Volume 43, Issue 4, April 2002
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Centers of mass and rotational kinematics for the relativistic body problem in the restframe instant form
View Description Hide DescriptionA relativistic kinematics for the body problem which solves all the problems raised until now on this topic is constructed by exploting the Wignercovariant restframe instant form of dynamics in the context of parametrized Minkowski theories. The Wigner hyperplanes, orthogonal to the total timelike fourmomentum of any body configuration, define the intrinsic rest frame and realize the separation of the centerofmass motion. The point chosen as origin in each Wigner hyperplane can be made to coincide with the covariant noncanonical Fokker–Pryce center of inertia. As is well known, the latter is distinct from the canonical pseudo fourvector describing the decoupled motion of the center of mass (which possess the same Euclidean covariance as the quantum Newton–Wigner threeposition operator) and from the noncanonical pseudofourvector known as Møller’s center of energy. Our approach leads to the splitting of the notion of center of mass into an external one, defined in terms of an external Poincaré group realization, and an internal one defined in terms of an internal unfaithful realization of the group inside the Wigner hyperplane. Because of the first class constraints defining the rest frame (vanishing of the internal threemomentum), the latter three internal concepts of center of mass weakly coincide. The resulting unique internal center of mass is thereby a gauge variable which, by a suitable gauge fixing, can be localized at the origin of the Wigner hyperplane. An adapted canonical basis of relative variables is constructed by means of the classical counterpart of the Gartenhaus–Schwartz transformation. The invariant mass of the body configuration is the Hamiltonian for the relative motions. Within this general framework, the rotational kinematics can be developed in terms of the same dynamical body frames, orientationshape variables, spin frame, and canonical spin bases already introduced in the case of the nonrelativistic body problem.

Path integrals for boundaries and topological constraints: A white noise functional approach
View Description Hide DescriptionUsing the Streit–Hida formulation where the Feynman path integral is realized in the framework of white noiseanalysis, we evaluate the quantum propagator for systems with boundaries and topological constraints. In particular, the Feynman integrand is given as generalized white noise functionals for systems with flat wall boundaries and periodic constraints. Under a suitable Gauss–Fourier transform of these functionals the quantum propagator is obtained for: (a) the infinite wall potential; (b) a particle in a box; (c) a particle constrained to move in a circle; and (d) the Aharonov–Bohm system. The energy spectrum and eigenfunctions are obtained in all four cases.

The fuzzy sphere ⋆product and spin networks
View Description Hide DescriptionWe analyze the expansion of the fuzzy sphere noncommutative product in powers of the noncommutativity parameter. To analyze this expansion we develop a graphical technique that uses spin networks. This technique is potentially interesting in its own right as introducing spin networks of Penrose into noncommutative geometry. Our analysis leads to a clarification of the link between the fuzzy sphere noncommutative product and the usual deformation quantization of the sphere in terms of the ⋆product.

Observable effects and parametrized scaling limits of a model in nonrelativistic quantum electrodynamics
View Description Hide DescriptionScaling limits of the Hamiltonian of a system of charged particles coupled to a quantized radiation field are considered. Ultraviolet cutoffs, are imposed on the radiation field and the Coulomb gauge is taken. It is the socalled Pauli–Fierz model in nonrelativistic quantum electrodynamics. We mainly consider two cases: (i) all the ultraviolet cutoffs are identical, (ii) supports of ultraviolet cutoffs have no intersection, The Hamiltonian acts on where F is a symmetric Fock space, and has the form Here denotes a particle Hamiltonian, a quadratic field operator, and an interaction term. The scaling is introduced as where κ is a scaling parameter and a parameter of the scaling. Performing a mass renormalization we consider the scaling limit of as in the strong resolvent sense. Then effective Hamiltonians in infected with reaction of effect of the radiation field is derived. In particular (1) effective Hamiltonians with an effective potential for and (2) effective Hamiltonians with an observed mass for are obtained.

On the Gauss law and global charge for quantum chromodynamics
View Description Hide DescriptionThe local Gauss law of quantum chromodynamics(QCD) on a finite lattice is investigated. It is shown that it implies a gauge invariant, additive law giving rise to a gauge invariant valued global charge in QCD. The total charge contained in a region of the lattice is equal to the flux through its boundary of a certain valued, additive quantity. Implications for continuum QCD are discussed.

Renormalization of Poincaré transformations in Hamiltonian semiclassical field theory
View Description Hide DescriptionSemiclassical Hamiltonian field theory is investigated from the axiomatic point of view. A notion of a semiclassical state is introduced. An “elementary” semiclassical state is specified by a set of classical field configurations and quantum states in this external field. “Composed” semiclassical states viewed as formal superpositions of “elementary” states are nontrivial only if the Maslov isotropic condition is satisfied; the inner product of “composed” semiclassical states is degenerate. The mathematical proof of Poincaré invariance of semiclassical field theory is obtained for “elementary” and “composed” semiclassical states. The notion of semiclassical field is introduced; its Poincaré invariance is also mathematically proved.

On the spectral properties of Hamiltonians without conservation of the particle number
View Description Hide DescriptionWe consider quantum systems with variable but finite number of particles. For such systems we develop geometric and commutator techniques. We use these techniques to find the location of the spectrum, to prove absence of singular continuous spectrum, and identify accumulation points of the discrete spectrum. The fact that the total number of particles is bounded allows us to give relatively elementary proofs of these basic results for an important class of manybody systems with nonconserved number of particles.

skyrmions from harmonic maps
View Description Hide DescriptionWe construct multiskyrmion fields of the Skyrme models by using harmonic maps of to the Grassmannian which we express in terms of rank2 projectors. Within this construction we derive some approximate spherically symmetric solutions of Skyrme models and show that their energies are marginally higher than those for the rankone cases. We also discuss the possibility of generating exact spherically symmetric solutions using this construction. In particular, we present arguments which suggest that the only solutions obtained in this way are embeddings.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


A parabolic approximation method with application to global wave propagation
View Description Hide DescriptionMotivated by the difficulty in using the splitting matrix method to obtain parabolic approximations to complicated wave equations, we have developed an alternative method. It is three dimensional, does not a priori assume a preferred direction or path of propagation in the horizontal, determines spreading factors, and results in equations that are energy conserving. It is an extension of previous work by several authors relating parabolic equations to the horizontal ray acoustics approximation. Unlike previous work it applies the horizontal ray acoustics approximation to the propagator rather than to the Green’s function or the homogenous field. The propagator is related to the Green’s function by an integral over the famous “fifth parameter” of Fock and Feynman. Methods for evaluating this integral are equivalent to narrowangle approximations and their wideangle improvements. When this new method is applied to simple problems it gives the standard results. In this paper it is described by applying it to a problem of current interest—the development of a parabolic approximation for modeling global underwater and atmospheric acoustic propagation. The oceanic or atmospheric waveguide is on an Earth that is modeled as an arbitrary convex solid of revolution. The method results in a parabolic equation that is energy conserving and has a spreading factor that describes field intensification for antipodal propagation. Significantly, it does not have the singularities in its rangesliced version possessed by many parabolic equations developed for global propagation. We then discuss two extensions of the method; first to propagation along refracted geodesics and second to a description involving discrete, local, normal modes.

Integration of a generalized Hénon–Heiles Hamiltonian
View Description Hide DescriptionThe generalized Hénon–Heiles Hamiltonian with an additional nonpolynomial term is known to be Liouville integrable for three sets of values of It has been previously integrated by genus two theta functions only in one of these cases. Defining the separating variables of the Hamilton–Jacobi equations, we succeed here, in the two other cases, to integrate the equations of motion with hyperelliptic functions.
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 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Generalized sineGordon/massive Thirring models and soliton/particle correspondences
View Description Hide DescriptionWe consider a real Lagrangian offcritical submodel describing the soliton sector of the socalled conformal affine Toda model coupled to matter fields. The theory is treated as a constrained system in the context of Faddeev–Jackiw and the symplectic schemes. We exhibit the parent Lagrangian nature of the model from which generalizations of the sineGordon (GSG) or the massive Thirring (GMT) models are derivable. The dual description of the model is further emphasized by providing the relationships between bilinears of GMT spinors and relevant expressions of the GSG fields. In this way we exhibit the strong/weak coupling phases and the (generalized) soliton/particle correspondences of the model. The case is also outlined.

The Lund–Regge surface and its motion’s evolution equation
View Description Hide DescriptionThe system of evolution equations for general motion of surfaces in orthogonal coordinates is analyzed to reduce the number of variables as well as equations. The explicit expression of the Lund–Regge surface is obtained. When the surface corresponds to the Lund–Regge equation, we prove that some components of velocity satisfy the linearizations of the Lund–Regge equation. The soliton solution is derived and one special case of the Lund–Regge surface is studied.

kconstraint for the modified Kadomtsev–Petviashvili system
View Description Hide DescriptionBy imposing constraint on the pseudodifferential operator the constrained modified Kadomtsev–Petviashvili (KP) hierarchy and their corresponding Lax pair are obtained from the linear problem and its adjoint of the modified KP system. Especially, the modified KdV system, the GNS system with derivative coupling, the Burgers system, and a new integrable system are presented as examples.
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 RELATIVITY AND GRAVITATION


Shearfree relativistic fluids and the absence of movable branch points
View Description Hide DescriptionThe problem of determining the metric for a nonstatic shearfree spherically symmetric fluid (either charged or neutral) reduces to the problem of determining a oneparameter family of solutions to a secondorder ordinary differential equation (ODE) containing two arbitrary functions and Choices for and are determined such that this ODE admits a oneparameter family of solutions that have poles as their only movable singularities. This property is strictly weaker than the Painlevé property and it is used to identify classes of solvable models. It is shown that this procedure systematically generates many exact solutions including the Vaidya metric, which does not arise from the standard Painlevé analysis of the secondorder ODE. Interior solutions are matched to exterior Reissner–Nordstrøm metrics. Some solutions given in terms of second Painlevé transcendents are described.
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 MISCELLANEOUS TOPICS IN MATHEMATICAL METHODS


Associated Lamé and various other new classes of elliptic potentials from sl(2,R) and related orthogonal polynomials
View Description Hide DescriptionUsing representations of sl(2,R) generators which yield associated Lamé Hamiltonians we obtain new classes of elliptic potentials. We explicitly calculate eigenstates and spectra for these potentials and construct the associated orthogonal polynomials. We show that in the proper limit these potentials reduce to wellknown exactly solvable potentials.

Irreducible modules of finite dimensional quantum algebras of type A at roots of unity
View Description Hide DescriptionProperly specializing the parameters contained in the maximal cyclic representation of the nonrestricted Atype quantum algebra at roots of unity, we find the unique primitive vector in it. In this case, the representation is no longer irreducible. We show that the submodule generated by the primitive vector is the unique irreducible submodule and can be identified with an irreducible highest weight module of the finite dimensional Atype quantum algebra, which is defined as the subalgebra of the restricted quantum algebra at roots of unity.

The real geometry of holomorphic fourmetrics
View Description Hide DescriptionThe real geometry of holomorphic fourmetrics is investigated. The almost product and complex structures, associated with the real eightmetrics corresponding to the real and imaginary parts of the holomorphic metrics, are studied. It is shown that halfflat holomorphic metrics, and the corresponding real eightmetrics, are associated with integrable almost product, complex and hyperKähler structures. Real and complex local coordinate descriptions are presented.

Quantum enveloping superalgebras and link invariants
View Description Hide DescriptionCorresponding to each finite dimensional simple basic classical Lie superalgebra, a new quantum enveloping superalgebra is introduced, which has the structure of a braided quasiHopf superalgebra. In the case of this quantum enveloping superalgebra is shown to be isomorphic to the standard Drinfeld–Jimbo quantum superalgebra as braided quasi Hopf superalgebras. The new quantum enveloping superalgebras are applied to construct link invariants, from which Vassiliev invariants can be readily extracted. This, in particular, provides a useful construction for the Vassiliev invariants associated with
