Volume 38, Issue 3, March 1997
Index of content:

Higher level WZW sectors from free fermions
View Description Hide DescriptionWe introduce a gauge group of internal symmetries of an ambient algebra as a new tool for investigating the superselection structure of WZW theories and the representation theory of the corresponding affine Lie algebras. The relevant ambient algebra arises from the description of these conformal field theories in terms of free fermions. As an illustration we analyze in detail the so(N) WZW theories at level two, which allows us in particular to construct explicit bases for the leveltwo irreducible highest weight modules. In this case there is actually a homomorphism from the representation ring of the gauge group to the subring of the WZW fusion ring that corresponds to the Neveu–Schwarz sector, even though the leveltwo observable algebra is smaller than the gauge invariant subalgebra of the field algebra.

The Dirac–Maxwell equations with cylindrical symmetry
View Description Hide DescriptionA reduction of the Dirac–Maxwell equations in the case of static cylindrical symmetry is performed. The behavior of the resulting system of o.d.e.s. is examined analytically and numerical solutions presented. There are two classes of solutions. The first type of solution is a Dirac field surrounding a charged “wire.’’ The Dirac field is highly localized, being concentrated in cylindrical shells about the wire. A comparison with the usual linearized theory demonstrates that this localization is entirely due to the nonlinearities in the equations which result from the inclusion of the “selffield.’’ The second class of solutions have the electrostatic potential finite along the axis of symmetry but unbounded at large distances from the axis.

The noncommutative constraints on the standard model à la Connes
View Description Hide DescriptionNoncommutative geometry applied to the standard model of electroweak and strong interactions was shown to produce fuzzy relations among masses and gauge couplings. We refine these relations and show then that they are exhaustive.

NonAbelian Berry’s phase in a slowly rotating electric field
View Description Hide DescriptionWe observe that the nonAbelian Berry’s potential can appear in the atomic system under the slowly rotating electric field (Stark effect). We calculate Berry’s phase using the perturbation theory in the weak external electric field. To elaborate the nature of nonAbelian character, we demonstrate the change of states during the adiabatic transportation for a particular set of paths.

Dia and paramagnetism for nonhomogeneous magnetic fields
View Description Hide DescriptionDiamagnetism of the magnetic Schrödinger operator and paramagnetism of the Pauli operator are rigorously proven for nonhomogeneous magnetic fields in the large field, in the large temperature and in the semiclassical asymptotic regimes. New counterexamples are presented which show that neither dia nor paramagnetism is true in a robust sense (without asymptotics). In particular, we demonstrate that the recent diamagnetic comparison result by Loss and Thaller [M. Loss and B. Thaller, Commun. Math. Phys. (submitted)] is essentially the best one can hope for.

On the statistical independence of algebras of observables
View Description Hide DescriptionWe reexamine various notions of statistical independence presently in use in algebraic quantum theory, establishing alternative characterizations for such independence, some of which are also valid without assuming that the observable algebras mutually commute. In addition, in the context which holds in concrete applications to quantum theory, the equivalence of three major notions of statistical independence is proven.

Geometries of relativistic strings and other pbranes
View Description Hide DescriptionWe discuss the geometric structure outside an infinitely long and straight relativistic bosonic string in various space–time dimensions. This structure should, in particular, be a good model for the geometry outside a gauge cosmic string in four dimensions. We study a new such geometry in some depth. A discussion of a possible internal geometry of a gauge cosmic string is also given. We show that space–time is flat outside a relativistic string only in four space–time dimensions. This is connected to the lack of a proper Newtonian limit of Einstein’s theory inside any source for the string geometry in four dimensions. On this background we discuss some features of the geometries induced by other pbranes in various dimensions. We also provide a study of the creation of a gauge cosmic string in the full Einstein theory in four dimensions, with results that differ considerably from similar earlier considerations.

On quantum field theory with nonzero minimal uncertainties in positions and momenta
View Description Hide DescriptionWe continue studies on quantum field theories on noncommutative geometric spaces, focusing on classes of noncommutative geometries which imply ultraviolet and infrared modifications in the form of nonzero minimal uncertainties in positions and momenta. The case of the ultraviolet modified uncertainty relation which has appeared from string theory and quantum gravity is covered. The example of Euclidean φ^{4}theory is studied in detail and in this example we can now show ultraviolet and infrared regularization of all graphs.

On the Born–Oppenheimer approximation of diatomic wave operators. II. Singular potentials
View Description Hide DescriptionThis paper is a continuation of an earlier paper [Commun. Math. Phys. 152, 73–95 (1993)]. Its purpose is to extend the quantitative estimates given previously for regular potentials to the case of potentials admitting some Coulombtype singularities. More precisely, considering the twocluster wave operators in diatomicscattering, we approximate them by socalled “adiabatic’’ wave operators as the mass of the nuclei tends to infinity.

Topological field theories and integrable models
View Description Hide DescriptionWe show that the classical nonAbelian pure Chern–Simons action is related to nonrelativistic models in (2+1) dimensions, via reductions of the gauge connection in Hermitian symmetric spaces. In such models the matter fields are coupled to gauge Chern–Simons fields, associated with the isotropy subgroup of the considered symmetric space. Thus a relation between the Chern–Simons theory and the Davey–Stewartson hierarchy is established in a natural way. The Bäcklund transformations are interpreted in terms of Chern–Simons constraints. Moreover, certain nonintegrable Heisenberg models can be embedded into the pure Chern–Simons theory. The main classical and quantum properties of these systems are discussed.

Inlaying vertex function and scattering amplitude
View Description Hide DescriptionScattering processes among strings are analyzed by using fundamental equations of three types, which divide the whole complex zplane into various types of N punctured ring domains plus various unpunctured ring domains, where internal strings freely propagate. In order to calculate scattering amplitudes (among physical particles) in Witten’s quantum string field theory, we derive and apply the “Gluing theorem,’’ mathematical proof of which is given (in operator forms) by constructing various (inlint) conformal mapping operators.

Group quantization on configuration space: Gauge symmetries and linear fields
View Description Hide DescriptionA new, configurationspace picture of a formalism of group quantization, the GAQ formalism, is presented in the context of a previous algebraic generalization. This presentation serves to make a comprehensive discussion in which other extensions of the formalism, principally to incorporate gauge symmetries, are developed as well. Both images are combined in order to analyze, in a systematic manner and with complete generality, the case of linear fields (Abelian current groups). To illustrate these developments we particularize them for several fields and, in particular, we carry out the quantization of the Abelian Chern–Simons models over an arbitrary closed surface in detail.

Chaos and geometrical resonance in the damped pendulum subjected to periodic pulses
View Description Hide DescriptionThe chaotic behavior of a damped pendulum driven by a periodic string of pulses is studied by means of Melnikov’s analysis. The reduction of homoclinic chaos, in the asymptotic case of infinite period driving, is explained in terms of geometrical resonance.

Dispersive nonlinear geometric optics
View Description Hide DescriptionWe construct infinitely accurate approximate solutions to systems of hyperbolic partial differential equations which model short wavelength dispersive nonlinear phenomena. The principal themes are the following. (1) The natural framework for the study of dispersion is wavelength ε solutions of systems of partial differential operators in ε∂. The natural εcharacteristic equation and εeikonal equations are not homogeneous. This corresponds exactly to the fact that the speeds of propagation, which are called group velocities, depend on the length of the wave number. (2) The basic dynamic equations are expressed in terms of the operator ε∂ _{t} . As a result growth or decay tends to occur at the catastrophic rate e^{ct/ε} . The analysis is limited to conservative or nearly conservative models. (3) If a phase φ(x)/ε satisfies the natural εeikonal equation, the natural harmonic phases, nφ(x)/ε, generally do not. One needs to impose a coherence hypothesis for the harmonics. (4) In typical examples the set of harmonics which are eikonal is finite. The fact that high harmonics are not eikonal suppresses the wave steepening which is characteristic of quasilinear wave equations. It also explains why a variety of monochromatic models are appropriate in nonlinear settings where harmonics would normally be expected to appear. (5) We study wavelength ε solutions of nonlinear equations in ε∂ for times O(1). For a given system, there is a critical exponent p so that for amplitudes O(ε^{p}), one has simultaneously smooth existence for t=O(1), and nonlinear behavior in the principal term of the approximate solutions. This is the amplitude for which the time scale of nonlinear interaction is O(1). (6) The approximate solutions have residual each of whose derivatives is O(ε^{n}) for all n>0. In addition, we prove that there are exact solutions of the partial differential equations, that is with zero residual, so that the difference between the exact solution and the approximate solutions is infinitely small. This is a stability result for the approximate solutions.

Bethe ansatz study for ground state of Fateev Zamolodchikov model
View Description Hide DescriptionA Bethe ansatz study of a selfdual spin lattice model, originally proposed by V. A. Fateev and A. B. Zamolodchikov, is undertaken. The connection of this model to the ChiralPotts model is established. Transcendental equations connecting the zeros of Fateev–Zamolodchikov transfer matrix are derived. The free energies for the ferromagnetic and the antiferromagneticground states are found for both even and odd spins.

Constrained KP models as integrable matrix hierarchies
View Description Hide DescriptionWe formulate the constrained KP hierarchy (denoted by cKP as an affine matrix integrable hierarchy generalizing the Drinfeld–Sokolov hierarchy. Using an algebraic approach, including the graded structure of the generalized Drinfeld–Sokolov hierarchy, we are able to find several new universal results valid for the cKP hierarchy. In particular, our method yields a closed expression for the second bracket obtained through Dirac reduction of any untwisted affine Kac–Moody current algebra. An explicit example is given for the case , for which a closed expression for the general recursion operator is also obtained. We show how isospectral flows are characterized and grouped according to the semisimple nonregular element of and the content of the center of the kernel of .

A solvable Hamiltonian system: Integrability and actionangle variables
View Description Hide DescriptionWe prove that the dynamical system characterized by the Hamiltonian H=λcos]} proposed and studied by Calogero [J. Math. Phys. 36, 9 (1994)] and Calogero and van Diejen [Phys. Lett. A 205, 143 (1995)] is equivalent to a system of noninteracting harmonic oscillators both classically and quantum mechanically. We find the explicit form of the conserved currents that are in involution. We also find the actionangle variables and solve the initial value problem in a very simple form.

Some rigorous results for the kinematic dynamo problem with general boundary conditions
View Description Hide DescriptionThe main energy inequalities for the induction equation are studied for a variety of boundary conditions. The results are applied to the evolution of the magnetic field and the vector potential within a conducting fluid disregarding the influence of these upon the velocity (the kinematic dynamo problem). Emphasis is placed upon the behavior of these fields at the singular limit of zero magnetic diffusivity.

Outer trapped surfaces and their apparent horizon
View Description Hide DescriptionWe give a new definition of “closed outer trapped surface’’ with respect to a hypersurface and show that the boundary of the trapped region (the apparent horizon) is a marginally trapped surface, i.e., has vanishing outer null expansion. While this is an important and well known result, there does not seem to exist a proof in the literature.

Multiplane gravitational lensing. III. Upper bound on number of images
View Description Hide DescriptionThe total number of lensed images of a light source undergoing gravitational lensing varies as the source traverses a caustic network. It is rigorously shown that for a pointlike light source not on any caustic, a threedimensional distribution of g point masses on g lens planes creates at most 2(2 ^{2(g−1)} −1) lensed images of the source (g⩾2). This complements previous work [Paper I, J. Math. Phys. 36, 4263 (1995)] that showed at least 2 ^{g} lensed images occur. Application of the upper bound to the global geometry of caustics is also presented. Our methods are based on a complex formulation of pointmass gravitational lensing and techniques from the theory of resultants. The latter yields a new approach to studying upper bounds on number of lensed images due to pointmass gravitational lens systems.