Volume 52, Issue 3, March 2011
Index of content:
- Quantum Mechanics (General and Nonrelativistic)
52(2011); http://dx.doi.org/10.1063/1.3559133View Description Hide Description
Quantum 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.
52(2011); http://dx.doi.org/10.1063/1.3562510View Description Hide Description
Noncommutative oscillators are first-quantized 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 single-particle 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.
52(2011); http://dx.doi.org/10.1063/1.3567168View Description Hide Description
The 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 Sobolev-type 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 well-posedness problem for the Gross–Pitaevskii hierarchy in this situation.
52(2011); http://dx.doi.org/10.1063/1.3566977View Description Hide Description
We 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.
52(2011); http://dx.doi.org/10.1063/1.3567422View Description Hide Description
We establish an explicit form of a non-Abelian SO(8) monopole in a 9-dimensional 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 16-dimensional harmonic oscillator and a 9-dimensional hydrogenlike atom in the field of the SO(8) monopole (MICZ-Kepler problem). Using the built connection the group of dynamical symmetry of the 9-dimensional MICZ-Kepler problem is found as SO(10, 2).
52(2011); http://dx.doi.org/10.1063/1.3559003View Description Hide Description
A Lyapunov operator is a self-adjoint 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.
52(2011); http://dx.doi.org/10.1063/1.3563580View Description Hide Description
We 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 many-body 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
52(2011); http://dx.doi.org/10.1063/1.3555801View Description Hide Description
We study the problem of error-free multiple unicast over directed acyclic networks in a quantum setting. We provide a new information-theoretic proof of the known result that network coding does not achieve a larger quantum information flow than what can be achieved by routing for two-pair communication on the butterfly network. We then consider a k-pair multiple unicast problem and for all k ⩾ 2 we show that there exists a family of networks where quantum network coding achieves k-times larger quantum information flow than what can be achieved by routing. Finally, we specify a graph-theoretic 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.
52(2011); http://dx.doi.org/10.1063/1.3560478View Description Hide Description
We 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 decoherence-free 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 positivity52(2011); http://dx.doi.org/10.1063/1.3559122View Description Hide Description
A 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 path-space formula for the nonself-adjoint Hamiltonian, the relation between Euclidean Osterwalder–Schrader positivity, the Krein metric, and Gauss’ law is examined.
52(2011); http://dx.doi.org/10.1063/1.3553456View Description Hide Description
We 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.
52(2011); http://dx.doi.org/10.1063/1.3567411View Description Hide Description
We 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 (θ0i = 0), we prove the CPT theorem using the noncommutative form of the Wightman functions. We also show that the spin-statistics theorem, demonstrated for the simplest case of a scalar field, holds in NC QFT within this formalism.
- General Relativity and Gravitation
52(2011); http://dx.doi.org/10.1063/1.3559917View Description Hide Description
In 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 well-defined 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.
52(2011); http://dx.doi.org/10.1063/1.3556608View Description Hide Description
A 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 problem52(2011); http://dx.doi.org/10.1063/1.3559064View Description Hide Description
In 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 asymptotics52(2011); http://dx.doi.org/10.1063/1.3559118View Description Hide Description
In 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.
52(2011); http://dx.doi.org/10.1063/1.3559065View Description Hide Description
We 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 B-type Kadomtsev–Petviashvili hierarchy: Hirota bilinear equations and Virasoro symmetry52(2011); http://dx.doi.org/10.1063/1.3559081View Description Hide Description
We 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.
52(2011); http://dx.doi.org/10.1063/1.3559128View Description Hide Description
The d’Alembert–Lagrange principle (DLP) is designed primarily for dynamical systems under ideal geometric constraints. Although it can also cover linear-velocity 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
52(2011); http://dx.doi.org/10.1063/1.3559145View Description Hide Description
The 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.