Volume 45, Issue 6, June 2004
Index of content:
 GENERAL RELATIVITY AND GRAVITATION


Nonsingular stiff fluid cosmologies
View Description Hide DescriptionIn this paper we analyze Abelian diagonal orthogonally transitive space–times with spacelike orbits for which the matter content is a stiff perfect fluid. The Einstein equations are cast in a suitable form for determining their geodesic completeness. A sufficient condition on the metric of these space–times is obtained, that is fairly easy to check and to implement in exact solutions. These results confirm that nonsingular space–times are abundant among stiff fluid cosmologies.

 METHODS OF MATHEMATICAL PHYSICS


The appearance of the resolved singular hypersurface in the classical phase space of the Lie group
View Description Hide DescriptionA classical phase space with a suitable symplectic structure is constructed together with functions which have Poisson brackets algebraically identical to the Lie algebrastructure of the Lie group In this phase space we show that the orbit of the generators corresponding to the simple roots of the Lie algebra give rise to fibers that are complex lines containing spheres. There are spheres on a fiber and they intersect in exactly the same way as the Cartan matrix of the Lie algebra. This classical phase space bundle, being compact, has a description as a variety. Our construction shows that the variety containing the intersecting spheres is exactly the one obtained by resolving the singularities of the variety in A direct connection between this singular variety and the classical phase space corresponding to the Lie group is thus established.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Perturbations of ground states in weakly interacting quantum spin systems
View Description Hide DescriptionWe consider a general weak bounded finite range perturbation of a general free quantum spin Hamiltonian on a lattice. We prove that if the free Hamiltonian has a nondegenerate ground state and a spectral gap, then the perturbation also has a ground state, and estimate the localization of the spectrum in the corresponding ground state representation.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Histories electromagnetism
View Description Hide DescriptionWorking within the HPO (History Projection Operator) Consistent Histories formalism, we follow the work of Savvidou on (scalar) field theory [J. Math. Phys. 43, 3053 (2002)] and that of Savvidou and Anastopoulos on (firstclass) constrained systems [Class. Quantum Gravt. 17, 2463 (2000)] to write a histories theory (both classical and quantum) of Electromagnetism. We focus particularly on the foliationdependence of the histories phase space/Hilbert space and the action thereon of the two Poincaré groups that arise in histories field theory. We quantize in the spirit of the Dirac scheme for constrained systems.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Symmetric informationally complete quantum measurements
View Description Hide DescriptionWe consider the existence in arbitrary finite dimensions of a positive operator valued measure (POVM) comprised of rankone operators all of whose operator inner products are equal. Such a set is called a “symmetric, informationally complete” POVM (SIC–POVM) and is equivalent to a set of equiangular lines in SIC–POVMs are relevant for quantum state tomography,quantum cryptography, and foundational issues in quantum mechanics. We construct SIC–POVMs in dimensions two, three, and four. We further conjecture that a particular kind of groupcovariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.

 DYNAMICAL SYSTEMS


Traveling wave solutions of a generalized modified Kadomtsev–Petviashvili equation
View Description Hide DescriptionA method to generate traveling wave solutions of the generalized modified Kadomtsev–Petviashvili equation is reported and several physical solutions, including conditions of their existence, are presented.

 METHODS OF MATHEMATICAL PHYSICS


Fine gradings of
View Description Hide DescriptionA grading of a Lie algebra is called fine if it cannot be further refined. Fine gradings provide basic information about the structure of the algebra. There are six fine gradings of the semisimple Lie algebra of type over the complex number field. An explicit description of all the fine gradings of is given in terms of the fourdimensional representation of the algebra.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Relativistic quantum field theory with a fundamental length
View Description Hide DescriptionSince there are indications (from string theory and concrete models) that one must consider relativistic quantum field theories with a fundamental length the question of a suitable framework for such theories arises. It is immediately evident that quantum field theory in terms of tempered distributions and even in terms of Fourier hyperfunctions cannot meet the (physical) requirements. We argue that quantum field theory in terms of ultrahyperfunctions is a suitable framework. For this we propose a set of axioms for the fields and for the sequence of vacuum expectation values of the fields, prove their equivalence, and we give a class of models (analytic, but not entire functions of free fields).

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Phasespace Green’s functions for modeling timeharmonic scattering from smooth inhomogeneous objects
View Description Hide DescriptionThe paper deals with inhomogeneous medium Green’s functions in the phasespace domain by which the phasespace (local) spectral distributions of the field, scattered by a high contrast object due a genetic timeharmonic incidence, are evaluated. Two forms of phasespace Green’s functions are considered: one that links induced sources in the configurationspace to phasespace distributions of the scattered field, while the other one directly links the phasespace distribution of the incident field to phasespace distributions of the scattered field. The scattering mechanism is described in terms of local samplings of the object function which are localized in the object domain according to the scattered and incidenceprocessing parameters. Applications in the field of inverse scattering may be expected to yield fast and efficient algorithms, due to the capability of analytically evaluating (forward) scatteringGreen’s functions.

 GENERAL RELATIVITY AND GRAVITATION


Quantizing the line element field
View Description Hide DescriptionA metric with signature can be constructed from a metric with signature and a doublesided vector field called the line element field. Some of the classical and quantum properties of this vector field are studied.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


The relativistic Dirac–Morse Green’s function
View Description Hide DescriptionUsing a recently developed approach for solving the threedimensional Dirac equation with spherical symmetry, we obtain the twopoint Green’s function of the relativistic Dirac–Morse problem. This is accomplished by setting up the relativistic problem in such a way that makes comparison with the nonrelativistic problem highly transparent and results in a mapping of the latter into the former. The relativistic bound states energy spectrum is easily obtained by locating the energy poles of the Green’s function components in a simple and straightforward manner.

 DYNAMICAL SYSTEMS


A technique to identify solvable dynamical systems, and a solvable generalization of the goldfish manybody problem
View Description Hide DescriptionA simple approach is discussed which associates to (solvable) matrix equations (solvable) dynamical systems, generally interpretable as (interesting) manybody problems, possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles. We then focus on cases in which the auxiliary variables can be altogether eliminated, reobtaining thereby (via this unified approach) wellknown solvable manybody problems, and moreover a (solvable) extension of the “goldfish” model.

 METHODS OF MATHEMATICAL PHYSICS


Symmetry classification of KdVtype nonlinear evolution equations
View Description Hide DescriptionGroup classification of a class of thirdorder nonlinear evolution equations generalizing KdV and mKdV equations is performed. It is shown that there are two equations admitting simple Lie algebras of dimension three. Next, we prove that there exist only four equations invariant with respect to Lie algebras having nontrivial Levi factors of dimension four and six. Our analysis shows that there are no equations invariant under algebras which are semidirect sums of Levi factor and radical. Making use of these results we prove that there are three, nine, thirtyeight, fiftytwo inequivalent KdVtype nonlinear evolution equations admitting one, two, three, and fourdimensional solvable Lie algebras, respectively. Finally, we perform a complete group classification of the most general linear thirdorder evolution equation.

differential calculi on certain noncommutative (super) spaces
View Description Hide DescriptionIn this paper, we construct a covariant differential calculus on a quantum plane with twoparametric quantum group as a symmetry group. The two cases and are completely established. We also construct differential calculi and nilpotent on super quantum spaces with one and twoparametric symmetry quantum supergroup.

 DYNAMICAL SYSTEMS


Soliton lattice and single soliton solutions of the associated Lamé and Lamé potentials
View Description Hide DescriptionWe obtain the exact nontopological soliton lattice solutions of the associated Lamé equation in different parameter regimes and compute the corresponding energy for each of these solutions. We show that in specific limits these solutions give rise to nontopological (pulselike) single solitons, as well as to different types of topological (kinklike) single soliton solutions of the associated Lamé equation. Following Manton, we also compute, as an illustration, the asymptotic interaction energy between these soliton solutions in one particular case. Finally, in specific limits, we deduce the soliton lattices, as well as the topological single soliton solutions of the Lamé equation, and also the sineGordon soliton solution.

On bosonic limits of two recent supersymmetric extensions of the Harry Dym hierarchy
View Description Hide DescriptionTwo generalized Harry Dym equations, recently found by Brunelli, Das, and Popowicz in the bosonic limit of new supersymmetric extensions of the Harry Dym hierarchy [J. Math. Phys. 44, 4756 (2003)], are transformed into previously known integrable systems: one, into a pair of decoupled KdV equations, the other one, into a pair of coupled mKdV equations from a biHamiltonian hierarchy of Kupershmidt.

 GENERAL RELATIVITY AND GRAVITATION


Classification of the pseudosymmetric space–times
View Description Hide DescriptionThe pseudosymmetry condition on a manifold is a generalization of the notion of spaces of constant curvature. A complete algebraic classification of the pseudosymmetric space–times based on the Petrov type of the Weyl tensor and the Segré type of the Ricci tensor is presented.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Calculation of the self force using the extendedobject approach
View Description Hide DescriptionWe present the extendedobject approach for the explanation and calculation of the selfforce phenomenon (often called “radiationreaction force”). In this approach, one considers a charged extended object of a finite size ε that accelerates in a nontrivial manner, and calculates the total force exerted on it by the electromagnetic field (whose source is the charged object itself). We show that at the limit this overall electromagnetic field yields a universal result, independent of the object’s shape, which agrees with the standard expression for the self force acting on a pointlike charge. Previous implementation of this approach ended up with expressions for the total electromagnetic force that include terms which do not have the form required by massrenormalization. (In the special case of a spherical charge distribution, this term was found to be 4/3 times larger than the desired quantity.) We show here that this problem was originated from a too naive definition of the notion of “total electromagnetic force” used in previous analyses. We then derive the correct notion of total electromagnetic force. This completely cures the problematic term, for any object’s shape, and yields the correct self force at the limit In particular, for a spherical charge distribution, the above “4/3 problem” is resolved.

 METHODS OF MATHEMATICAL PHYSICS


Hydrodynamic reductions of multidimensional dispersionless PDEs: The test for integrability
View Description Hide DescriptionA dimensional dispersionless PDE is said to be integrable if its component hydrodynamic reductions are locally parametrized by arbitrary functions of one variable. The most important examples include the fourdimensional heavenly equation descriptive of selfdual Ricciflat metrics and its sixdimensional generalization arising in the context of selfdual Yang–Mills equations. Given a multidimensional PDE which does not pass the integrability test, the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which, when added to the equation, make it an integrable system in fewer dimensions. As an example of this phenomenon we discuss the second commuting flow of the dispersionless KP hierarchy. Considered separately, this is a fourdimensional PDE which does not pass the integrability test. However, the method of hydrodynamic reductions generates additional differential constraints which reconstruct the full dimensional dispersionless KP hierarchy.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


2D conformal field theories and holography
View Description Hide DescriptionIt is known that the chiral part of any 2D conformal field theory defines a 3D topological quantum field theory: quantum states of this TQFT are the CFT conformal blocks. The main aim of this paper is to show that a similar CFT/TQFT relation exists also for the full CFT. The 3D topological theory that arises is a certain “square” of the chiral TQFT. Such topological theories were studied by Turaev and Viro; they are related to 3D gravity. We establish an operator/state correspondence in which operators in the chiral TQFT correspond to states in the Turaev–Viro theory. We use this correspondence to interpret CFT correlation functions as particular quantum states of the Turaev–Viro theory. We compute the components of these states in the basis in the Turaev–Viro Hilbert space given by colored 3valent graphs. The formula we obtain is a generalization of the Verlinde formula. The later is obtained from our expression for a zero colored graph. Our results give an interesting “holographic” perspective on conformal field theories in two dimensions.
