Volume 38, Issue 9, September 1997
Index of content:

Towards a Coulomb gas of instantons of the S0(4)×U(1) Higgs model on
View Description Hide DescriptionThe Higgs model on is extended by a term so that the action receives a nonvanishing contribution from the interactions of twoinstantons and threeinstantons, and can be expressed as the inverse of the Laplacian on in terms of the mutual distances of the instantons. The oneinstanton solutions of both the basic and the extended models have been studied in detail numerically.

Extension of the Barut–Girardello coherent state and path integral
View Description Hide DescriptionWe extend the Barut–Girardello coherent state for the representation of SU(1,1) to the coherent state for a representation of and construct the measure. We also construct a path integral formula for some Hamiltonian.

An affine string vertex operator construction at an arbitrary level
View Description Hide DescriptionAn affine vertex operator construction at an arbitrary level is presented which is based on a completely compactified chiral bosonic string whose momentum lattice is taken to be the (Minkowskian) affine weight lattice. This construction is manifestly physical in the sense of string theory, i.e., the vertex operators are functions of Del Giudice–Di Vecchia–Fubini (DFF) “oscillators” and the Lorentzgenerators, both of which commute with the Virasoro constraints. We therefore obtain explicit representations of affine highest weight modules in terms of physical (DDF) string states. This opens new perspectives on the representation theory of affine Kac–Moody algebras, especially in view of the simultaneous treatment of infinitely many affine highest weight representations of arbitrary level within a single state space as required for the study of hyperbolic Kac–Moody algebras. A novel interpretation of the affine Weyl group as the “dimensional null reduction” of the corresponding hyperbolic Weyl group is given, which follows upon reexpression of the affine Weyl translations as Lorentz boosts.

Fermion doubling on a lattice and topological aspects of chiral anomaly
View Description Hide DescriptionThe problem of fermion doubling on a lattice has been discussed here from the specific geometrical properties of a lattice structure and topological aspects of chiral anomaly. It is argued that there cannot be chiral anomaly on a lattice and as such there cannot be any conserved charge. This unveils the root cause of fermion doubling, and the unwanted fermions just reflect the geometrical properties of a lattice and may be viewed as to represent the “fictitious” chiral spinors associated with the lattice structure which make chiral fermions anomaly free.

The BRST operator of quantum symmetries: The quantum analogs of Donaldson invariants
View Description Hide DescriptionWe construct the BRST operator of quantum symmetries and show that the nilpotency of this operator can either be derived from the Hopf axiom structure of the quantum group symmetries or from the Jacobi identity of their quantum Lie algebra. We extend this BRST operator to the topological transformations and we investigate the properties of invariant polynomials of curvatures from which we derive the descent equations for Donaldson invariants.

The Dirac operator and gamma matrices for quantum Minkowski spaces
View Description Hide DescriptionGamma matrices for quantum Minkowski spaces are found. The invariance of the corresponding Dirac operator is proven. We introduce momenta for spin particles and get (in certain cases) formal solutions of the Dirac equation.

Supersymmetry and supercoherent states of a nonrelativistic free particle
View Description Hide DescriptionCoordinate atypical representation of the orthosymplectic superalgebra in a Hilbert superspace of square integrable functions constructed in a special way is given. The quantum nonrelativistic free particle Hamiltonian is an element of this superalgebra which turns out to be a dynamical superalgebra for this system. The supercoherent states, defined by means of a supergroup displacement operator, are explicitly constructed. These are the coordinate representation of the known atypical abstract super group coherent states. We interpret obtained results from the classical mechanics viewpoint as a model of classical particle which is immovable in the even sector of the phase superspace and is in rectilinear movement (in the appropriate coordinate system) in its odd sector.

Debye potentials for Maxwell and Dirac fields from a generalization of the Killing–Yano equation
View Description Hide DescriptionBy using conformal Killing–Yano tensors, and their generalizations, we obtain scalar potentials for both the sourcefree Maxwell and massless Dirac equations. For each of these equations we construct, from conformal Killing–Yano tensors, symmetry operators that map any solution to another.

A continued fraction representation for the effective conductivity of a twodimensional polycrystal
View Description Hide DescriptionA continued fraction representation for the effective conductivity tensor of a twodimensional polycrystal is derived. This representation is in terms of a sequence of positive definite symmetric matrices which characterize the underlying geometric structure of the material. The proof is accomplished by considering a particular basis for the Hilbert space of fields in the composite in which the linear operators relevant to determining the effective conductivity take simple forms as infinite matrices. These infinite matrices are then used in the variational definition of the effective conductivity to formulate the continued fraction. This continued fraction is used to derive upper and lower bounds on

On 3+1 decompositions with respect to an observer field via differential forms
View Description Hide Description3+1 decompositions of differential forms on a Lorentzian manifold +−−−) with respect to arbitrary observer field and the decomposition of the standard operations acting on them are studied, making use of the ideas of the theory of connections on principal bundles. Simple explicit general formulas are given as well as their application to the Maxwell equations.

On a first integral of the Kepler problem
View Description Hide DescriptionA quadratic first integral of the Kepler problem, obtained by Benenti (“Orthogonal separable dynamical systems,” 5th International Conference on Differential Geometry and its Applications, 24–28 August 1992, Silesian University at Opava) through separation in elliptic coordinates, is shown to be intimately connected with the prequantization of the Kepler manifold, thus acquiring a physical interpretation.

Dynamics of Brownian particles in a potential
View Description Hide DescriptionLet be the density of a cloud of particles, and the temperature at at time Let and measurable uniformly positive and exponentially bounded in be given. We study the coupled system with initial data and We show that there is a unique solution for small times if the conditions and hold.

A class of homogeneous nonlinear evolution equations with stable, localized solutions in any dimension
View Description Hide DescriptionA new set of nonlinear evolution equations is introduced and studied. These equations derive from a local Lagrangian and are (i) homogeneous and (ii) invariant under the Galilei group or the Lorentz group (including time reversal). Some of them have confined solutions with a solitonlike behavior, irrespective of the space dimension. Moreover, these solutions are shown to be stable against small and localized perturbations. Another family of localized solutions is worked out, and briefly discussed with regard to the elusive integrability properties of the new equations.

Signs and approximate magnitudes of Lyapunov exponents in continuous time dynamical systems
View Description Hide DescriptionMethods for algebraically determining the signs and the magnitudes of Lyapunov exponents of a given dynamical system are studied. The existence of zero Lyapunov exponents for the Toda, Hénon–Heiles, and Rössler systems are shown. The approximate Lyapunov spectra of Lorenz and Rössler systems are computed.

A note on fractional KdV hierarchies
View Description Hide DescriptionOne of the cornerstones of the theory of integrable systems of KdV type has been the remark that the GD (Gel’fand–Dickey) equations are reductions of the full Kadomtsev–Petviashvilij (KP) theory. In this paper we address the analogous problem for the fractional KdV theories, by suggesting candidates of the “KP theories” lying behind them. These equations are called “ hierarchies,” and are obtained as reductions of a bigger dynamical system, which we call the “central system.” The procedure allowing passage from the central system to the equations, and then to the fractional equations, is discussed in detail in the paper. The case of is given as a paradigmatic example.

Exact solution of a boundary value problem in semiconductor kinetic theory
View Description Hide DescriptionAn explicit solution of the stationary onedimensional halfspace boundary value problem for the linear Boltzmann equation is presented in the presence of an arbitrarily high constant external field. The collision kernel is assumed to be separable, which is also known as “relaxation time approximation;” the relaxation time may depend on the electron velocity. Our method consists in a transformation of the halfspace problem into a nonnormal singular integral equation, which has an explicit solution.

Algebra of pseudodifferential operators and symmetries of equations in the Kadomtsev–Petviashvili hierarchy
View Description Hide DescriptionPoint symmetries are obtained for all equations in the KP hierarchy. The Lie algebra for each equation is infinite dimensional and involves several arbitrary functions of the corresponding time The symmetry algebra is a semidirect sum of a Virasoro algebra and a Kac–Moody one. The “positive” part of this algebra is embedded into the known algebra of KP symmetries and into the free fermion algebra. The corresponding action on the taufunction is presented. The negative part of the point symmetries does not fit into the free fermion algebra, but is embedded into a algebra, based on the algebra of pseudodifferential operators.

Painlevé analysis of new higherdimensional soliton equation
View Description Hide DescriptionIn this note, we prove that the recently proposed new higherdimensional nonlinear partial differential equation admits the Painlevé property. We briefly discuss the integrability properties of the equation.

Integrable discretizations of the spin Ruijsenaars–Schneider models
View Description Hide DescriptionIntegrable discretizations are introduced for the rational and hyperbolic spin Ruijsenaars–Schneider models. These discrete dynamical systems are demonstrated to belong to the same integrable hierarchies as their continuoustime counterparts. Explicit solutions are obtained for arbitrary flows of the hierarchies, including the discrete time ones.

Vandermondelike determinants and fold Darboux/Bäcklund transformations
View Description Hide DescriptionWe define Vandermondelike determinants and analyze their structure. The resulting scheme is wellsuited to achieve a remarkable compactness and transparency in soliton formulas or, more generally, in formulas for fold Darboux or Bäcklund transformations.