Volume 43, Issue 1, January 2002
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Coupling onedimensional timedependent classical and quantum transport models
View Description Hide DescriptionA transient model for onedimensional chargetransport in an open quantum system is proposed. In the semiclassical limit, it reduces to the inflow boundary value problem for the classical transport equation. On this basis, the coupling of classical and quantum transport models through an interface is investigated. Suitable interface conditions are derived through asymptotic formulas involving the quantum reflection–transmission coefficients and time delays.

Geometric amplitude, adiabatic invariants, quantization, and strong stability of Hamiltonian systems
View Description Hide DescriptionConsidered is a linear set of ordinary differential equations with a matrix depending on a set of adiabatically varying parameters. Asymptotic solutions have been constructed. As has been shown, an important characteristic determining the qualitative portrait of the system is the real part of Berry’s complex geometric phase, which we call the geometric amplitude. For systems with purely imaginary eigenvalues the equivalence has been proven in the adiabatic approximation of system sets without geometric amplitude, Hamiltonian systems, quantizable systems, and strongly stable systems. Classification of the systems without geometric amplitude is given with respect to the kind of matrix of the initial system.

Planar Dirac electron in Coulomb and magnetic fields: A Bethe ansatz approach
View Description Hide DescriptionThe Dirac equation for an electron in two spatial dimensions in the Coulomb and homogeneous magnetic fields is an example of the socalled quasiexactly solvable models. The solvable parts of its spectrum were previously solved from the recursion relations. In this work we present a purely algebraic solution based on the Bethe ansatz equations. It is realized that, unlike the corresponding problems in the Schrödinger and the Klein–Gordon cases, here the unknown parameters to be solved for in the Bethe ansatz equations include not only the roots of the wave function assumed, but also a parameter from the relevant operator. We also show that the quasiexactly solvable differential equation does not belong to the classes based on the algebra

Noncommuting limits in homogenization theory of electromagnetic crystals
View Description Hide DescriptionWe study the homogeneous properties of metallic 2D photonic crystals. We give a rigorous proof that the limits when the ratio period over wavelength tends to zero and the conductivity tends to infinity do not commute.

Covariant geometric quantization of nonrelativistic timedependent mechanics
View Description Hide DescriptionWe provide geometric quantization of the vertical cotangent bundle equipped with the canonical Poissonstructure and treated as a momentum phase space of nonrelativistic timedependent mechanics. We show that this quantization is equivalent to fiberwise quantization of symplectic fibers of and that the quantum algebra of timedependent mechanics is an instantwise algebra. Quantization of the classical evolution equation defines a connection on this instantwise algebra and describes quantum evolution in timedependent mechanics as a parallel transport.

Foundations for relativistic quantum theory. I. Feynman’s operator calculus and the Dyson conjectures
View Description Hide DescriptionIn this paper, we provide a representation theory for the Feynman operator calculus. This allows us to solve the general initialvalue problem and construct the Dyson series. We show that the series is asymptotic, thus proving Dyson’s second conjecture for quantum electrodynamics. In addition, we show that the expansion may be considered exact to any finite order by producing the remainder term. This implies that every nonperturbative solution has a perturbative expansion. Using a physical analysis of information from experiment versus that implied by our models, we reformulate our theory as a sum over paths. This allows us to relate our theory to Feynman’s path integral, and to prove Dyson’s first conjecture that the divergences are in part due to a violation of Heisenberg’s uncertainly relations.

Spiked harmonic oscillators
View Description Hide DescriptionA complete variational treatment is provided for a family of spikedharmonic oscillator Hamiltonians for arbitrary A compact topological proof is presented that the set of known exact solutions for constitutes an orthonormal basis of the Hilbert space Closedform expressions are derived for the matrix elements of H with respect to S. These analytical results, and the inclusion of a further free parameter, facilitate optimized variational estimation of the eigenvalues of H to high accuracy.

The noncommutative harmonic oscillator in more than one dimension
View Description Hide DescriptionThe noncommutative harmonic oscillator, with noncommutativity not only in position space but also in phase space, in arbitrary dimension is examined. It is shown that the ⋆genvalue problem, which replaces the Schrödinger problem in this case, can be decomposed into separate harmonic oscillator equations for each dimension. The twodimensional noncommutative harmonic oscillator (four noncommutative phasespace dimensions) is investigated in greater detail. The requirement of the existence of rotationally symmetric solutions leads to a two parameter harmonic oscillator which is completely solved in this case. The angular momentum operator is derived and its ⋆genvalue problem is shown to be equivalent to the usual eigenvalue problem of the ⋆genfunction related wave function. The ⋆genvalues of the angular momentum are found to depend on the energy difference of the oscillations in the two dimensions. Furthermore two examples of a symmetric noncommutative harmonic oscillators are analyzed. The first is the noncommutative twodimensional Landau problem with harmonic oscillator potential, which shows degeneracy in the energy levels for certain critical values of the noncommutativity parameters, and the second is the threedimensional harmonic oscillator with noncommuting coordinates and momenta.

The supersymmetric technique for randommatrix ensembles with zero eigenvalues
View Description Hide DescriptionThe supersymmetric technique is applied to computing the average spectral density near zero energy in the large limit of the randommatrix ensembles with zero eigenvalues: odd, and the chiral ensembles (classes and The supersymmetric calculations reproduce the existing results obtained by other methods. The effect of zero eigenvalues may be interpreted as reducing the symmetry of the zeroenergy supersymmetric action by breaking a certain Abelian symmetry.

Supersymmetric spin networks
View Description Hide DescriptionIn this article we study the construction of supersymmetric spin networks, which has a direct interpretation in context of the representation theory of the superalgebra. In particular we analyze a special kind of spin network associated with superalgebra It turns out that the set of corresponding spin network states forms an orthogonal basis of the Hilbert space and this argument holds even in the qdeformed case. The spin networks are also discussed briefly. We expect they could provide useful techniques to quantum supergravity and gauge field theories from the point of nonperturbative view.

The Fejér average and the mean value of a quantity in a quasiclassical wave packet
View Description Hide DescriptionThe mean value of a quantity in an equally weighted wave packet was recently found in the classical limit to be the Fejér average of partial sums of Fourier series expansion of the classical quantity, and the number of stationary states in it is equal to that of partial sums. The incompleteness of the Fejér average in representing a classical quantity enables us to define a classical uncertainty relation which turns out to be the counterpart of the quantum one. In this paper, two typical quantum systems, a harmonic oscillator and a particle in an infinite square well, are used to illustrate the abovementioned points.

Discrete Kaluza–Klein from scalar fluctuations in noncommutative geometry
View Description Hide DescriptionWe compute the metric associated with noncommutative spaces described by a tensor product of spectral triples. Wellknown results of the twosheets model (distance on a sheet, distance between the sheets) are extended to any product of two spectral triples. The distance between different points on different fibers is investigated. When one of the triples describes a manifold, one finds a Pythagorean theorem as soon as the direct sum of the internal states (viewed as projections) commutes with the internal Dirac operator. Scalar fluctuations yield a discrete Kaluza–Klein model in which the extra component of the metric is given by the internal part of the geometry. In the standard model, this extra component comes from the Higgs field.

PseudoHermiticity versus symmetry: The necessary condition for the reality of the spectrum of a nonHermitian Hamiltonian
View Description Hide DescriptionWe introduce the notion of pseudoHermiticity and show that every Hamiltonian with a real spectrum is pseudoHermitian. We point out that all the symmetric nonHermitian Hamiltonians studied in the literature belong to the class of pseudoHermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudoHermiticity. We explore the basic properties of general pseudoHermitian Hamiltonians, develop pseudosupersymmetric quantum mechanics , and study some concrete examples, namely the Hamiltonian of the twocomponent Wheeler–DeWitt equation for the FRWmodels coupled to a real massive scalar field and a class of pseudoHermitian Hamiltonians with a real spectrum.

Asymptotics of bound states and bands for laterally coupled waveguides and layers
View Description Hide DescriptionThe asymptotics (in the width of windows) of eigenvalues and bands for twodimensional waveguides and threedimensional layers coupled through small windows is obtained. The technique is matching of asymptotic expansions of the solutions of boundary value problems.

Remarks on the lattice Green’s function: The Glasser case
View Description Hide DescriptionWe have investigated the lattice Green’s function for the Glasser cubic lattice. Expressions for its density of states, phase shift, and scattering cross section in terms of complete elliptic integrals of the first kind are derived.

Infinite infrared regularization and a state space for the Heisenberg algebra
View Description Hide DescriptionWe present a method for the construction of a Krein space completion for spaces of test functions, equipped with an indefinite inner product induced by a kernel which is more singular than a distribution of finite order. This generalizes a regularization method for infrared singularities in quantum field theory, introduced by Morchio and Strocchi, to the case of singularities of infinite order. We give conditions for the possibility of this procedure in terms of local differential operators and the Gelfand–Shilov test function spaces, as well as an abstract sufficient condition. As a model case we construct a maximally positive definite state space for the Heisenberg algebra in the presence of an infinite infrared singularity.

Lorentz invariant Lagrangians
View Description Hide DescriptionThe aim of this article is to study certain Lorentz invariant Lagrangians. The first of these Lagrangians could be related to a particle of spin moving in a particular Yang–Mills gauge field. The second Lagrangian is related to the relativistic Newton–Coulomb problem. For each of these Lagrangians, we write the corresponding wave equations and determine the negative energy levels. The article concludes with the construction of a class of Lagrangians associated with pairs of particles, one of which has zero mass.

Fedosov quantization on symplectic ringed spaces
View Description Hide DescriptionWe expose the basics of the Fedosov quantization procedure, placed in the general framework of symplectic ringed spaces. This framework also includes some Poissonmanifolds with nonregular Poisson structures, presymplectic manifolds, complex analytic symplectic manifolds, etc.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Planar trajectories in a monopole field
View Description Hide DescriptionMotion of a Yang–Mills particle in a monopole field is proposed and planar orbits are observed. The planar orbits are studied further with some numerical analysis of the equations of motion.

A planar Runge–Lenz vector
View Description Hide DescriptionFollowing Dahl’s method an exact Runge–Lenz vector M with two components and is obtained as a constant of motion for a two particle system with charges and whose electromagnetic interaction is based on Chern–Simons electrodynamics. The Poisson bracket but is modified by the appearance of the product as central charges.
