Volume 40, Issue 12, December 1999
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Quantum description of rigidly or adiabatically constrained fourparticle systems and supersymmetry
View Description Hide DescriptionA general formalism for the quantum description of manybody systems is developed, using the principalaxis hyperspherical parametrization of coordinates. This formalism is applied to fourparticle systems for which the exact kinetic energy operator is derived using a model constraints and dynamical constraints. Then, using the supersymmetry and shape invariance approach, we obtained in a closed form the eigenvalues and eigenfunctions of a wide class of noncentral potentials for the adiabatically constrained fourparticle systems.

Coulomb wave functions with complex values of the variable and the parameters
View Description Hide DescriptionThe motivation for the present paper lies in the fact that the literature concerning the Coulomb wave functions and is a jungle in which it may be hard to find a safe way when one needs general formulas for the Coulomb wave functions with complex values of the variable ρ and the parameters L and η. For the Coulomb wave functions and certain linear combinations of these functions we discuss the connection with the Whittaker function, the Coulomb phase shift, Wronskians, reflection formulas integral representations, series expansions, circuital relations and asymptotic formulas on a Riemann surface for the variable ρ. The parameters L and η are allowed to assume complex values.

Phaseintegral formulas for Coulomb wave functions with complex values of the variable and the parameters
View Description Hide DescriptionPhaseintegral formulas for the Coulomb wave functions and and certain linear combinations of these functions, with complex values of the variable ρ and the parameters L and η, are obtained explicitly up to the fifth order of the phaseintegral approximation for two different choices of the base function.

Collective dynamics of solitons and inequivalent quantizations
View Description Hide DescriptionThe collective dynamics of solitons with a coset space as moduli space is studied. It is shown that the collective band for a vibrational state is given by the inequivalent coset space quantization corresponding to the representation of H carried by the vibration. To leading order the collective dynamics is free motion in coupled to background gauge fields determined by the vibrational state.

The osp(1,2)covariant Lagrangian quantization of reducible massive gauge theories
View Description Hide DescriptionThe osp(1,2)covariant Lagrangian quantization of irreducible gauge theories is generalized to Lstage reducible theories. The dependence of the generating functional of Green’s functions on the choice of gauge in the massive case is discussed and Ward identities related to osp(1,2) symmetry are given. Massive firststage theories with closed gauge algebra are studied in detail. The generalization of the Chapline–Manton model and topological Yang–Mills theory to the case of massive fields are considered as examples.

Ground states of a model in nonrelativistic quantum electrodynamics. I
View Description Hide DescriptionThe system of a one charged nonrelativistic particle with external potentials minimally coupled to a massless quantized radiation field is considered. An ultraviolet cutoff is imposed on the quantized radiation field and the charged particle has spin 1/2. The class of external potentials considered in this paper contains Coulomb potentials. It is shown that the ground states of the system exist provided that a coupling constant is in a region.

Spherically symmetric solutions of the Skyrme models
View Description Hide DescriptionRecently we have presented an ansatz which allows us to construct skyrmion fields from the harmonic maps of to In this paper we examine this construction in detail and use it to construct, in an explicit form, new static spherically symmetric solutions of the Skyrme models. We also discuss some properties of these solutions.

Two algebraic properties of thermal quantum field theories
View Description Hide DescriptionWe establish the Schlieder and the Borchers property for thermal field theories. In addition, we provide some information on the commutation and localization properties of projection operators.

Nonrelativistic quantum Hamiltonian for Lorentz violation
View Description Hide DescriptionA method is presented for deriving the nonrelativistic quantum Hamiltonian of a free massive fermion from the relativistic Lagrangian of the Lorentzviolating standardmodel extension. It permits the extraction of terms at arbitrary order in a Foldy–Wouthuysen expansion in inverse powers of the mass. The quantum particle Hamiltonian is obtained and its nonrelativistic limit is given explicitly to third order.

Casimir energy of a ball and cylinder in the zeta function technique
View Description Hide DescriptionA simple method is proposed to construct the spectral zeta functions required for calculating the electromagnetic vacuum energy with boundary conditions given on a sphere or on an infinite cylinder. When calculating the Casimir energy in this approach no exact divergencies appear and no renormalization is needed. The starting point of the consideration is the representation of the zeta functions in terms of contour integral, further the uniform asymptotic expansion of the Bessel function is essentially used. After the analytic continuation, needed for calculating the Casimir energy, the zeta functions are presented as infinite series containing the Riemann zeta function with rapidly falling down terms. The spectral zeta functions are constructed exactly for a material ball and infinite cylinder placed in a uniform endless medium under the condition that the velocity of light does not change when crossing the interface. As a special case, perfectly conducting spherical and cylindrical shells are also considered in the same line. In this approach one succeeds, specifically, in justifying, in mathematically rigorous way, the appearance of the contribution to the Casimir energy for cylinder which is proportional to

Quasilinearization method and its verification on exactly solvable models in quantum mechanics
View Description Hide DescriptionThe proof of the convergence of the quasilinearization method of Bellman and Kalaba, whose origin lies in the theory of linear programming, is extended to large and infinite domains and to singular functionals in order to enable the application of the method to physical problems. This powerful method approximates solution of nonlinear differential equations by treating the nonlinear terms as a perturbation about the linear ones, and is not based, unlike perturbation theories, on existence of some kind of small parameter. The general properties of the method, particularly its uniform and quadratic convergence, which often also is monotonic, are analyzed and verified on exactly solvable models in quantum mechanics. Namely, application of the method to scattering length calculations in the variable phase method shows that each approximation of the method sums many orders of the perturbation theory and that the method reproduces properly the singular structure of the exact solutions. The method provides final and reasonable answers for infinite values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist.

Picard–Fuchs equations and Whitham hierarchy in supersymmetric Yang–Mills theory
View Description Hide DescriptionIn general, Whitham dynamics involves infinitely many parameters called Whitham times, but in the context of supersymmetric Yang–Mills theory it can be regarded as a finite system by restricting the number of Whitham times appropriately. For example, in the case of gauge theory without hypermultiplets, there are r Whitham times and they play an essential role in the theory. In this situation, the generating meromorphic oneform of the Whitham hierarchy on the Seiberg–Witten curve is represented by a finite linear combination of meromorphic oneforms associated with these Whitham times, but it turns out that there are various differential relations among these differentials. Since these relations can be written only in terms of the Seiberg–Witten oneform, their consistency conditions are found to give the Picard–Fuchs equations for the Seiberg–Witten periods.

Wigner–Weyl correspondence and semiclassical quantization in spherical coordinates
View Description Hide DescriptionThe Wigner–Weyl quantumtoclassical correspondence rule is nonunique with respect to coordinate choice. This ambiguity can be exploited to improve the accuracy of semiclassical approximations. For instance, the wellknown Langer modification was recently derived by applying a coordinate transformation to the radial Schrödinger equation prior to using the Wigner–Weyl rule—albeit only by presuming exact quantum solutions for all nonradial degrees of freedom [J. J. Morehead, J. Math. Phys. 36, 5431 (1995)]. In this paper, the full classical Hamiltonian is derived in all degrees of freedom, using a (hyper)spherical coordinate Wigner–Weyl correspondence with a Langerlike modification of polar angles. For central force Hamiltonians, the new result is radially equivalent to that of Langer, and to the standard Cartesian form. The new correspondence is superior with respect to all angular momentum operators however, in that the resultant semiclassical eigenvalues are exact—a desirable goal, evidently achieved here for the first time.

On separable Schrödinger equations
View Description Hide DescriptionWe classify dimensional Schrödinger equations for a particle interacting with the electromagnetic field that are solvable by the method of separation of variables. As a result, we get 11 classes of the vector potentials of the electromagnetic field providing separability of the corresponding Schrödinger equations. It is established, in particular, that the necessary condition for the Schrödinger equation to be separable is that the magnetic field must be independent of the spatial variables. Next, we prove that any Schrödinger equation admitting variable separation into secondorder ordinary differential equations can be reduced to one of the 11 separable Schrödinger equations mentioned above and carry out variable separation in the latter. Furthermore, we apply the results obtained for separating variables in the Hamilton–Jacobi equation.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


On the explicit solutions of the elliptic Calogero system
View Description Hide DescriptionLet be the coordinates of N particles on the circle, interacting with the integrable potential where ℘ is the Weierstrass elliptic function. We show that every symmetric elliptic function in is a meromorphic function in time. We give explicit formulas for these functions in terms of genus theta functions.

SU(N) skyrmions and harmonic maps
View Description Hide DescriptionHarmonic maps from to are introduced to construct lowenergy configurations of the Skyrme model. We show that one of such maps gives an exact, topologically trivial, solution of the model. We study various properties of these maps and show that, in general, their energies are only a little higher than the energies of the corresponding embeddings. Moreover, we show that the baryon and energy densities of the configurations with baryon number are more symmetrical than their analogs, thus suggesting that there exist solutions of the model with these symmetries. We also show that any solution embedded into the Skyrme model becomes a topologically trivial solution of this model.

QuasiLagrangian systems of Newton equations
View Description Hide DescriptionSystems of Newton equations of the form with an integral of motion quadratic in velocities are studied. These equations generalize the potential case (when the identity matrix) and they admit a curious quasiLagrangian formulation which differs from the standard Lagrange equations by the plus sign between terms. A theory of such quasiLagrangian Newton (qLN) systems having two functionally independent integrals of motion is developed with focus on twodimensional systems. Such systems admit a biHamiltonian formulation and are proved to be completely integrable by embedding into fivedimensional integrable systems. They are characterized by a linear, secondorder partial differential equationPDE which we call the fundamental equation. Fundamental equations are classified through linear pencils of matrices associated with qLN systems. The theory is illustrated by two classes of systems: separable potential systems and driven systems. New separation variables for driven systems are found. These variables are based on sets of nonconfocal conics. An effective criterion for existence of a qLN formulation of a given system is formulated and applied to dynamical systems of the Hénon–Heiles type.

 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


Relativistic gas: Moment equations and maximum wave velocity
View Description Hide DescriptionFor a rarefied relativistic gas we consider the Nmoment equations associated with the relativistic Boltzmann–Chernikov equation and we require the compatibility with the entropy principle thus obtaining a closed symmetric hyperbolic system. This interesting form permits one to deduce a lower and an upper bound for the maximum velocity of a wave propagating in a monoatomic or a degenerate gas of fermions or bosons and to prove that when this number N increases this velocity tends to the speed of light.

Integral representations of thermodynamic 1PI Green’s functions in the worldline formalism
View Description Hide DescriptionThe issue discussed is a thermodynamic version of the Bern–Kosower master amplitude formula, which contains all necessary oneloop Feynman diagrams. It is demonstrated how the master amplitude at finite values of temperature and chemical potential can be formulated within the framework of the worldline formalism. In particular we present an elegant method of how to introduce a chemical potential for a loop in the master formula. Various useful integral formulas for the master amplitude are then obtained. The nonanalytic property of the master formula is also derived in the zero temperature limit with the value of chemical potential kept finite.

 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


A field theory approach to Lindstedt series for hyperbolic tori in three time scales problems
View Description Hide DescriptionInteracting systems consisting of two rotators and a pendulum are considered, in a case in which the uncoupled systems have three very different characteristic time scales. The abundance of unstable quasiperiodic motions in phase space is studied via the Lindstedt series regarded as a sum of Feynman graphs and studied with renormalization group techniques based on Eliasson’s work on KAM tori. The result is a strong improvement, compared to our previous results, on the domain of validity of bounds that imply existence of invariant tori, large homoclinic angles, long heteroclinic chains, and driftdiffusion in phase space.
