Volume 40, Issue 8, August 1999
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Asymptotics of the scattering coefficients for a generalized Schrödinger equation
View Description Hide DescriptionThe generalized Schrödinger equation is considered, where P and Q are integrable potentials with finite first moments and F satisfies certain conditions. The behavior of the scattering coefficients near zeros of F is analyzed. It is shown that in the socalled exceptional case, the values of the scattering coefficients at a zero of F may be affected by The location of the kvalues in the complex plane where the exceptional case can occur is studied. Some examples are provided to illustrate the theory.

On the dynamics of the Holstein model from the anticontinuous limit
View Description Hide DescriptionWe consider the Holstein model describing an electron interacting with a lattice of identical oscillators. We remark that the on site system (i.e., the system in which the interaction between the different sites of the lattice vanishes) is integrable and anisocronous. This allows us to apply some recent Nekhoroshevtype results to show that corresponding to the majority of initial data in which the electron probability is concentrated on a finite number of sites, the electron probability distribution is approximatively constant for times growing exponentially with the inverse of the coupling parameter. Moreover, for the same times, the total energy of the oscillator system is approximatively constant.

Phasespace representation of quantum state vectors: The relativestate approach and the displacementoperator approach
View Description Hide DescriptionPhasespace representation of quantum state vectors has been recently formulated by means of the relativestate method developed by the present author [J. Math. Phys. 39, 1744 (1998)]. It is, however, pointed out by Mo/ller that the displacementoperator method provides another basis of phasespace representation of quantum state vectors [J. Math. Phys. (to appear)]. Hence the relation between the relativestate approach and the displacementoperator approach is discussed, both of which yield equivalent phasespace representations.

A onedimensional model for nlevel atoms coupled to an electromagnetic field
View Description Hide DescriptionA model for nlevel atoms coupled to quantized electromagnetic fields in a fibrillar geometry is constructed. In the slowly varying envelope and rotating wave approximations, the equations of motion are shown to satisfy a zero curvature representation, implying integrability of the quantum system.

Magnetic monopole in the Feynman’s derivation of Maxwell equations
View Description Hide DescriptionIn 1992, Dyson published Feynman’s proof of the homogeneous Maxwell equations assuming only the Newton’s law of motion and the commutation relations between position and velocity for a nonrelativistic particle. Recently Tanimura gave a generalization of this proof in a relativistic context. Using the Hodge duality we extend his approach in order to derive the two groups of Maxwell equations with a magnetic monopole in flat and curved spaces.

The generalized Moyal–Nahm and continuous Moyal–Toda equations
View Description Hide DescriptionWe present in detail a class of solutions to the 4D SU(∞) Moyal–antiselfdual Yang–Mills (ASDYM) equations (an effective 6D theory) that are related to reductions of the generalized Moyal–Nahm equations using the Ivanova–Popov ansatz. The former yields solutions to the ASDYM/SDYM equations for arbitrary gauge groups in four dimensions. A further dimensional reduction of the above effective 6D equations yields solutions to the Moyal–antiselfdual gravitational equations in four dimensions. The selfdual Yang–Mills/selfdual gravity case requires a separate study. The SU(2) Toda lattice and SU(∞) (continuous) Moyal–Toda lattice equations are derived from the Moyal–Nahm equations. An explicit map taking the Moyal heavenly form (after a rotational Killing symmetry reduction of the Moyal heavenly equations) into the SU(2) Toda lattice field is found. Finally, the generalized Moyal–Nahm equations are conjectured that contain the (continuous) SU(∞) Moyal–Toda lattice equations, after a suitable reduction process. Embeddings of the different types of Moyal–Toda lattice equations into the Moyal–Nahm equations are described.

Central charge and the Andrews–Bailey construction
View Description Hide DescriptionFrom the equivalence of the bosonic and fermionic representations of finitized characters in conformal field theory, one can extract mathematical objects known as Bailey pairs. Recently Berkovich, McCoy, and Schilling have constructed a “generalized” character formula depending on two parameters and using the Bailey pairs of the unitary model By taking appropriate limits of these parameters, they were able to obtain the characters of model model and the unitary model with central charge In this letter we computed the effective central charge associated with this “generalized” character formula using a saddle point method. The result is a simple expression in dilogarithms which interpolates between the central charges of these unitary models.

Modular invariance on the torus and Abelian Chern–Simons theory
View Description Hide DescriptionThe implementation of modular invariance on the torus as a phase space at the quantum level is discussed in a grouptheoretical framework. Unlike the classical case, at the quantum level some restrictions on the parameters of the theory should be imposed to ensure modular invariance. Two cases must be considered, depending on the cohomology class of the symplectic form on the torus. If it is of integer cohomology class n, then full modular invariance is achieved at the quantum level only for those wave functions on the torus which are periodic if n is even, or antiperiodic if n is odd. If the symplectic form is of rational cohomology class a similar result holds—the wave functions must be either periodic or antiperiodic on a torus r times larger in both directions, depending on the parity of nr. Application of these results to the Abelian Chern–Simons theory is discussed.

Free field approach to the dilute models
View Description Hide DescriptionWe construct a free field realization of vertex operators of the dilute models along with the Felder complex. For we also study an structure in terms of the deformed Virasoro currents.

Exact solutions of the Dirac equation in a nonfactorizable metric
View Description Hide DescriptionWe present the covariant generalization of the Dirac equation in a nonfactorizable metric and give the corresponding exact solutions in terms of special functions as well as the explicit form of the spinor solution. Then we treat the particular case of the Weyl equation for the neutrinos.

Oneloop stresstensor renormalization in curved background: The relation between ζfunction and pointsplitting approaches, and an improved pointsplitting procedure
View Description Hide DescriptionWe conclude the rigorous analysis of a previous paper [V. Moretti, Commun. Math. Phys. 201, 327 (1999)] concerning the relation between the (Euclidean) pointsplitting approach and the local ζfunction procedure to renormalize physical quantities at oneloop in (Euclidean) Quantum Field Theory in curved space–time. The case of the stress tensor is now considered in general Ddimensional closed manifolds for positive scalar operators Results obtained formally in previous works [in the case and ] are rigorously proven and generalized. It is also proven that, in static Euclidean manifolds, the method is compatible with Lorentziantime analytic continuations. It is proven that the result of the ζfunction procedure is the same obtained from an improved version of the pointsplitting method which uses a particular choice of the term in the Hadamard expansion of the Green’s function, given in terms of heatkernel coefficients. This version of the pointsplitting procedure works for any value of the field mass m. If D is even, the result is affected by an arbitrary oneparameter class of (conserved in absence of external source) symmetric tensors, dependent on the geometry locally, and it gives rise to the general correct trace expression containing the renormalized field fluctuations as well as the conformal anomaly term. Furthermore, it is proven that, in the case and the given procedure reduces to the Euclidean version of Wald’s improved pointsplitting procedure provided the arbitrary mass scale present in the ζfunction is chosen opportunely. It is finally argued that the found pointsplitting method should work generally, also dropping the hypothesis of a closed manifold, and not depending on the ζfunction procedure. This fact is indeed checked in the Euclidean section of Minkowski space–time for where the method gives rise to the correct Minkowski stress tensor for automatically.

Inverse problem for an inhomogeneous Schrödinger equation
View Description Hide DescriptionLet Assume that the potential is real valued and compactly supported: for and that produces no bound states. Let and be the data. It is shown that under the above assumptions these data determine uniquely.

Spinmass content of the Bhabha particle and the group chain
View Description Hide DescriptionThe relativistic particles described by the Bhabha equation (Bhabha particles) can possess several values of masses and spins In order to find the spinmass content of such particles corresponding to the general irrep of the group which is the symmetry group of the Bhabha equation, the noncanonical chain of groups is considered. In this paper, we construct the basis of the irrep in a boson realization form using the method of elementary permissible diagrams. This is achieved by the introduction of special “symplectic” bosons. The maximum and minimum values of spin are established for each value of τ. The multiplicity of degenerate pairs can be easily calculated with the help of the generating function obtained in this paper.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Timeasymptotic travelingwave solutions to the nonlinear Vlasov–Poisson–Ampère equations
View Description Hide DescriptionWe consider the Vlasov–Poisson–Ampère system of equations, and we seek solutions for the electric field that are periodic in space and asymptotically almost periodic in time. Introducing the representation (where T and A are, respectively, the transient and timeasymptotic parts of E) enables us to decompose the nonlinear Poisson equation into a transient equation and a timeasymptotic equation. We then study the latter in isolation as a bifurcation problem for A with the initial condition and T as parameters. We show that the Fréchet derivative at a generic bifurcation point has a nontrivial null space determined by the roots of a Vlasov dispersion relation. Hence, the bifurcationanalysis leads to a general solution for A given (at leading order) by a discrete superposition of travelingwave modes, whose frequencies and wave numbers satisfy the Vlasov dispersion relation, and whose amplitudes satisfy a system of nonlinear algebraic equations. In applications, there is usually a finite number of roots to the dispersion relation, and the equations for the timeasymptotic wave amplitudes reduce to a finite dimensional bifurcation problem in terms of the amplitude of the initial condition.

Conformal behavior of the Lorentz–Dirac equation and Machian particle dynamics
View Description Hide DescriptionIt is shown that the selfinteraction force on a pointlike electric charge in curved spacetime has conformal weight −1. Motivated by this result, a conformally covariant version of the Lorentz–Dirac equation is presented, where the particle mass is treated as a position and timedependent quantity. This feature suggests that the underlying dynamics is Machian.

Soliton stability in a field theory
View Description Hide DescriptionWe investigate the stability of the coupled soliton solutions of a twocomponent vector fieldmodel, in contraposition to similar solutions of a model recently introduced. We demonstrate that the coupled soliton solutions of the model are classically unstable.

 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


On the replica symmetric equations for the Hopfield model
View Description Hide DescriptionWe prove the central limit theorem for the cavity field in the case of the Hopfield model.

 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Relation between the Kadometsev–Petviashvili equation and the confocal involutive system
View Description Hide DescriptionThe special quasiperiodic solution of the dimensional Kadometsev–Petviashvili equation is separated into three systems of ordinary differential equations, which are the second, third, and fourth members in the wellknown confocal involutive hierarchy associated with the nonlinearized Zakharov–Shabat eigenvalue problem. The explicit theta function solution of the Kadometsev–Petviashvili equation is obtained with the help of this separation technique. A generating function approach is introduced to prove the involutivity and the functional independence of the conserved integrals which are essential for the Liouville integrability.

A new integrable Davey–Stewartsontype equation
View Description Hide DescriptionA new integrable nonlinear partial differential equation (PDE) in dimensions is derived starting from the Konopelchenko–Dubrovsky equation. We use an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling and obtain a new integrable Davey–Stewartsontype equation. In order to demonstrate the integrability of the new equation by the inverse scattering method, we apply the reduction technique to the Lax pair of the Konopelchenko–Dubrovsky equation and find the corresponding Lax pair of the new equation. The new equation reduces to the Davey–Stewartson or the nonlinear Schrodinger equation by appropriate limits.

 RELATIVITY AND GRAVITATION


Equivalence classes of perturbations in cosmologies of Bianchi types I and V: Formulation
View Description Hide DescriptionIn this paper we deal for the first time with gaugeinvariant perturbations of anisotropic cosmological models of Bianchi types I and V from a unified point of view. Motivated by Ehlers’ pioneering concepts, the key idea is to identify the gaugeinvariant perturbations with the equivalence classes of tangents to oneparameter families of exact solutions to Einstein’s field equations. For cases where these models are filled with a nonbarotropic perfect fluid, we show that a set of 26 “geometrically” independent, not identically vanishing gaugeinvariant variables, denoted collectively by D and referred to as the complete set of basic variables, can be used to extract the equivalence classes of tangents from D in a unique way. The set D is complete because it has the following property: any gaugeinvariant quantity is obtainable linearly from the basic variables through purely algebraic and differential operations. Mathematically, this approach to the gauge problem is a nontrivial example of the general scheme that we have described in our two previous papers [Int. J. Theor. Phys. 36, 1787, 1817 (1997)], and the new concepts developed were also applied to the construction of a complete set of basic gaugeinvariant variables for the cases of a fixed background de Sitter space–time and an almostRobertson–Walker universe model. Arguments are given that there are a number of advantages to be gained by replacing the coordinatebased method of Bardeen or the covariant formalism of Ellis and Bruni by the present one.
