Volume 39, Issue 7, July 1998
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Analytic regularization of the model at finite temperature and density
View Description Hide DescriptionSimultaneously using the dimensional and analytic regularization methods we reexamine the Gross–Neveu model at finite temperature and density (chemical potential) in a dimensional space–time. The regularized effective potential is presented at the (large approximation. It is found that in the effective potential is finite.

The doublet representation of nonHilbert eigenstates of the Hamiltonian. II
View Description Hide DescriptionApplying the doublet representation we analyze the solutions of a Hamiltonian system which has eigenstates with complex eigenvalues. The example of the Friedrichs model allows us to show how the appearance of solutions with nonHilbert initial conditions is linked to the energy degeneration of the Hamiltonian spectrum. We discuss the difficulties of giving a physical meaning to the growing or decaying nonHilbert solutions. We also suggest a way to circumvent the problem of the anomalous probabilities related to both complex energy eigenvalues and degeneration of the spectrum.

Multicomponent Wentzel–Kramers–Brillouin approximation on arbitrary symplectic manifolds: A star product approach
View Description Hide DescriptionIt is known that in the Wentzel–Kramers–Brillouin approximation of multicomponent systems like the Dirac equation or Born–Oppenheimer approximation, an additional phase appears apart from the Berry phase. So far, this phase was only examined in special cases, or under certain restrictive assumptions, namely, that the eigenspaces of the matrix or endomorphism valued symbol of the Hamiltonian form trivial bundles. We give a completely global derivation of this phase which does not depend on any choice of local trivializing sections. This is achieved using a star product approach to quantization. Furthermore, we give a systematic and global approach to a reduction of the problem to a problem defined completely on the different “polarizations.” Finally, we discuss to what extent it is actually possible to reduce the problem to a really scalar one, and make some comments on obstructions to the existence of global quasiclassical states.

QED in external field with space–time uniform invariants: Exact solutions
View Description Hide DescriptionWe study exact solutions of Dirac and Klein–Gordon equations and Green functions in dimensional QED and in an external electromagnetic field with constant and homogeneous field invariants. The cases of even and odd dimensions are considered separately; they are essentially different. We direct attention to the asymmetry of the quasienergy spectrum, which appears in odd dimensions. The in and out classification of the exact solutions as well as the completeness and orthogonality relations is strictly proven. Different Green functions in the form of sums over the exact solutions are constructed. The Fock–Schwinger proper time integral representations of these Green functions are found. As physical applications, we consider the calculations of different quantum effects related to the vacuum instability in the external field. For example, we present mean values of particles created from the vacuum, the probability of the vacuum remaining a vacuum, the effective action, and the expectation values of the current and energymomentum tensor.

Exponential stability for the energy exchange dynamics of the Holstein model
View Description Hide DescriptionWe investigate the dynamics of the energy exchange between excitonic and vibrational degrees of freedom in the context of the Holstein model. Using Nekhoroshevtype arguments we derive bounds on the energy exchange for weak exciton–vibron coupling. It is shown that for large differences of the excitonic and vibronic frequencies the energy exchange is suppressed up to times growing exponentially with the ratio of the frequencies. Although the excitonic and vibronic energies are each for itself conserved there is equipartition of the excitonic energy among all lattice sites entailing an extended exciton. However, in the limit of large exciton–vibron coupling the excitonic actions are frozen separately, which leads to exciton localization. The action freezing is reflected in breather solutions.

Adler–Kostant–Symes scheme for face and Calogero–Moser–Sutherlandtype models
View Description Hide DescriptionWe give the construction of quantum Lax equations for IRF models and the difference version of the Calogero–Moser–Sutherland model introduced by Ruijsenaars. We solve the equations using factorization properties of the underlying face Hopf algebras/elliptic quantum groups. This construction is in the spirit of the Adler–Kostant–Symes method and generalizes our previous work to the case of face Hopf algebras/elliptic quantum groups with dynamical matrices.

On infravacua and superselection theory
View Description Hide DescriptionIn the Doplicher, Haag, and Roberts theory of superselection sectors, one usually considers states which are local excitations of some vacuum state. Here, we extend this analysis to local excitations of a class of “infravacuum” states appearing in models with massless particles. We show that the corresponding superselection structure, the statistics of superselection sectors and the energymomentum spectrum are the same as with respect to the vacuum state. (The latter result is obtained with a novel method of expressing the shape of the spectrum in terms of properties of local charge transfer cocycles.) These findings provide evidence to the effect that infravacua are a natural starting point for the analysis of the superselection structure in theories with longrange forces.

An increasing entropy for a free quantum particle
View Description Hide DescriptionFor a quantummechanical particle without interaction, a linear manifold of states is identified that has a preferred time direction pointing to the future. States with the broken timereversal symmetry are singled out by their behavior under dilations. At positive times such states can be described in terms of a density operator with the property that the trace of its square decreases as the time increases. This density operator determines an entropy that approaches its least upper bound when the time tends to infinity. The expectation value of the dilation operator was negative in the distant past and will be positive in the remote future. This is the irreversible aspect of the time evolution that causes the entropy to increase, although there is no approach to equilibrium. The density operator with increasing entropy is obtained from the usual density operator by an invertible transformation that is compared with the Λtransformation in the Prigogine theory of irreversible behavior in systems and large Hamiltonian systems with many resonances.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


KodairaSpencer deformation of complex structures and Lagrangian field theory
View Description Hide DescriptionSimilar to the Beltrami parametrization of complex structures on a (compact) Riemann surface, we use in this paper the KodairaSpencer deformation theory of complex structures on a (compact) complex manifold of higher dimension. According to the NewlanderNirenberg theorem, a smooth change of local complex coordinates is implemented with respect to an integrable complex structure parametrize by a Beltrami differential. The question of constructing a local field theory on a complex compact manifold is addressed and the action of smooth diffeomorphisms is studied in the BRS algebraic approach. The BRS cohomology for the diffeomorphisms gives generalized Gel’fandFuchs cocycles provided that the KodairaSpencer integrability condition is satisfied. The diffeomorphism anomaly is computed and turns out to be holomorphically split as in the bidimensional Lagrangian conformal models. Moreover, its algebraic structure is much more complicated than the one proposed in a recent paper [Losev et al. Nuc. Phys. B 484, 196 (1997)].

Timedependent plane wave and multipole expansions of the electromagnetic field
View Description Hide DescriptionA new and conceptually simple derivation is presented of the timedependent multipole expansion of the electromagnetic field radiated by a timevarying, localized, volume chargecurrent distribution. The analysis is based on a new timedependent plane wave representation of the electromagnetic field (i.e., an “angular spectrum expansion” in the time domain), also derived in the paper. Expressions are given for the timedependent plane wave spectra in terms of a fourfold Radon transform representation of the transverse current distribution. Two alternative expressions for the timedependent multipole moments are derived; the first gives them in terms of the spectral amplitude vectors of the corresponding timedependent plane wave representation while the second gives them in terms of a weighted radialtemporal average of the current distribution. Thus the analysis also sheds light on the relationship between the timedependent plane wave and multipole expansions in their common domains of validity.

Arnol’d’s transformation—An example of a generalized canonical transformation
View Description Hide DescriptionThe duality transformation described by Arnol’d, which relates the orbits for different central potentials through a change of variables, is shown to be an example of a generalized canonical transformation, in an extended phase space that includes time as a dynamical variable.
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 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


Lattice approximation of quantum statistical traces at a complex temperature
View Description Hide DescriptionWe show that the addition of an imaginary part to the temperature in the Gibbs ensemble will not destroy the convergence of the standard lattice approximation scheme for quantum mechanical systems with Hamiltonians of the type provided the potential function is real and bounded from below and it satisfies the condition for all As a byproduct we obtain an explicit bound for the realtemperature lattice kernels and a simple condition for the convergence of the realtemperature lattice expectation values of observables given by polynomially bounded functions.

Generalized plasma dispersion functions
View Description Hide DescriptionGeneralized plasma dispersion functions have recently arisen in the derivation of an exact expression for the quadratic response tensor for a plasma with constituent species having Maxwellian velocity distributions. A range of mathematical properties satisfied by the generalized plasma dispersion functions are derived here, including recursion relations, series expansions, and approximate analytic forms, including Padé approximants. These properties are of central importance to the application of the exact quadratic response tensor to descriptions of secondorder processes in weak and strong turbulence theory, such as threewave interactions in warm plasmas.
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 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Advection–diffusion around a curved obstacle
View Description Hide DescriptionAdvection and diffusion of a substance around a curved obstacle is analyzed when the advection velocity is large compared to the diffusion velocity, i.e., when the Peclet number is large. Asymptotic expressions for the concentration are obtained by the use of boundary layer theory, matched asymptotic expansions, etc. The results supplement and extend previous ones for straight obstacles. They apply to electrophoresis, the flow of ground water,chromatography,sedimentation, etc.

A linearizing transformation for the Korteweg–de Vries equation; generalizations to higherdimensional nonlinear partial differential equations
View Description Hide DescriptionIt is shown that the Korteweg–de Vries (KdV) equation can be transformed into an ordinary linear partial differential equation in the wave number domain. Explicit solutions of the KdV equation can be obtained by subsequently solving this linear differential equation and by applying a cascade of (nonlinear) transformations to the solution of the linear differential equation. It is also shown that similar concepts apply to the nonlinear Schrödinger equation. The role of symmetry is discussed. Finally, the procedure which is followed in the onedimensional cases is successfully applied to find special solutions of higherdimensional nonlinear partial differential equations.

extended superelliptic integrable systems
View Description Hide DescriptionA supersymmetric extension for an arbitrary number of odd parameters of the stationary KdV equation is studied; we call this an extended super KdV. It is shown that it is related to superelliptic curves, and thereafter the complete ring of superdifferential operators commuting with the associated Schrödinger operator is constructed.

Matrix formulation of Hamiltonian structures of constrained KP hierarchy
View Description Hide DescriptionWe give a matrix formulation of the Hamiltonian structures of constrained KP hierarchy. First, we derive from the matrix formulation the Hamiltonian structure of the oneconstraint KP hierarchy, which was originally obtained by Oevel and Strampp. We then generalize the derivation to the multiconstraint case and show that the resulting bracket is actually the second Gelfand–Dickey bracket associated with the corresponding Lax operator. The matrix formulation of the Hamiltonian structure of the oneconstraint KP hierarchy in the form introduced in the study of matrix model is also discussed.

Second harmonic generation: Hamiltonian structures and particular solutions
View Description Hide DescriptionFor the equations describing second harmonic generation (SHG), we find a linear system different from one found by Kaup (but connected with it by a gauge transformation), which enables us to consider the SHG equations as related to the first negative flow in the coupled KdV hierarchy and to find therefore the Hamiltonian structures and the recursion operator. With the help of the recursion operator we write down the simplest commuting flows and consider their stationary points in order to find particular solutions of the SHG equations. We also investigate some particular invariant solutions of the SHG equations that can be of physical interest, and find their connections with invariant solutions of the Tzitzéica equation. The latter are either connected with Painlevé III or solvable in elliptic functions.

Motion of curves and surfaces and nonlinear evolution equations in (2+1) dimensions
View Description Hide DescriptionIt is shown that a class of important integrable nonlinear evolution equations in (2+1) dimensions can be associated with the motion of space curves endowed with an extra spatial variable or equivalently, moving surfaces. Geometrical invariants then define topological conserved quantities. Underlying evolution equations are shown to be associated with a triad of linear equations. Our examples include Ishimori equation and Myrzakulov equations which are shown to be geometrically equivalent to DaveyStewartson and ZakharovStrachan (2+1) dimensional nonlinear Schrödinger equations, respectively.

The secular solutions of the linearized Korteweg–de Vries equation
View Description Hide DescriptionWe study the inhomogeneous linearized Korteweg–de Vries (KdV) equation. It is solved by the inverse scattering transform method. The secularproducing terms on the righthand side (rhs) are characterized in several ways: first we give a mathematical characterization as resonant terms. Second, the secularproducing terms are interpreted as conserved densities of the KdV equation. Third, it is checked that the removal of all linear terms from the rhs, polynomial in the solution of KdV, ensures the boundness of the solution of the linearized equation. Fourth, considering this solution itself as the rhs, we determine which part of it is secular producing, and which part is not.
