Volume 41, Issue 2, February 2000
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Quantum fields in nonstatic background: A histories perspective
View Description Hide DescriptionFor a quantum field living on a nonstatic space–time no instantaneous Hamiltonian is definable, for this generically necessitates a choice of inequivalent representation of the canonical commutation relations at each instant of time. This fact suggests a description in terms of timedependent Hilbert spaces, a concept that fits naturally in a (consistent) histories framework. Our primary tool for the construction of the quantum theory in a continuoustime histories format is the recently developed formalism based on the notion of the history group. This we employ to study a model system involving a scalar field in a cavity with moving boundaries. The instantaneous (smeared) Hamiltonian and a decoherence functional are then rigorously defined so that finite values for the timeaveraged particle creation rate are obtainable through the study of energy histories. We also construct the Schwinger–Keldysh closedtimepath generating functional as a “Fourier transform” of the decoherence functional and evaluate the corresponding npoint functions.

On the concept of Einstein–Podolsky–Rosen states and their structure
View Description Hide DescriptionIn this paper the notion of an EPR state for the composite S of two quantum systems relative to and a set O of bounded observables of is introduced in the spirit of the classical examples of Einstein–Podolsky–Rosen and Bohm. We restrict ourselves mostly to EPR states of finite norm. The main results are contained in Theorems 3–6 and imply that if EPR states of finite norm relative to exist, then the elements of O have discrete probability distributions and the Von Neumann algebra generated by O is essentially imbeddable inside by an antiunitary map. The EPR states then correspond to the different imbeddings and certain additional parameters, and are explicitly given by formulas which generalize the famous example of Bohm. If O generates all bounded observables, must be of finite dimension and can be imbedded inside by an antiunitary map, and the EPR states relative to are then in canonical bijection with the different imbeddings of inside moreover they are then given by formulas which are exactly those of the generalized Bohm states. The notion of EPR states of infinite norm is also explored and it is shown that the original state of Einstein–Podolsky–Rosen can be realized as a renormalized limit of EPR states of finite quantum systems considered by Weyl, Schwinger, and many others. Finally, a family of states of infinite norm generalizing the Einstein–Podolsky–Rosen example is explicitly given.

Hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the Jordan–Schwinger map
View Description Hide DescriptionThe aim of this paper is to show that there is a Hopf structure of the parabosonic and parafermionic algebras and this Hopf structure can generate the wellknown Hopf algebraic structure of the Lie algebras, through a realization of Lie algebras using the parabosonic (and parafermionic) extension of the Jordan–Schwinger map. The differences between the Hopf algebraic and the graded Hopf superalgebraic structure on the parabosonic algebra are discussed.

Ground states of a model in nonrelativistic quantum electrodynamics. II
View Description Hide DescriptionThe system of Nnonrelativistic spineless particles minimally coupled to a massless quantized radiation field with an ultraviolet cutoff is considered. The Hamiltonian of the system is defined for arbitrary coupling constants in terms of functional integrals. It is proved that the ground state of the system with a class of external potentials, if they exist, is unique. Moreover an expression of the ground state energy is obtained and it is shown that the ground state energy is a monotonously increasing, concave, and continuous function with respect to the square of a coupling constant.

Isospectral deformation of some shape invariant potentials
View Description Hide DescriptionHere we generalize the isospectral deformation of the discrete eigenspectrum of Eleonsky and Korolev [Phys. Rev. A 55, 2580 (1997)] to continuous eigenspectrum of some wellknown shapeinvariant potentials. We show that the isospectral deformations preserve their shape invariance properties. Hence, using the preserved shape invariance property of the deformed potentials, we obtain both discrete and continuous eigenspectrum of the deformed Rosen–Morse, Natanzon, Rosen–Morse with added Dirac delta term, and Natanzon with added Dirac delta term potentials, respectively. It is shown that deformation does not change their other pecurialities, such as the reflectionless property of the Rosen–Morse potential and the penetrationless property of the Natanzon one.

Calculation of the level splitting of energy and the decay rate of some onedimensional potentials by instantons method
View Description Hide DescriptionUsing the popular Langer–Polyakov–Coleman instanton method, the decay rate of metastable states and the level splitting of energy, due to tunneling of some onedimensional potentials have been calculated. The operators appearing in the prefactor of the decay rate or the level splitting of energy of these kinds of potentials possess the shape invariance symmetry, hence using this property, the determinant of these operators have been calculated analytically via heat kernel method.

Construction of quasitwo and higherdimensional quantum integrable models
View Description Hide DescriptionA class of quantum integrable quasitwo and higherdimensional quantum spin as well as strongly correlated electron systems with localized interactions are proposed. The basic idea of construction is to introduce interchain interactions in an array of spin chains or onedimensional Hubbard models through twisting transformation. The models allow explicit quantum Rmatrix, Lax operator, and exact eigenvaluesolution.

On 2D Euler equations. I. On the energy–Casimir stabilities and the spectra for linearized 2D Euler equations
View Description Hide DescriptionIn this paper, we study a linearized twodimensional Euler equation. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. Another important invariant besides the Hamiltonian for each invariant subsystem is found and is utilized to prove an “unstable disk theorem” through a simple energy–Casimir argument [Holm et al., Phys. Rep. 123, 1–116 (1985)]. The eigenvalues of the linear Hamiltonian system are of four types: real pairs purely imaginary pairs quadruples and zero eigenvalues. The eigenvalues are computed through continued fractions. The spectral equation for each invariant subsystem is a Poincarétype difference equation, i.e., it can be represented as the spectral equation of an infinite matrix operator, and the infinite matrix operator is a sum of a constantcoefficient infinite matrix operator and a compact infinite matrix operator. We have obtained a complete spectral theory.

Quantum interaction the construction of quantum field defined as a bilinear form
View Description Hide DescriptionWe construct the solution of the quantum wave equation as a bilinear form which can be expanded over Wick polynomials of the free field, and where is defined as the normal ordered product with respect to the free field. The constructed solution is correctly defined as a bilinear form on where is a dense linear subspace in the Fock space of the free field. On the diagonal of the Wick symbol of this bilinear form satisfies the nonlinear classical wave equation.

Cluster expansion for explicit estimates
View Description Hide DescriptionWe apply the tree expansion to the polynomial selfinteracting quantum fields in two dimension. In the range of small coupling constants, we show that the expansion is also convergent when the full propagator is considered and the theory is regularized by normal ordering. Explicit estimates of the convergency ranges are given.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


An approach to Mel’nikov theory in celestial mechanics
View Description Hide DescriptionUsing a completely analytic procedure—based on a suitable extension of a classical method—we discuss an approach to the Poincaré–Mel’nikov theory, which can be conveniently applied also to the case of nonhyperbolic critical points, and even if the critical point is located at the infinity. In this paper, we concentrate our attention on the latter case, and precisely on problems described by Keplertype potentials in one or two degrees of freedom, in the presence of general timedependent perturbations. We show that the appearance of chaos (possibly including Arnol’d diffusion) can be proved quite easily and in a direct way, without resorting to singular coordinate transformations, such as the McGehee or blowingup transformations. Natural examples are provided by the classical Gyldén problem, originally proposed in celestial mechanics, but also of interest in different fields, and by the general threebody problem in classical mechanics.

Monopole Skyrmions
View Description Hide DescriptionA systematic numerical study of the classical solutions to the combined system consisting of the Georgi–Glashow model and the SO(3) gauged Skyrme model is presented. The gauging of the Skyrme system permits a lower bound on the energy, so that the solutions of the composite system can be topologically stable. The solutions feature some very interesting bifurcation patterns, and it is found that some branches of these solutions are unstable.

On the geodesic form of secondorder dynamic equations
View Description Hide DescriptionIt is shown that any secondorder dynamic equation on a configuration bundle of nonrelativistic mechanics is equivalent to a geodesic equation with respect to a (nonlinear) connection on the tangent bundle The case of quadratic dynamic equations is analyzed in detail. The equation for Jacobi vector fields is constructed and investigated by the geometric methods.

A new procedure for specifying nonradiating current distributions and the fields they produce
View Description Hide DescriptionThis paper reports a new procedure for specifying monochromatic nonradiating (NR) current distributions (NR sources) and the electric and magnetic fields they produce (NR fields). Vector spherical harmonics and a Fourier–Bessel series are used to derive a new vector sphericalwave expansion for continuous NR fields confined within a spherical volume. The analysis yields complete orthogonal sets in terms of which all such NR fields can be expanded. By making use of a Maxwell operator representation for NR current distributions, we obtain a new series expansion for NR current distributions confined within a spherical volume. The analysis also yields complete sets for such NR current distributions. The developed theory is illustrated with special cases.
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 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


Convergence of approximation for the neutron transport equation with reflective boundary condition
View Description Hide DescriptionIn this paper the spherical harmonic function method for the plane geometry neutron transport equation with reflective boundary condition is discussed. The existence and uniqueness for the solution of the spherical harmonic approximation equation are studied, and then the convergence of the solution of the equation as to the solution of the neutron transport equation is proved.
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 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Lie–Poisson structure for the homogeneous motion of selfgravitating compressible fluids
View Description Hide DescriptionIn this paper we apply the theory developed by Marsden, Ratiu, and Weinstein for the reduction of a Hamiltonian system defined on the cotangent bundle of a Lie group to a Hamiltonian system in the coalgebra of a semidirect product to study the motion of a selfgravitating homogeneous compressible ideal fluid with a variable ellipsoidal boundary, assuming that the motions are given by invertible linear transformations. The relation between the Lie–Poisson equations obtained and the classical Dyson equations is discussed, and the Hamiltonian structure for the homogeneous expansion of a free nonrotating ellipsoid is derived.
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 RELATIVITY AND GRAVITATION


Gravoelectric dual of the Kerr solution
View Description Hide DescriptionBy decomposing the Riemann curvature into electric and magnetic parts, we define the gravoelectric duality transformation by interchange of active and passive electric parts which amounts to interchange of the Ricci and Einstein tensors. It turns out that the vacuum equation is duality invariant. We obtain solutions dual to the Kerr solution by writing an effective vacuum equation in such a way that it still admits the Kerr solution but is not duality invariant. The dual equation is then solved to obtain the dualKerr solution which can be interpreted as the Kerr black hole sitting in a string dust universe.

Polyakov conjecture on the supertorus
View Description Hide DescriptionWe prove the Polyakov conjecture on the supertorus We determine an iterative solution at any order of the superconformal Ward identity and we show that the Polyakov action that describes the (1,0) 2D(twodimensional)supergravity, resums this perturbative series. The resolution of the superBeltrami equation for the Wess–Zumino field from which the Polyakov action is expressed, is done by using on the one hand the Cauchy kernel techniques on defined in H. Kachkachi and M. Kachkachi, Class. Quantum Grav. 11, 493 (1994) and on the other hand, the formalism developed in M. Kachkachi and S. Kouadik, J. Math. Phys. 38, 4336 (1997). Hence, we determine the npoints Green functions from the Polyakov action expressed as a functional integral of the Beltrami superfield

On Killing vector fields and Newman–Penrose constants
View Description Hide DescriptionAsymptotically flat space–times with one Killing vector field are studied. The Killing equations are solved asymptotically using polyhomogeneous expansions (i.e., series in powers of and and solved order by order. The solution to the leading terms of these expansions yields the asymptotic form of the Killing vector field. The possible classes of Killing fields are discussed by analyzing their orbits on null infinity. The integrability conditions of the Killing equations are used to obtain constraints on the components of the Weyl tensor and on the shear (σ). The behavior of the solutions to the constraint equations is studied. It is shown that for Killing fields that are nonsupertranslational the characteristics of the constraint equations are the orbits of the restriction of the Killing field to null infinity. As an application, the particular case of boostrotation symmetric space–times is considered. The constraints on are used to study the behavior of the coefficients that give rise to the Newman–Penrose constants, if the space–time is nonpolyhomogeneous, or the logarithmic Newman–Penrose constants, if the space–time is polyhomogeneous.
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 MISCELLANEOUS TOPICS IN MATHEMATICAL METHODS


Semiclassical limits of extended Racah coefficients
View Description Hide DescriptionWe explore the geometry and asymptotics of extended Racah coefficients. The extension is shown to have a simple relationship to the Racah coefficients for the positive discrete unitary representation series of SU(1,1) which is explicitly defined. Moreover, it is found that this extension may be geometrically identified with two types of Lorentzian tetrahedra for which all the faces are timelike. The asymptotic formulas derived for the extension are found to have a similar form to the standard Ponzano–Regge asymptotic formulas for the SU(2) symbol and so should be viable for use in a state sum for three dimensional Lorentzian quantum gravity.
