bluuebunny/crossref_metadata_2025 huggingface dataset:
C*-algebraic scattering theory and explicitly solvable quantum field theories
Authors: Warchall Henry A.
Published: July 2002

A general theoretical framework is developed for the treatment of a class of quantum field theories that are explicitly exactly solvable, but require the use of C*-algebraic techniques because time-dependent scattering theory cannot be constructed in any one natural representation of the observable algebra. The purpose is to exhibit mechanisms by which inequivalent representations of the observable algebra can arise in quantum field theory, in a setting free of other complications commonly associated with the specification of dynamics. One of two major results is the development of necessary and sufficient conditions for the concurrent unitary implementation of two automorphism groups in a class of quasifree representations of the algebra of the canonical commutation relations (CCR). The automorphism groups considered are induced by one-parameter groups of symplectic transformations on the classical phase space over which the Weyl algebra of the CCR is built; each symplectic group is conjugate by a fixed symplectic transformation to a one-parameter unitary group. The second result, an analog to the Birman–Belopol’skii theorem in two-Hilbert-space scattering theory, gives sufficient conditions for the existence of Mo/ller wave morphisms in theories with time-development automorphism groups of the above type. In a paper which follows, this framework is used to analyze a particular model system for which wave operators fail to exist in any natural representation of the observable algebra, but for which wave morphisms and an associated S matrix are easily constructed.
Completeness relation for energy-dependent separable potentials
Authors: Garcilazo Humberto
Published: July 2002

We point out that the eigenfunctions of energy-dependent separable potentials, which are commonly used in the relativistic three-body problem, form a complete set of states. The completeness property is important, since it is necessary in order to satisfy the optical theorem, and consequently to conserve probability. We show that there exists a large family of energy-dependent separable potentials whose eigenfunctions form a complete set. Although the eigenfunctions of these potentials are not mutually orthogonal, it is shown that in general they are linearly independent.
A mean-field solution of the reflection of a spherical acoustic wave from a rough interface
Authors: Lerche I. , Hill N. R.
Published: July 2002

We use mean-field methods to calculate the reflection by a rough interface of incident acoustic waves emanating from a point source. The calculation is accurate to second order in roughness height. For the special cases of very long and very short roughness wavelengths, we find closed-form expressions for the reflected field. We give special attention to the head-wave arrival and find the roughness can attenuate or enhance head-wave arrival amplitude depending on the velocities and densities of the media. The roughness can also cause a delay of the head-wave arrival though the apparent velocity is not changed (to second-order accuracy). As a prerequisite to the rough-interface calculation, we consider a smooth interface and find an asymptotic method of calculating the reflected field which avoids severe distortions of the path of integration.
Lie transformation group solutions of the nonlinear one-dimensional Vlasov equation
Authors: Abraham-Shrauner B.
Published: July 2002

The solution of the exact, nonlinear one-dimensional Vlasov equation with a space- and time-dependent electric field is reduced to the solution of a nonlinear, first-order ordinary differential equation with two subsidiary equations. The reduction holds for any electric field nonlinear in the spatial coordinate or for a subclass of electric fields linear in the spatial coordinate and is equivalent to the solution of a generalized Bernstein–Greene–Kruskal (BGK) Vlasov equation with a velocity-dependent Lorentz force. The Lie method for the solution of differential equations invariant under a transformation group has been used to calculate the group generator, the canonical variables, and the generalized BGK Vlasov equation. Analytical forms for the functional dependence of the Vlasov equation one-particle distribution function are given.
The Lorentz group and the Thomas precession. II. Exact results for the product of two boosts
Authors: Salingaros Nikos
Published: July 2002

The product of two Lorentz boosts in different directions is equal to the product of a pure boost and a spatial rotation. To second order, the resulting boost is simply the sum of the individual boosts, and the rotation is responsible for the Thomas precession. Here the resulting boost and the rotation correction are calculated exactly. The calculation of higher-order corrections to the sum of two finite Lorentz boosts is not usually done, and here previous exact results are critically compared. The complete expression is, to the best of the author’s knowledge, a new result.
On the complete integrability of the supersymmetric nonlinear Schrödinger equation
Authors: Roy Chowdhury A. , Naskar Minati
Published: July 2002

The Painlevé criterion has been applied to the supersymmetric nonlinear Schrödinger equation. This particular system of fermionic and bosonic fields shows up a rich spectrum of resonances and it can be explicitly proved that the expansion coefficients at the resonance positions can remain arbitrary. At this point is is worth noting that even when the extra nonlinear field (which is fermionic in this case) is considered to be bosonic, the resulting system turns out to satisfy the Painlevé test so that this second system may be thought of as a new completely integrable system whose Lax pair is still to be found.
Vacuum polarization in curved backgrounds deduced from Hadamard kernels
Authors: Castagnino Mario A. , Harari Diego D. , Núñez Carmen A.
Published: July 2002

The point-splitting method of regularization is applied to alternative Hadamard kernels, for scalar fields in curved backgrounds, in order to study the polarization term in the vacuum expectation value (VEV) of the energy-momentum tensor. An ansatz to define this term is introduced and verified in several cases. The renormalized VEV of the energy-momentum tensor of a scalar massive conformally coupled field in a Robertson–Walker universe is explicitly computed. Two terms appear, the polarization term, which coincides with the one computed using the above-mentioned ansatz, and a creation of particle term.
New classes of symmetries for partial differential equations
Authors: Bluman George W. , Reid Gregory J. , Kumei Sukeyuki
Published: July 2002

New classes of symmetries for partial differential equations are introduced. By writing a given partial differential equation S in a conserved form, a related system T with potentials as additional dependent variables is obtained. The Lie group of point transformations admitted by T induces a symmetry group of S. New symmetries may be obtained for S that are neither point nor Lie–Bäcklund symmetries. They are determined by a completely algorithmic procedure. Significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation.
Gravity as a gauge theory with Cartan connection
Authors: Zardecki Andrew
Published: July 2002

A gauge formulation of gravity, based on the notion of the connection of Cartan, is given. The Cartan connection involves two principal fiber bundles P and P′ with groups G and G′, respectively; G′ is a subgroup of G and can be regarded as a symmetry group to which G is broken. When the differential form defining the connection in P gives absolute parallelism in P′, one speaks about the Cartan connection. General geometric framework is specialized to the case in which G is the de Sitter group SO(4,1) and G′ is the Lorentz group SO(3,1). The action of the Yang–Mills type is similar to, but not identical with, the action derived earlier by Townsend. The field equations in P are translated into a system of coupled equations for curvature and torsion in P′. Under contraction of SO(4,1) to the Poincaré group ISO(3,1), and for vanishing torsion, the equations become Yang’s equation and Einstein’s equation in vacuum. The BRS invariance of the theory, supplemented by Faddeev–Popov and gauge fixing terms, is analyzed in some detail.
Extended wave solutions in an integrable chiral model in (2+1) dimensions
Authors: Leese Robert
Published: July 2002

There is a modification of the SU(2) chiral model, which is integrable in (2+1) dimensions [J. Math. Phys. 29, 386 (1988)]. In addition to localized lumps, it admits extended wave solutions, which move at constant velocity. The interaction of two waves causes each to experience a phase shift. In the interaction between a wave and a lump, the wave suffers no phase shift, but the lump changes shape.
Constraints on the total and kinetic energy of ground states in a class of potential models
Authors: Common A. K. , Martin A.
Published: July 2002

It is shown that the ground state energy in a central potential is a concave (convex) function of angular momentum l when this potential is a concave (convex) function of r2. A result stronger than concavity is obtained when the Laplacian of the potential is negative. New bounds in the kinetic energy of these states are derived in terms of energy differences and for s states in terms of the wave function at the origin. These bounds can be used to give good estimates of the ground state energies for power law potentials.
Intervals of electrohydrodynamic Rayleigh–Taylor instability. I. Effect of a tangential periodic field
Authors: Elshehawey Elsayed F.
Published: July 2002

The intervals of electrohydrodynamic Rayleigh—Taylor instability influenced by a periodic tangential field are considered. It is shown that a linear model of the interface is governed by Hill’s differential equation. Characteristic values and intervals of stability are discussed. The special case of the Mathieu differential equation type is obtained.
Collisions of gravitational and electromagnetic waves that do not develop curvature singularities
Authors: Papacostas Taxiarchis , Xanthopoulos Basilis C.
Published: July 2002

A five-parameter family of solutions is investigated, describing the collision of plane-fronted impulsive gravitational and shock electromagnetic waves. In the interaction region, to the future of the collision, it is a locally known solution of the Einstein–Maxwell electrovacuum equations of Petrov type D. The collision results in the formation of a Cauchy horizon. Extensions of the space-time are constructed beyond the Cauchy horizon and beyond certain two-dimensional surfaces that are mere coordinate singularities. It is found that in the extended space-time the following may occur: (i) no curvature singularities, (ii) two-dimensional spacelike curvature singularities, and (iii) two-dimensional timelike curvature singularities, according to the ranges of the parameters of the solution.
On the violation of causality in relativistic quantum mechanics
Authors: Kosiński Piotr , Maślanka Pawel/
Published: July 2002

The estimates on causality violating part of the wave function are given. The problem of acausal behavior of charge and current densities is discussed.
Algebras of diffeomorphisms of the N-torus
Authors: Ramos E. , Sah C. H. , Shrock R. E.
Published: July 2002

The general algebra 𝒟(N,V) of infinitesimal diffeomorphisms of the N-torus (S1)N involving generators Lv,m depending on a structure vector v∈CN and a vector-valued index m∈ZN is constructed. Several results are proved for this algebra. Special cases of the algebra were previously presented in Ramos–Shrock [E. Ramos and R. E. Shrock, Int. J. Mod. Phys. A 4, 4279 (1989).] The concept of the Q rank of the space V of structure vectors, denoted rQ(V), is defined and certain related ‘‘Cartan matrices’’ are introduced. It is shown that 𝒟(N,V), as an ungraded algebra, is simple if and only if rQ(V)=N, i.e., the Q rank of V is maximal. A classification under isomorphisms is given for the algebra and is shown to reduce to a classification of the Cartan matrices. The space of structure vectors is isomorphic to the unique ‘‘Cartan subalgebra.’’ A number of properties concerning the central extensions of these algebras are then proved. For the case dim V=1, rQ(V)=N, it is shown that the central extension is unique. For the case dim V=1,rQ(V)<N, a new and greatly enlarged central extension is constructed involving a central charge function from ZN−r to C (essentially equivalent to an infinite number of central charge parameters), rather than a single central charge parameter. Finally, it is proved that for dim V≥2, this algebra has no nontrivial central extension.
Exponentially evolving densities
Authors: Verosky John M.
Published: July 2002

The notion of a conserved density is generalized to those densities whose integrals evolve exponentially. In the formal differential algebra used to find them, only a slight modification of the conserved density case is needed. Various examples for quasilinear evolution equations are given and an application to finding higher-order conserved densities by taking the Poisson bracket of exponentially evolving densities is indicated.
The linearized gauge field propagator in an inhomogeneous medium
Authors: Tang P. , Wilets L.
Published: July 2002

In an inhomogeneous medium with μ=κ−1, the linearized gauge field propagator is presented in terms of spherical harmonics for the Coulomb gauge. Explicit expressions for the propagator are given in the case of the MIT bag for both the Coulomb and Lorentz gauges. Gauge invariance is shown explicitly for the one gluon interaction in the MIT bag. This work supplements and corrects the earlier work by Bickeböller, Goldflam, and Wilets [J. Math. Phys. 26, 1810 (1985)].
Low-frequency moments in inverse scattering theory
Authors: Dassios George
Published: July 2002

The low-frequency moments of the scattering amplitude are utilized in order to identify the capacity, the center, and the orientation of an acoustically soft scatterer.
Dispersion relations for causal Green’s functions: Derivations using the Poincaré–Bertrand theorem and its generalizations
Authors: Davies K. T. R. , Davies R. W. , White G. D.
Published: July 2002

A famous theorem by Poincaré and Bertrand formally describes how to interchange the order of integration in a double integral involving two principal-value factors. This theorem has important applications in many-body physics, particularly in the evaluation of response functions (or ‘‘loop integrals’’) at either zero or finite temperatures. Of special interest is the loop containing an integration with respect to the energy of two causal propagators. It is shown that such a response function with two boson or two fermion lines behaves statistically like a boson, while the response function containing a boson and a fermion behaves like a fermion. Examples are given of typical loop integrals occurring in the solution of Dyson’s equations for nuclear matter in the presence of delta, nucleon, and pion interactions. A ‘‘form factor’’ that is essential for the convergence of the nucleon–pion loop integral is chosen to have little effect on the analogous nucleon–delta loop integral. The Poincaré–Bertrand (PB) theorem is then generalized to multiple integrals and higher-order poles. From the generalization of the theorem to triple integrals, it is shown that causality is rigorously maintained, at zero temperature, for convolutions with respect to the time of products of Green’s functions and thus for Dyson’s equations. Also, for finite temperature, the three-propagator loop integral satisfies the statistics appropriate for the loop as a whole, in direct analogy with the result for the two-propagator loop. The intimate connection between the PB theorem and analyticity (or causality) is clearly demonstrated. Although this work considers explicitly only nuclear physics examples, the results are relevant to other fields where many-body theory is used.
Proof of the noninteraction theorem for a system of N relativistic particles in direct interaction
Authors: Li Fu-Bin
Published: July 2002

This paper proves a noninteraction theorem for a system of N relativistic particles in direct interaction, and each particle has internal Grassmann degrees of freedom for describing spin.
Internal and external metrics for a perfect fluid cylinder in general relativity
Authors: Davidson W.
Published: July 2002

A solution to Einstein’s equations describing a perfect fluid cylinder of finite radius is presented. The proper density μ and pressure p of the fluid are physically well behaved in the radial coordinate range 0≤r≤r1. On the axis (r=0) the solution is regular and μ and p are finite and positive. As r increases μ and p decrease steadily through positive values, p vanishing at r=r1. The ratio p/μ (<1) is also monotonically decreasing, as is also the velocity of sound a (<1) in the fluid. The equation of state is p= (3)/(7)  μ−Nμ3/10, where N is a positive constant. The matching metric for the vacuum exterior to the cylinder is given, so that the space-time is complete and nonsingular.
Inverse scattering on the line for a Dirac system
Authors: Hinton D. B. , Jordan A. K. , Klaus M. , Shaw J. K.
Published: July 2002

The whole-line version of the Gelfand–Levitan–Marchenko (GLM) equation for a Dirac system is studied. A new derivation of the GLM equation is given, under weaker hypotheses than Frolov’s earlier treatment [Sov. Math. Dokl. 13, 1468 (1972)], and the complete inversion is carried out in some explicit cases in which a spectral gap is present. Previous calculations of this type are restricted either to a scalar potential or degenerate gap. Applications are discussed in connection with optical couplers and soliton equations.
Uniform approximation and coherent state path integrals
Authors: Fricke S. H. , Balantekin A. B. , Uzer T.
Published: July 2002

One possible formulation of mean-field approximations is in terms of coherent state path integrals. In this paper, the construction of uniform semiclassical approximation to coherent state integrals is investigated. As a specific example, the method is applied to the forced harmonic oscillator.
Two-point functions and stress-energy tensors of p-forms in de Sitter and anti-de Sitter spaces
Authors: Folacci A.
Published: July 2002

It is obtained, in four-dimensional de Sitter and anti-de Sitter spaces, the two-point functions of the quantum p-forms corresponding to maximally symmetric and Hadamard vacua. It is then determined the associated renormalized stress-energy tensors.
Graded representations of the Temperley–Lieb algebra, quantum supergroups, and the Jones polynomial
Authors: Zhang R. B.
Published: July 2002

It is shown that from each self-dual representation of a quantum supergroup with nonvanishing q-superdimension, a graded representation of the Temperley–Lieb algebra can be constructed, which, in turn, gives rise to a solution of the graded Yang–Baxter equation. As examples, the vector irrep of Uq (osp(M/2n)) and all the finite-dimensional irreps of Uq (osp(1/2)) are studied, and the corresponding graded Ř-matrices are obtained explicitly. A general method is also given to obtain link polynomials from graded Ř-matrices. Applied to the ones associated with graded representations of the Temperley–Lieb algebra, the method yields the Jones polynomial.
The ’t Hooft reparametrization respects Borel summability
Authors: Kopper Christoph
Published: July 2002

In a renormalizable field theory with one coupling constant, one may redefine the coupling in the sense of formal power series such that the β-function in the new coupling consists of the first two terms only. It is shown that this change of coupling is Borel summable if the original β-function was Borel summable.
Recursive equation of character expansion coefficient in lattice gauge theory
Authors: Tanimura N. , Tanimura O.
Published: July 2002

The strict mathematical derivation is given for the recursion relations of the coefficients of the character expansion of the single-link action in the SU(2), U(2), and SU(3) groups. This relies on the immediate Schwinger–Dyson variation on the character. It will be suggested that the independent recursion relations for the U(N) and SU(N) groups are associated with the group invariants, i.e., the Casimir invariants and the unit generator [for U(N)].
Conduction in a two-phase plane with diamond-shaped tiling
Authors: Helsing Johan , Grimvall Göran , Bao Ke-da
Published: July 2002

The effective conductivity of a two-phase two-dimensional composite with diamond-shaped tiling is considered. This analysis, based on a projection of the boundary conditions on linear combinations of solutions to the electrostatic equation that are orthonormal on the boundary, generalizes results by Keller [J. Math. Phys. 28, 2516 (1987)] and others. Numerical results are given for several conductivity ratios of the two phases and for varying obtuse angles of the tiles. Special emphasis is given to very large and very small conductivity differences, and very elongated tiles.
On the unitary irreducible representations of SO(n,1) and U(n,1) in the scalar product with the nonintertwining operator
Authors: Maekawa Takayoshi
Published: July 2002

The irreducible representations of the noncompact groups SO(n,1) and U(n,1) are discussed in the space of functions on the compact groups SO(n) and U(n), and then it is shown that the Hermitian form scalar product with a nonintertwining operator can be introduced for each of the complementary and the discrete series of the unitary irreducible representations (UIR) of the groups SO(n,1) and U(n,1). The positive definite scalar product can be introduced for all classes except for a discrete series in the case of U(n,1).
The Riemann–Hilbert problem for nonsymmetric systems
Authors: Greenberg W. , Zweifel P. F. , Paveri-Fontana S.
Published: July 2002

A comparison of the Riemann–Hilbert problem and the Wiener–Hopf factorization problem arising in the solution of half-space singular integral equations is presented. Emphasis is on the factorization of functions lacking the reflection symmetry usual in transport theory.
Self-duality and the metric in a family of neighbors of Einstein’s equations
Authors: Bengtsson Ingemar
Published: July 2002

In four space-time dimensions, there exists an infinite-parameter family of neighbors of Einstein’s equations. The action is formulated in terms of an SO(3) vector potential. It is proven that the space-time metric can be computed from the SO(3) field strength by means of a parameter-independent formula with a clear geometrical meaning.
A scaling limit of the Glauber dynamics for a class of neural network models
Authors: Koch Hans
Published: July 2002

The time evolution of d mean field variables is considered for networks of N neurons whose connection matrices JN have d distinct rows. Certain assumptions are made about the large N behavior of JN, which guarantee the convergence of a free-energy density function. These assumptions are known to be satisfied, e.g., in the Hopfield model with p stored patterns, for d=2p. It is proved that in a scaling limit, where N tends to infinity and d stays fixed, the time evolution approaches that of a diffusion process in Rd. This process describes in detail, and for times up to ℴ(N3/2) iterations, the dynamics of the mean field fluctuations near a local minimum of the free-energy density.
Chiral anomalies for vortex potentials in two dimensions and a decompactification limit
Authors: Bassetto A. , Griguolo L.
Published: July 2002

A complete set of eigenfunctions on a two-dimensional sphere is obtained for the Dirac operator corresponding to a vortexlike Abelian potential. Then a decompactification limit is studied from the sphere to R2, in which the potential leads to a constant field configuration. The chiral anomaly is studied by using first-order perturbation theory and the crucial role played by zero modes in this context is carefully discussed.
Universal R matrices and invariants of quantum supergroups
Authors: Zhang R. B. , Gould M. D.
Published: July 2002

A general method is developed for constructing invariants of quantum supergroups using universal R matrices. Applied to Uq(gl(2/1)), this method yields the invariants of this quantum supergroup in explicit form. The eigenvalues of these invariants in arbitrary irreducible highest weight representations are computed.
Note on the unitary irreducible representations of U(n) in the scalar product with the nonintertwining operator
Authors: Maekawa T.
Published: July 2002

It is shown that a discrete series of the unitary irreducible representations of U(n,1) by Gel’fand and Graev cannot be treated in the space of functions on the compact subgroup U(n)⊆U(n,1).
Curiosities of arithmetic gases
Authors: Bakas Ioannis , Bowick Mark J.
Published: July 2002

Statistical mechanical systems with an exponential density of states are considered. The arithmetic analog of parafermions of arbitrary order is constructed and a formula for boson-parafermion equivalence is obtained using properties of the Riemann zeta function. Interactions (nontrivial mixing) among arithmetic gases using the concept of twisted convolutions are also introduced. Examples of exactly solvable models are discussed in detail.
The boundary of a boundary principle in field theories and the issue of austerity of the laws of physics
Authors: Kheyfets Arkady , Miller Warner A.
Published: July 2002

The boundary of a boundary principle has been suggested by J. A. Wheeler as a realization of the austerity idea in field theories. This principle is described in three basic field theories—electrodynamics, Yang–Mills theory, and general relativity. It is demonstrated that it supplies a unified geometric interpretation of the source current in each of the three theories in terms of a generalized E. Cartan moment of rotation. The extent to which the boundary of a boundary principle represents the austerity principle is discussed. It is concluded that it works in a way analogous to thermodynamic relations and it is argued that deeper principles might be needed to comprehend the nature of austerity.
The weak coupling limit for quadratic interaction
Authors: Lu Y. G.
Published: July 2002

The weak coupling limit of a ‘‘system+reservoir’’ model with nonlinear (quadratic) interaction is discussed. In a previous paper [Accardi and Lu, Stoch. Process. Phys. Geom., 1–26], it has been shown that, if the matrix elements of the van Hove rescaled evolution are taken with respect to the same collective vectors used for linear interactions, then the limiting evolution is not unitary. In the present paper, we choose a new type of collective vectors and obtain, in the weak coupling limit, a unitary evolution satisfying a quantum stochastic differential equation. A remarkable feature of the limiting process is that the quadratic nature of the interaction shows up in a nontrivial choice of the Hilbert space where the limiting quantum noise takes its values, which is qualitatively new with respect to all the models considered up to now.
O(2,1)-like little group for spacelike four-momenta and localized light waves in continuous media
Authors: Kim Y. S. , Yeh Leehwa
Published: July 2002

While a light wave travels in a nonvacuum medium with speed less than the speed of light in vacuum, its energy-momentum four-vector is spacelike. Thus the little group for this light wave is locally isomorphic to O(2,1). It is shown that the transformation of this little group produces observable effects on the light wave, in contrast to the case of the E(2)-like little group for photons in vacuum. It is also shown that the little group can be defined for a localized light wave consisting of a superposition of waves with different frequencies traveling in the same direction.
Ferromagneticity of simplicial fields on two-dimensional compact manifolds
Authors: Albeverio Sergio , Hohler Erling G. B. , Zegarlinski Boguslav
Published: July 2002

Smooth triangulations of a compact smooth connected two-dimensional Riemannian manifold M are considered. The q-simplicial fields are defined with values in the space of q-cochains and a natural Gaussian measure is defined giving their distribution, with covariance defined essentially in terms of the combinatorial Laplacian Δc. In the continuum limit this measure for q=0 is the free quantum field measure over M. In this case it is shown that for a certain collection of triangulations there exists a sequence of subdivisions of each triangulation such that the corresponding measure is ferromagnetic. It is also shown that for sufficiently fine subdivisions −Δ+m2I, m≳0 has nonpositive off-diagonal elements. The proofs are obtained by a result on triangulations by simplexes with acute angles. It is also proven that the probability measures describing quantum fields on M with polynomial, trigonometric, or exponential interactions satisfy FKG inequalities.
Gauge equivalence theory of the noncompact Ishimori model and the Davey–Stewartson equation
Authors: Leo R. A. , Martina L. , Soliani G.
Published: July 2002

The gauge equivalence between a noncompact version of the Ishimori spin model and the Davey–Stewartson equation is established. Explicit relationships connecting the corresponding two sets of fields involved in these systems are obtained via any pair of complex functions satisfying an equation of the Schrödinger type for a free particle. Using these formulas, two examples of classes of nontrivial exact singular solutions to the Davey–Stewartson equation are given. One of them is of the closed stringlike type, while the other is doubly periodic and is expressed in terms of Riemann theta functions. The role played by the symmetry group associated with the gauge equivalent equations under consideration is also clarified.
Lattices and Θ-function identities. I: Theta constants
Authors: Gannon Terry , Lam C. S.
Published: July 2002

The gluing construction of lattices is used to generate and study a number of theta constant identities. It is shown, for example, that the Jacobi identity, a well-known degree 4 identity, can be derived from a degree 2 identity. A list of all quadratic identities in θ3 derivable from this lattice method, containing over 30 algebraically independent identities, and conjectured to yield all quadratic identities in θ3 derivable by any means, is included. In contrast, only two independent quadratic identities in θ3 have been found elsewhere in the mathematical literature. The theta constants of lattices and their glue classes are investigated, and the fact that the theta constant of each glue class is a root of a polynomial whose coefficients are linear combinations of theta constants of lattices is shown.
Cosmological singularity at the Planck scale
Authors: Magnon Anne
Published: July 2002

We present quantum charges which could have contributed to galaxy seeding in the early universe, and onset of the expansion field. The evaporation of such charges due to topological modifications (and dissipation of strings) is shown to be related to the onset, in the infinite past, of a singularity which could disappear in the Planckian regime, due to dislocation of the geometry.
Generalized intermediate long-wave hierarchy in zero-curvature representation with noncommutative spectral parameter
Authors: Degasperis A. , Lebedev D. , Olshanetsky M. , Pakuliak S. , Perelomov A. , Santini P. M.
Published: July 2002

The simplest generalization of the intermediate long-wave hierarchy (ILW) is considered to show how to extend the Zakharov–Shabat dressing method to nonlocal, i.e., integro-partial differential, equations. The purpose is to give a procedure of constructing the zero-curvature representation of this class of equations. This result obtains by combining the Drinfeld–Sokolov formalism together with the introduction of an operator-valued spectral parameter, namely, a spectral parameter that does not commute with the space variable x. This extension provides a connection between the ILWk hierarchy and the Saveliev–Vershik continuum graded Lie algebras. In the case of ILW2 the Fairlie–Zachos sinh-algebra was found.
The Clebsch–Gordan problem of the SL(2,R) coherent states
Authors: Basu Debabrata
Published: July 2002

The Clebsch–Gordan problems of the Barut–Girardello, and Perelomov coherent states of SL(2,R) are studied using the associated Hilbert spaces as the respective carrier spaces of the representations of the group. For the Barut–Girardello coherent states this Hilbert space is a subspace of the Bargmann–Segal Hilbert space B(C2) called the ‘‘reduced Bargmann space.’’ The generators of the group in this realization are essentially the boson operators of Holman and Biedenharn which provide a convenient starting point of the problem. For the Perelomov coherent states the associated Hilbert space turns out to be Bargmann’s canonical carrier space for the realization of the discrete series of representations, namely, the Hilbert space of functions analytic inside the open unit disc. The scalar product, the principal vector, and a complete orthonormal set in these Hilbert spaces are constructed and used for the explicit evaluation of the Clebsch–Gordan coefficients. For each of the coherent state systems the product state turns out to be the principal vector and, therefore, the coupled state itself is the Clebsch–Gordan coefficient. For the Barut–Girardello coherent states this is, apart from normalization, the product of a Bessel function and d-function. For the Perelomov coherent states, on the other hand, this closely resembles the Clebsch–Gordan coefficient of the SU(2) coherent states.
On N=3 super Korteweg–de Vries equation
Authors: Bellucci Stefano , Ivanov Evgenyi , Krivonos Sergey
Published: July 2002

A one-parameter family is constructed of N=3 supersymmetric and SO(3) symmetric extensions of the Korteweg–de Vries (KdV) equation as a Hamiltonian flow on a N=3 superconformal algebra and it is argued that it is nonintegrable for any choice of the parameter. Then a modified N=3 super KdV equation is proposed which possesses the higher order conserved quantities and so is a candidate for an integrable system. Upon reduction to N=2, it yields the recently discussed ‘‘would-be integrable’’ version of the N=2 super KdV equation. In the bosonic core it contains a coupled system of the KdV-type equation and a three-component generalization of the mKdV equation. A Hamiltonian formulation of the new N=3 super KdV equation as a flow on some contraction of the direct sum of two N=3 superconformal algebras is given.
On the problem of a local extension of the quantum formalism
Authors: Angelidis Thomas D.
Published: July 2002

The suggested impossibility of a consistent local extension of the quantum formalism is reviewed in the context of the Einstein–Podolsky–Rosen–Bohm (EPRB) ideal experiment. A consistent local theory Th(𝒢) extending the quantum formalism is proposed. The postulates of Th(𝒢) stipulate a finer state specification than that given by the quantum state ‖γ1,γ2≳. This describes how the spontaneous annihilation process itself breaks the spherical symmetry of the singlet state ‖γ1,γ2≳ into a pair of back-to-back photons having only axial symmetry about the direction of their motion. The theory Th(𝒢) shows that local action suffices to explain all that the quantum formalism predicts for the EPRB ideal experiment as well as some other results.
Spin system, gauge theory, and renormalization group equation
Authors: Goswami Gautam , Bandyopadhyay Pratul
Published: July 2002

The gauge theoretical formulation of a spin system is studied here. It is pointed out that a quantum spin can be associated with a gauge bundle where the gauge group is SU(2). The Ising spin is obtained by breaking the group to U(1) coupled with a Z2 symmetry giving rise to a Z2 gauge theory in three dimension. The gauge structure of a spin system helps us to reformulate the block-variable transformation in terms of a gauge transformation.
Null geodesics in the gravitational field of a rotating, radiating body
Authors: Hadley R. H. , Mallett R. L.
Published: July 2002

A solution of the Einstein field equations for a rotating radiating body was presented by Carmeli and Kaye in 1977 [Ann. Phys. 103, 97 (1977)]. Their paper presented a new line element along with the corresponding energy-momentum tensor for the radiation field. The Newman–Penrose formalism [J. Math. Phys. 3, 566 (1962)] was used to solve and analyze the new solution which lead to the tetrad components of the Ricci, Maxwell, and Weyl tensors. However, no attempt was made to solve the geodesic equation in this new metric. The present paper starts with a linearized form of the Carmeli and Kaye line element and solves the resulting differential equations to a particular order for a null geodesic. The deflection angle is found from the equations of motion and compared in detail for the two cases of a rotating, highly luminous quasar and our sun.
Thomas–Fermi theory in magnetic fields of arbitrary strength
Authors: Pfalzner S. , March N. H.
Published: July 2002

The Thomas–Fermi theory is generalized to apply to an inhomogeneous electron assembly in an applied constant magnetic field of arbitrary field strength. The tool used is the canonical density matrix satisfying the Bloch equation. A suitable inverse Laplace transform then generates the ground-state electron density. Known results can be recovered analytically in the weak field and in the high field limits. However, for intermediate fields, results are presented for specimen field strengths and for model atomic potentials, using a numerical Laplace inversion technique.
Order and spacings of energy levels for the Klein–Gordon equation
Authors: Grosse H. , Martin A. , Stubbe J.
Published: July 2002

One can prove and extend some results about the order and spacings of energy levels for the Klein–Gordon equation with vector-like and scalar-like potentials.
The abstract complex Lorentz transformation group with real metric. I. Special relativity formalism to deal with the holomorphic automorphism group of the unit ball in any complex Hilbert space
Authors: Ungar Abraham A.
Published: July 2002

Viewing the unit ball in a complex Hilbert space as a set of relativistically admissible complex velocities, the holomorphic automorphisms of the ball emerge as ‘‘complex relativistic velocity additions’’ and ‘‘complex rotations’’ (that is, unitary transformations). Hence, newly discovered properties of the abstract real Lorentz group [A. A. Ungar, Am. J. Phys. 59, 824 (1991); A. A. Ungar, Am. J. Phys. 60, 815 (1992)], which turn out to provide powerful tools for the study of the holomorphic automorphisms of the unit ball in any complex Hilbert space, are of interest in the literature [W. Rudin, Function Theory in the Unit Ball of Cn (Springer-Verlag, New York, 1980) ]. A particularly useful and important tool turns out to be provided by Thomas gyration, which is an abstraction of the well-known Thomas precession of special relativity theory, and which gives rise to a gyrogroup structure underlying the ball. Results of this article set the stage for the study, in a subsequent article, of complex Lorentz transformation groups, U(1,n), in a way analogous to the study of Galilean groups. The dimension n is finite or infinite, and a real parameter c is involved in such a way that in the limit of large c, c→∞, the Lorentz groups that we study reduce to their Galilean counterparts.
Lattice solitons directly by the bilinear method
Authors: Chow Kwok W.
Published: July 2002

Recently a doubly periodic solution of the Kadomtsev–Petviashvili equation was deduced by summing over component solitons. The same solution was derived directly by the Hirota bilinear method. This alternate route enables one to obtain a new solution. Such a mechanism can also be applied to wavepacket dynamics, e.g., the Davey–Stewartson equation.
Hamiltonian properties of the canonical symmetry
Authors: Leznov A. N. , Razumov A. V.
Published: July 2002

The properties of the canonical symmetry are investigated. The densities of the local conservation laws for the nonlinear Schrödinger equation are shown to change under the action of the canonical symmetry by total space derivatives. It is also shown how the canonical symmetry can be used to look for the hierarchy of the Hamiltonian operators relevant to the system under consideration. It appears that only the invariance condition can be used to solve the problem.
SO(5)q and contraction: Chevalley basis representations for q-generic and root of unity
Authors: Chakrabarti A.
Published: July 2002

Representations of SO(5)q are constructed explicitly on the Chevalley basis. The classical representations, once obtained on this basis, are very simply deformed for arbitrary q, generic, and root of unity. This is in sharp contrast to the classical canonical Gelfand–Zetlin (GZ) representations which lead to problems, studied elsewhere, in q deformations for orthogonal (though not for unitary) algebras. Our results are limited to irreducible representations labeled by three variable parameters, the maximal number being four. Within this restriction it is shown how to construct all representations for all q’s. The contraction of the Chevalley generators (e2,f2) leads to an easily obtained complete Hopf algebra. The structure of the contracted algebra is quite different, even at the classical level, from the familiar E(4), the four-dimensional Euclidean algebra, obtained on contracting the generator J45 in the GZ formalism. The contracted representations are given. The SO(5)q representations have their own interest. Moreover they lead to a consistent Hopf algebra and representations for the nonsemisimple contracted case, for all q.
The q-gamma, q-beta functions, and q-multiplication formula
Authors: Chung Won-Sang , Kang Hye-Jung
Published: July 2002

In this article the q-deformed gamma function and q-deformed beta function are discussed in detail. Three definitions for the q-gamma function are obtained and the q-deformed duplication and multiplication formulas are proven. The q-beta function is defined and its useful properties are derived.
A Fock space representation for the quantum Lorentz gas
Authors: Maassen H. , Tip A.
Published: July 2002

A Fock space representation is given for the quantum Lorentz gas, i.e., for random Schrödinger operators of the form H(ω)=p2+Vω=p2+∑ φ(x−xj(ω)), acting in ℋ=L2(Rd), with Poisson distributed xjs. An operator H is defined in 𝒦=ℋ⊗𝒫=ℋ⊗L2(Ω,P(dω))=L2(Ω,P(dω);ℋ) by the action of H(ω) on its fibers in a direct integral decomposition. The stationarity of the Poisson process allows a unitarily equivalent description in terms of a new family {H(k)‖k∈Rd}, where each H(k) acts in 𝒫 [A. Tip, J. Math. Phys. 35, 113 (1994)]. The space 𝒫 is then unitarily mapped upon the symmetric Fock space over L2(Rd,ρdx), with ρ the intensity of the Poisson process (the average number of points xj per unit volume; the scatterer density), and the equivalent of H(k) is determined. Averages now become vacuum expectation values and a further unitary transformation (removing ρ in ρdx) is made which leaves the former invariant. The resulting operator HF(k) has an interesting structure: On the nth Fock layer we encounter a single particle moving in the field of n scatterers and the randomness now appears in the coefficient √ρ in a coupling term connecting neighboring Fock layers. We also give a simple direct self-adjointness proof for HF(k), based upon Nelson’s commutator theorem. Restriction to a finite number of layers (a kind of low scatterer density approximation) still gives nontrivial results, as is demonstrated by considering an example.
Nonlinear motions against the Newtonian uniform expansion background: The case of the unperturbed density
Authors: Silbergleit A. S.
Published: July 2002

A class of nonlinear perturbations of the Newtonian uniform expansion background is studied, for which only the velocity field is assumed to be perturbed. The general closed form solution to the corresponding equations of motion is found, and several particular families of solutions are analyzed. All the flows of this class are proved to be essentially rotational. A major part of liquid particles constituting the flows have a nonzero angular momentum and move along parabolas, the rest move along straight lines. The analysis is carried out by means of both the Euler and Lagrange variables.
Unitary continuous representations of compact quantum groups
Authors: Drabant Bernhard , Weich Wolfgang
Published: July 2002

Generalizing the notion of continuous Hilbert space representations of compact topological groups we define unitary continuous corepresentations of C*-completions of compact quantum group Hopf algebras on arbitrary Hilbert spaces. It is proved that the unitary continuous corepresentations decompose in finite-dimensional irreducible corepresentations.
Exact matrix expansions for group elements of SU(N)
Authors: Kusnezov Dimitri
Published: July 2002

Some exact closed form expressions are presented for exponentials of matrices of su(n) in the fundamental representation. As a consequence, the same formulas apply to embedded subalgebras 𝒜⊆su(n) in higher dimensional representations. Explicit formulas are given for n<5, as well as applications to Skyrmions in the Callan–Klebanov bound state approach.
Duals of quasitriangular Hopf superalgebras and the classical limit
Authors: Pittner L. , Uray P.
Published: July 2002

The duality of quasitriangular Hopf superalgebras and superbialgebras with so-called main commutation relations (MCR), which is established by means of bilinear forms with values in a finite-dimensional representation of the universal R-matrix R, is used to calculate relations of quantum supercoordinates for several examples. The tensor product of representations of R, with respect to the comultiplication, then leads to the construction of an accordingly deformed tensor product of MCR-type superbialgebras. The correspondingly deformed tensor product of quantum vector superspaces is then established. In the classical limit of Lie superbialgebras one similarly obtains supercommutation relations, which may be understood, in case of quantum groups with deformation parameter q ≠ 1, as MCR in the first order of ln q. Again the deformed tensor product of such semiclassical MCR-type Lie superbialgebras is constructed.
On the initial value problem for the modified Benjamin–Ono equation
Authors: Scoufis George , Cosgrove Christopher M.
Published: July 2002

The initial value problem for the modified Benjamin–Ono equation is solved by exploiting the Fokas–Ablowitz inverse scattering transform for a certain complex extension of the Benjamin–Ono equation. The asymptotics of the Jost functions for small (positive) values of the spectral parameter are sharpened and lead to a previously overlooked nongeneric case.
Angular momentum and Killing potentials
Authors: Glass E. N.
Published: July 2002

When the Penrose–Goldberg (PG) superpotential is used to compute the angular momentum of an axial symmetry, the Killing potential Qμν(φ) for that symmetry is needed. Killing potentials used in the PG superpotential must satisfy Penrose’s equation. It is proved for the Schwarzschild and Kerr solutions that the Penrose equation does not admit a Qμν(φ) at finite r and therefore the PG superpotential can only be used to compute angular momentum asymptotically.