions for a nondegenerate Hopf bifurcation in a generalized Lagrange–Poisson problem
View Description Hide DescriptionIn this paper we provide sufficient conditions for the existence of a nondegenerate Hamiltonian Hopf bifurcation at the equilibria corresponding to the rotation around the vertical axis of a symmetric gyrostat with a fixed point under the effect of an axially symmetric potential.

Existence of dark solitons in a class of stationary nonlinear Schrödinger equations with periodically modulated nonlinearity and periodic asymptotics
View Description Hide DescriptionIn this paper, we give a proof of the existence of stationary dark solitonsolutions or heteroclinic orbits of nonlinear equations of Schrödinger type with periodic inhomogeneous nonlinearity. The result is illustrated with examples of dark solitons for cubic and photorefractive nonlinearities.

Darbouxian integrals for generalized Raychaudhuri equations
View Description Hide DescriptionWe give a complete characterization of the Darbouxian first integrals of a generalized Raychaudhuri equation which appears in modern string cosmology and which has the form , where α, β, γ, δ are real parameters. Our approach uses the Darboux theory of integrability.

On the constrained Btype Kadomtsev–Petviashvili hierarchy: Hirota bilinear equations and Virasoro symmetry
View Description Hide DescriptionWe derive the bilinear equations of the constrained BKP hierarchy from the calculus of pseudodifferential operators. The full hierarchy equations can be expressed in Hirota's bilinear form characterized by the functions ρ, σ, and τ. Besides, we also give a modification of the original Orlov–Schulman additional symmetry to preserve the constrained form of the Lax operator for this hierarchy. The vector fields associated with the modified additional symmetry turn out to satisfy a truncated centerless Virasoro algebra.

d’Alembert–Lagrange analytical dynamics for nonholonomic systems
View Description Hide DescriptionThe d’Alembert–Lagrange principle (DLP) is designed primarily for dynamical systems under ideal geometric constraints. Although it can also cover linearvelocity constraints, its application to nonlinear kinematic constraints has so far remained elusive, mainly because there is no clear method whereby the set of linear conditions that restrict the virtual displacements can be easily extracted from the equations of constraint. On recognition that the commutation rule traditionally accepted for velocity displacements in Lagrangian dynamics implies displaced states that do not satisfy the kinematic constraints, we show how the property of possible displaced states can be utilized ab initio so as to provide an appropriate set of linear auxiliary conditions on the displacements, which can be adjoined via Lagrange's multipliers to the d’Alembert–Lagrange equation to yield the equations of state, and also new transpositional relations for nonholonomic systems. The equations of state so obtained for systems under general nonlinear velocity and acceleration constraints are shown to be identical with those derived (in Appendix A) from the quite different Gauss principle. The present advance therefore solves a long outstanding problem on the application of DLP to ideal nonholonomic systems and, as an aside, provides validity to axioms as the Chetaev rule, previously left theoretically unjustified. A more general transpositional form of the Boltzmann–Hamel equation is also obtained.

Classical Mechanics and Classical Fields

Periodic orbits and nonintegrability of generalized classical Yang–Mills Hamiltonian systems
View Description Hide DescriptionThe averaging theory of first order is applied to study a generalized Yang–Mills system with two parameters. Two main results are proved. First, we provide sufficient conditions on the two parameters of the generalized system to guarantee the existence of continuous families of isolated periodic orbits parameterized by the energy, and these families are given up to first order in a small parameter. Second, we prove that for the nonintegrable classical Yang–Mills Hamiltonian systems, in the sense of Liouville–Arnold, which have the isolated periodic orbits found with averaging theory, cannot exist in any second first integral of class . This is important because most of the results about integrability deals with analytic or meromorphic integrals of motion.
