Volume 40, Issue 7, July 1999
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Quantum field theory of partitions
View Description Hide DescriptionGiven a sequence of numbers it is always possible to find a set of Feynman rules that reproduce that sequence. For the special case of the partitions of the integers, the appropriate Feynman rules give rise to graphs that represent the partitions in a clear pictorial fashion. These Feynman rules can be used to generate the Bell numbers and the Stirling numbers that are associated with the partitions of the integers.

Connections and metrics respecting purification of quantum states
View Description Hide DescriptionStandard purification interlaces Hermitian and Riemannian metrics on the space of density operators with metrics and connections on the purifying Hilbert–Schmidt space. We discuss connections and metrics which are well adopted to purification, and present a selected set of relations between them. A connection, as well as a metric on state space, can be obtained from a metric on the purification space. We include a condition, with which this correspondence becomes one to one. Our methods are borrowed from elementary representation and fiber space theory. We lift, as an example, solutions of a von Neumann equation, write down holonomy invariants for cyclic ones, and “add noise” to a curve of pure states.

On formpreserving transformations for the timedependent Schrödinger equation
View Description Hide DescriptionIn this paper we point out a close connection between the Darboux transformation and the group of point transformations which preserve the form of the timedependent Schrödinger equation (TDSE). In our main result, we prove that any pair of timedependent real potentials related by a Darboux transformation for the TDSE may be transformed by a suitable point transformation into a pair of timeindependent potentials related by a usual Darboux transformation for the stationary Schrödinger equation. Thus, any (real) potential solvable via a timedependent Darboux transformation can alternatively be solved by applying an appropriate formpreserving point transformation of the TDSE to a timeindependent potential. The preeminent role of the latter type of transformations in the solution of the TDSE is illustrated with a family of quasiexactly solvable timedependent anharmonic potentials.

Lifetimes of impurity states in crossed magnetic and electric fields
View Description Hide DescriptionWe study the quantum dynamics of localized impurity states created by a point interaction for an electron moving in two dimensions under the influence of a perpendicular magnetic field and an inplane weak electric field. All impurity states are unstable in presence of the electric field. Their lifetimes are computed and shown to grow in a Gaussian way as the electric field tends to zero.

Optimal cloning of pure states, testing single clones
View Description Hide DescriptionWe consider quantum devices for turning a finite number N of dlevel quantum systems in the same unknown pure state σ into systems of the same kind, in an approximation of the Mfold tensor product of the state σ. In a previous paper it was shown that this problem has a unique optimal solution, when the quality of the output is judged by arbitrary measurements, involving also the correlations between the clones. We show in this paper, that if the quality judgment is based solely on measurements of single output clones, there is again a unique optimal cloning device, which coincides with the one found previously.

Tunneling of a massless field through a 3D Gaussian barrier
View Description Hide DescriptionWe propose a method for the approximate computation of the Green function of a scalar massless field φ subjected to potential barriers of given size and shape in space–time. This technique is applied to the case of a 3D Gaussian ellipsoidlike barrier, placed on the axis between two pointlike sources of the field. Instead of the Green function we compute its temporal integral, that gives the static potential energy of the interaction of the two sources. Such interaction takes place in part by tunneling of the quanta of φ across the barrier. We evaluate numerically the correction to the potential in dependence on the barrier size and on the barriersources distance.

Generalized adiabatic product expansion: A nonperturbative method of solving the timedependent Schrödinger equation
View Description Hide DescriptionWe outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly nonHermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding timedependent Schrödinger equation. We apply this method to study the dynamics of a general nondegenerate twolevel quantum system, a timedependent classical harmonic oscillator, and a degenerate system consisting of a spin 1 particle interacting with a timedependent electric field through the Stark Hamiltonian

Geometric algebra and the causal approach to multiparticle quantum mechanics
View Description Hide DescriptionIt is argued that geometricalgebra, in the form of the multiparticle spacetimealgebra, is well suited to the study of multiparticle quantum theory, with advantages over conventional techniques both in ease of calculation and in providing an intuitive geometric understanding of the results. This is illustrated by comparing the geometricalgebra approach for a system of two spin1/2 particles with the nonrelativistic approach of Holland [Phys. Rep. 169, 294 (1988)].
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


A differentialgeometric interpretation of Kirchhoff’s elastic rods
View Description Hide DescriptionIn this article, Kirchhoff’s elasticity theory of rods is revisited but from a viewpoint of Riemannian geometry. By means of the Cayley–Klein parameter, the theory under clampedend conditions can be regarded as a geometry of paths on some geometrically distorted threesphere produced by a constrained elastic energy functional. Using this geometric formulation, the uniqueness of elasticae with prescribed initial values of the strains can be easily shown. On the other hand, a family of elasticae with prescribed values of orthonormal frames at two endpoints is demonstrated to be parametrized by an open set in In particular, a criterion of the nonuniqueness of elasticae satisfying clampedend conditions is given in terms of a geometric concept—conjugate points.

Symplecticenergymomentum preserving variational integrators
View Description Hide DescriptionThe purpose of this paper is to develop variational integrators for conservativemechanical systems that are symplectic and energy and momentum conserving. To do this, a space–time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy. Criteria for the solvability of the time steps and some numerical examples are given.

Magnetohydrodynamic boundary layer on a flat plate: Further analytic results
View Description Hide DescriptionFurther analytic results are deduced with the magnetohydrodynamic boundary layer equations for a flat plate. The asymptotic behavior of the solutions is deduced using the scaling group method. Then, an analytic perturbative procedure is used to determine an approximate solution that exhibits this asymptotic behavior.
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 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


The Weierstrass–Enneper system for constant mean curvature surfaces and the completely integrable sigma model
View Description Hide DescriptionThe integrability of a system which describes constant mean curvature surfaces by means of the adapted Weierstrass–Enneper inducing formula is studied. This is carried out by using a specific transformation which reduces the initial system to the completely integrable twodimensional Euclidean nonlinear sigma model. Through the use of the apparatus of differential forms and Cartan theory of systems in involution, it is demonstrated that the general analytic solutions of both systems possess the same degree of freedom. Furthermore, a new linear spectral problem equivalent to the initial Weierstrass–Enneper system is derived via the method of differential constraints. A new procedure for constructing solutions to this system is proposed and illustrated by several elementary examples, including a multisoliton solution.

Unitary deformations and complex soliton equations
View Description Hide DescriptionThe generalized Laxequation leaves the spectrum of invariant if and are selfadjoint operators. Consequently such equations possess many conversed quantities. With the help of this scheme, complex equations of Korteweg–de Vries type are derived.

A finitedimensional integrable system associated with the threewave interaction equations
View Description Hide DescriptionUnder a constraint between the potentials and the eigenfunctions, the AKNS matrix spectral problem and its adjoint spectral problem associated with the threewave interaction equations are nonlinearized so as to be a new finitedimensional Hamiltonian system. A general scheme for generating involutive systems of conserved integrals and their two new generators are proposed, by which the finitedimensional Hamiltonian system is further proved to be completely integrable in the Liouville sense. Moreover, the involutive solutions of the threewave interaction equations are given.

Phase space reduction and Poisson structure
View Description Hide DescriptionLet be a Gprincipal fiber bundle. The action of G on the cotangent bundle is free and Hamiltonian. By Liberman and Marle [Symplectic Geometry and Analytical Mechanics (Reidel, Dortrecht, 1987)] and Marsden and Ratiu [Lett. Math. Phys. 11, 161 (1981)] the quotient space is a Poissonmanifold. We will determine the Poisson bracket on the reduced Poissonmanifold and its symplectic leaves.
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 RELATIVITY AND GRAVITATION


Cauchy analysis of the linearized skew sector of the massive nonsymmetric gravitational theory
View Description Hide DescriptionCauchy analysis of the linearized field equations of the skew sector of the massive nonsymmetric gravitational theory shows that small perturbations give rise to bounded accelerations thereby ensuring good asymptotic behavior for the skew part of the fundamental tensor.

Almostcomplex and almostproduct Einstein manifolds from a variational principle
View Description Hide DescriptionIt is shown that the firstorder (Palatini) variational principle for a generic nonlinear metricaffine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almostproduct Einstein structure or to an almostcomplex antiHermitian Einstein structure on a manifold. It is proved that a real antiHermitian metric on a complex manifold satisfies the Kähler condition on the same manifold treated as a real manifold if and only if the metric is the real part of a holomorphic metric. A characterization of antiKähler Einstein manifolds and almostproduct Einstein manifolds is obtained. Examples of such manifolds are considered.

Null particle solutions in threedimensional (anti) de Sitter spaces
View Description Hide DescriptionWe obtain a class of exact solutions representing null particles moving in threedimensional (anti) de Sitter spaces by boosting the corresponding static point source solutions given by Deser and Jackiw. In de Sitter space the resulting solution describes two null particles moving on the (circular) cosmological horizon, while in antide Sitter space it describes a single null particle propagating from one side of the universe to the other. We also boost the Bañados–Teitelboim–Zanelli black hole solution to the ultrarelativistic limit and obtain the solution for a spinning null particle moving in antide Sitter space. We find that the ultrarelativistic geometry of the black hole is exactly the same as that resulting from boosting the Deser–Jackiw solution when the angular momentum of the hole vanishes. A general class of solutions is also obtained which represents several null particles propagating in the Deser–Jackiw background. The differences between the threedimensional and fourdimensional cases are also discussed.

Canonical general relativity: Matter fields in a general linear frame
View Description Hide DescriptionBuilding on the results of previous work [M. A. Clayton, “Canonical general relativity: Diffeomorphism constraints and spatial frame transformations,” J. Math. Phys. 39, 3805–3816 (1998)], we demonstrate how matter fields are incorporated into the general linear frame approach to general relativity. When considering the Maxwell oneform field, we find that the system that leads naturally to canonical vierbein general relativity has the extrinsic curvature of the Cauchy surface represented by gravitational as well as nongravitational degrees of freedom. Nevertheless the metric compatibility conditions are undisturbed, and this apparent derivativecoupling is seen to be an effect of working with (possibly orthonormal) linear frames. The formalism is adapted to consider a Dirac Fermion, where we find that a milder form of this apparent derivativecoupling appears.

Reductive decompositions and Einstein–Yang–Mills equations associated to the oscillator group
View Description Hide DescriptionAll of the homogeneous Lorentzian structures on the oscillator group equipped with a biinvariant Lorentzian metric, and then the associated reductive pairs, are obtained. Some of them are solutions of the Einstein–Yang–Mills equations.
