Volume 36, Issue 3, March 1995
Index of content:

High‐ and low‐energy estimates for the Dirac equation
View Description Hide DescriptionThe scattering theory for the Dirac equation with radial potential is studied. The leading term at high energy is computed and Parzen’s theorem is proved. In the case of zero mass, the behavior at low energy is analyzed, which turns out to be different from the low‐energy behavior for positive mass, and an appropriate version of Levinson’s theorem is proved under the assumption that the potential is integrable over (0,∞).

Algebraic treatment of the Kaluza–Klein monopole system
View Description Hide DescriptionIn analogy to the Kepler problem, the Green’s function for a particle in the background of the Kaluza–Klein monopole is constructed through the algebraic approach, with the help of the Kustaanheimo–Stiefel variables and the generators of the SO(2,1) group in the exponential representation of Schwinger. The bound and continuum states are also obtained.

Improved high‐energy bounds on absorptive parts of elastic scattering amplitudes
View Description Hide DescriptionImproved rigorous high‐energy upper bounds are established on the absorptive parts of the elastic scattering amplitudes for the momentum‐transfer‐squared variable t positive and within the Lehmann–Martin ellipse. These bounds are used to set upper bounds on Regge trajectories and also to extend the region in the complex t plane where the amplitudes cannot have zeros. No assumption is made about the high‐energy behavior of the total cross sections.

The double‐well splitting of the low energy levels for the Schrödinger operator of discrete φ^{4}‐models on tori
View Description Hide DescriptionThe formula is obtained for the asymptotics of the splitting of the first two eigenvalues of the Schrödinger operator that stands for the discrete multidimensional φ^{4}‐model on tori. The behavior of the main term is investigated in the limit of the large number of particles.

On curved vortex solutions in the Abelian Higgs model
View Description Hide DescriptionIt is proven that curved Abrikosov–Nielsen–Olesen vortexsolutions of the Abelian Higgs model exist around world sheets given by solutions of Nambu–Goto equations with additional constraints. The explicit form of these constraints is also found.

Quantization of the electromagnetic field on static space–times
View Description Hide DescriptionA quantization of the electromagnetic four‐potential in the Lorentz gauge on static space–times with compact Cauchy surfaces is developed herein. For the classical problem, the field solution is obtained by use of a Hilbert space approach in which time evolution reduces to a unitary mapping of Cauchy data. For the quantum problem, a representation of the CCRs on a Fock space is constructed. The gauge condition is imposed as a constraint on state vectors and determines a physical subspace. The field equations are recovered on the physical Fock space (Gupta–Bleuler method).

Flat connections and nonlocal conserved quantities in irrational conformal field theory
View Description Hide DescriptionIrrational conformal field theory (ICFT) includes rational conformal field theory as a small subspace, and the affine‐Virasoro Ward identities describe the biconformal correlators of ICFT. The Ward identities are reformulated as an equivalent linear partial differential system with flat connections and new nonlocal conserved quantities. As examples of the formulation, the system of flat connections is solved for the coset correlators, the correlators of the affine‐Sugawara nests, and the high‐level n‐point correlators of ICFT.

A noncommutative geometric model with horizontal family symmetry
View Description Hide DescriptionThe important role of multiple fermionic families in noncommutative geometry is investigated by introducing a horizontal symmetry into the model building scheme. It is demonstrated that this can be done consistently.

Boson mapping of symplectic algebras with Abelian subalgebra mapped as coordinates
View Description Hide DescriptionAn extended Dyson and Holstein‐Primakoff boson mapping is constructed for coordinate systems. Only functions which depend on even powers in the coordinates are considered. It is shown that the Holstein‐Primakoff boson mapping for coordinate systems is different from the one using creation and annihilation operators, with some advantages in application. As an example the mapping is applied to the schematic model of constant gauge fields in Quantum‐Chromo‐Dynamics with SU(2) for color. Other possible applications related to field theory are briefly discussed.

Price’s bound on the structure factor: Derivation and comparison with some exact results
View Description Hide DescriptionSome years ago, P. J. Price obtained an important (but not well recognized) bound on the structure factor in the ground state. A more general version is derived by means of sum rules, showing a limitation imposed on by the fsum rule. The condition for merging of the bound with the structure factor turns out to be the existence of a single branch in the excitation spectrum. Price’s bound is tested in exactly solvable many‐body models, most of which are models for Fermi particles, some not satisfying the fsum rule. This analysis sheds light on the structure factors of these models. Also, Price’s bound is compared with other bounds obtained by convexity theory. Finally, by formulating it in terms of a moment, the existence of Price’s bound in a wider class of models is established.

Dirac and reduced quantization: A Lagrangian approach and application to coset spaces
View Description Hide DescriptionA Lagrangian treatment of the quantization of first class Hamiltonian systems with constraints and Hamiltonian linear and quadratic in the momenta, respectively, is performed. The ‘‘first reduce and then quantize’’ and the ‘‘first quantize and then reduce’’ (Dirac’s) methods are compared. A source of ambiguities in this latter approach is pointed out and its relevance on issues concerning self‐consistency and equivalence with the ‘‘first reduce’’ method is emphasized. One of the main results is the relation between the propagator obtained à laDirac and the propagator in the full space. As an application of the formalism developed, quantization on coset spaces of compact Lie groups is presented. In this case it is shown that a natural selection of a Dirac quantization allows for full self‐consistency and equivalence. Finally, the specific case of the propagator on a two‐dimensional sphere S ^{2} viewed as the coset space SU(2)/U(1) is worked out.

Symmetries, parametrization, and group structure of transfer matrices in quantum scattering theory
View Description Hide DescriptionUseful and new representations of the transfer matrices are explicitly obtained for systems with and without time reversal or spin rotation symmetries. In the symplectic, orthogonal, and unitary cases, the whole group of transfer matrices contains a compact and a noncompact subgroup, and the generalized Bargman’s polar parametrization is obtained in a straightforward way. It is shown that the transfer matrices used by Luttinger, to prove the Saxon–Hutner conjecture on the forbidden energy levels, belong to the noncompact Sp(2,C) subgroup.

Symplectic symmetry of Hubbard model and η pairing
View Description Hide DescriptionBy means of symplectic group Sp(2M), the symplectic symmetry of the Hubbard model and the η pairing are investigated to approach the eigenstates which have the largest spin multiplicities. The behavior of the off‐diagonal long‐range order of these eigenstates is examined by the criterion proposed by Yang and the energy diagram of the corresponding eigenvalues is elucidated by the η pairing mechanism. A basis set which is expressible in terms of η and η_{ a } pairings is proposed.

Local differential geometry as a representation of the supersymmetric oscillator
View Description Hide DescriptionThe choice of a coordinate chart on an analytical R ^{ n }(R ^{ n } _{ a }) provides a representation of the n‐dimensional supersymmetric oscillator. The 1‐parameter group of dilations provides a Euclidean evolution moving the system through a sequence of charts, that at each instant supply a Hilbert space by Cartan’s exterior algebra endowed with a suitable scalar product. Stationary states and coherent states are eigenstates of the Lie derivatives generating the dilations and the translations, respectively.

Transformation of the solutions of the Maxwell equations on a Lorentzian manifold by means of a certain class of diffeomorphisms
View Description Hide DescriptionPoincaré invariance of the Maxwell equations in Minkowski space enables one to generate in a usual way new solutions (corresponding to new sources) from a given one by means of the pullback operation. A method is proposed generalizing this to a wider class of diffeomorphisms f:M → M of a Lorentzian manifold (M,g). It results in the simple explicit formulas for obtaining the new field F ^{ f } and its source j ^{ f } from a given pair (F,j). Some applications, e.g., a uniformly rotating or radially pulsating source, are presented.

Canonical partition function for the hydrogen atom via the Coulomb propagator
View Description Hide DescriptionThe electronic partition function for the hydrogen atom is derived by integration over the recently‐available Coulomb propagator. This provides a resolution to an old paradox in statistical mechanics: the apparent divergence of the hydrogen partition function. Electronic excitation does not contribute significantly to the standard‐state partition function until temperatures of the order of 5000 K. Thereafter, the continuum, with its immense density of states, makes the dominant contribution. From the discrete and continuum contributions to the partition function, a modification of the Saha–Boltzmann equation for the ionization equilibrium in atomic hydrogen is derived.

Polylogarithmic analysis of chemical potential and fluctuations in a D‐dimensional free Fermi gas at low temperatures
View Description Hide DescriptionThe chemical potential and fluctuations in number of particles in a D‐dimensional free Fermi gas at low temperatures are obtained by means of polylogarithms. This idea is extended to show that the density of any ideal gas, whether Fermi, Bose, or classical, can be expressed in polylogarithms. The densities of different statistics correspond to different domains of polylogarithms in such a way that there emerges a unifying picture. The density of the classical ideal gas represents a fixed point of polylogarithms. Inequalities for polylogarithms are used to provide a precise bound on errors in the fermion chemical potential at low temperatures.

Lazutkin coordinates and invariant curves for outer billiards
View Description Hide DescriptionThe outer billiard ball map (OBM) is defined from and to the exterior of a domain, Ω, in the plane as taking a point, q, to another point, q _{1}, when the line segment with endpoints q and q _{1} is tangent to the boundary, ∂Ω (with a chosen orientation), and the point of tangency with the boundary divides the segment in half. Let C be an invariant circle for the OBM on Ω, with ∂Ω smooth with positive curvature. After computing the loss of derivatives between ∂Ω and C, it is shown via KAM theory that in this setting the OBM has uncountably many invariant circles in any neighborhood of the boundary. One is also led to an infinitesimal obstruction for the evolution property, an obstruction which, among closed smooth convex curves, is only removed for ellipses.

Hamiltonian structure and integrability of the stationary Kuramoto–Sivashinsky equation
View Description Hide DescriptionIn this article, the stationary Kuramoto–Sivashinsky (KS) equation is studied. It is shown, first, that this equation can be transformed into a one‐dimensional Hamiltonian system describing the motion of a particle in a time‐dependent potential. Second, working with this formalism and coming back, then, to the initial KS equation, a subset of initial conditions is derived for which the corresponding solutions u(x) will behave like the dominant pole 120/(x−x _{0})^{3}, deduced from the Painlevé analysis, and will explode in a finite ‘‘time.’’ Finally, special attention is given to the well‐known Kuramoto–Tsuzuki solution and it is shown that provided the independent variable x is initialized in a specified domain, this solution explodes also like 120/(x−x _{0})^{3}.

Integrable dynamics of a discrete curve and the Ablowitz–Ladik hierarchy
View Description Hide DescriptionIt is shown that the following elementary geometric properties of the motion of a discrete (i.e., piecewise linear) curve select the integrable dynamics of the Ablowitz–Ladik hierarchy of evolution equations: (i) the set of points describing the discrete curve lie in the sphere S ^{3}; (ii) the distance between any two subsequent points does not vary in time; (iii) the equations of the dynamics do not depend explicitly on the radius of the sphere. These results generalize to a discrete context the previous work on continuous curves [A. Doliwa and P. M. Santini, Phys. Lett. A 185, 373 (1994)].