bluuebunny/crossref_metadata_2025 huggingface dataset:
Inhomogeneous boson realization of indecomposable representations of Lie algebras
Authors: Fu Hong-Chen , Sun Chang-Pu
Published: July 2002

By making use of the differential realization of Lie algebras in the space of inhomogeneous polynomials of a certain number of variables, the corresponding inhomogeneous boson realization of Lie algebras is given. A new kind of indecomposable representations of Lie algebras are studied on the universal enveloping algebra of Heisenberg–Weyl algebra, its subspaces and its quotient spaces. The finite-dimensional representations are naturally obtained on the subspaces of Fock space. As an example, the indecomposable and irreducible representations of the Lie algebra su(2) are discussed in detail.
On boson condensation considering a generalized Casimir example
Authors: Voigt K.
Published: July 2002

From the concept of generalized condensation [M. Van den Berg, J. T. Lewis, and J. V. Pulè, Helv. Phys. Acta 59, 1271 (1986)] it is known that two critical densities ρc and ρm exist for a free boson gas. Density ρc is the classical one and ρm is the critical density below which there can be no macroscopic occupation of ground state. A free boson gas is studied in a weak external potential which behaves asymptotically like ‖x1‖α1+‖x2‖α2+⋅⋅⋅ +‖xd‖αd near the origin. It is shown that there are only two possibilities to get ρc<ρm<∞, namely, α1=α2=∞ and d≥3 (this corresponds to Dirichlet boundary conditions), and α1=2 and d≥2 (i.e., a harmonic oscillator).
Unified prescription for the generation of electroweak and gravitational gauge field Lagrangian on a principal fiber bundle
Authors: Verwimp Theo
Published: July 2002

It is essentially known that useful gauge field Lagrangians arise as Weil polynomials of the curvature of the gauge connection. The deeper implications and details of this fact are worked out in two widely differing cases. The Glashow–Weinberg–Salam gauge field Lagrangian for electroweak theory and the Townsend–Zardecki action for gravitation are obtained from the same type of ‘‘Yang–Mills’’ Weil form on a principal fiber bundle over space-time, with symmetry group U(2) and SO(2,3), respectively. The unified geometrical approach given here shows that fiber bundle reduction and symmetry breaking are essential not only in electroweak theory but also in the SO(2,3) gauge theory for gravitation. In fact, the process of symmetry breaking in electroweak theory and the soldering of the anti-de Sitter bundle, essential in the interpretation of SO(2,3) gauge theory as a theory for gravitation, are corresponding geometrical concepts.
Conformal mapping and vertex operators in the light-cone gauge
Authors: Naito Seichi , Nishimoto Masahiro
Published: July 2002

Using Fourier coefficients of Neumann functions for the tree scattering in the light-cone gauge, an open-string field operator Ψp(X(x),{α−m}) generating all ‘‘off-shell’’ vertex operators, which are reduced in the mass-shell case to already proposed general vertex operators, is introduced.
On a conjecture about the invariance of the collision between solitary waves
Authors: Peng Zhang
Published: July 2002

For the invariance of the collision between solitary waves, some famous scientists conjectured: that if there exist infinitely many independent conserved quantities, then the solitary wave solutions of one nonlinear differential equation, should be solitons, in other words, after nonlinear interactions the shapes and velocities of solitary waves are identical to the original ones. This paper attempts to prove the above conjecture.
Compatibility conditions for Hamiltonian pairs
Authors: Cooke David B.
Published: July 2002

Conditions necessary for pairs of Hamiltonian operators of first through fifth order to form Hamiltonian pairs are computed. In the case of bi-Hamiltonian systems involving a Hamiltonian pair, one operator of which is a first-, third-, or a certain type of fifth-order operator, a technical condition in Magri’s Theorem can be dropped.
Metrics and dual operators
Authors: Harnett Gerald
Published: July 2002

For a four-dimensional vector space it is possible to define independently of a metric the notion of a dual operator acting on bivectors. It is shown that the map which takes a conformal class of metrics together with an orientation to the induced dual operator is an isomorphism. For each metric signature type, this map is equivalent to an isomorphism of certain homogeneous spaces.
Higher-order averaging for nonperiodic systems
Authors: Sáenz A. W.
Published: July 2002

In this paper, a recently developed improved averaging method for periodic nonlinear systems of ordinary differential equations (ODEs) is extended to a wide class of nonperiodic ODE systems of nonlinear type. The paper presents the first systematic treatment of Nth-order averaging for nonperiodic systems of the present degree of generality. Our main results are: (i) a basic existence, uniqueness, and approximation theorem for Nth-order averaging for the latter systems, including explicit, rigorous error bounds expressed in terms of suitable order functions; (ii) a proof that our (N+1)st error estimate is smaller (o(1)) than the corresponding Nth-order estimate when the relevant perturbation parameter ε is sufficiently small (this is not obvious, since our error bounds are not generally proportional to powers of ε); (iii) a proof that (ii) holds when the exact order functions are replaced by appropriate upper bounds which are simpler to calculate in practice. For the example of a linear oscillator with time-dependent friction, it is shown that our second-order averaging estimates are generally better than those given by the approach of Sanders and Verhulst. Nth-order averaging for a large class of C∞ quasiperiodic systems is also discussed, generalizing earlier work of Perko.
The pseudodipole as the limit of a resonant cylindrical dipole
Authors: Freeman D. Kent
Published: July 2002

It is shown that the pseudodipole can be derived as the limit of an inductively center-loaded cylindrical dipole, as the size of the dipole becomes small while the inductance becomes large. The two-term approximation is used for the cylindrical dipole current.
Nonlocal symmetries and recursion operator of the Landau–Lifshitz equation
Authors: van Bemmelen Theo , Kersten Paul
Published: July 2002

The Lie algebra of symmetries of the Landau–Lifshitz equation (LL) is discussed. Point, higher-order, and nonlocal symmetries of the LL are computed. The recursion operator is constructed from a nonlocal symmetry, reflecting the bi-Hamiltonian structure of the LL. The same recursion operator of the LL has been derived by Barouch, Fokas, and Papageorgiou [J. Math. Phys. 29, 2628 (1988)], starting from the Lax pair.
Asymptotics of determinants from functional integration
Authors: Wentworth Richard
Published: July 2002

The expression for the determinant of the Laplace operator is used in terms of functional integration to compute the asymptotic behavior on degenerating Riemann surfaces. In the case of the Arakelov metric, the information is sufficiently precise to give a value for the absolute constants appearing in bosonization formulas.
Generalized tops and moment maps to loop algebras
Authors: Harnad J. , Hurtubise J.
Published: July 2002

Using a combination of Hamiltonian reduction under continuous and discrete symmetry groups, the Lax equations for the motion of generalized tops are derived as part of a general approach to moment map embeddings of integrable Hamiltonian systems in loop algebras.
Energy eigenvalues for an arbitrary potential well with N minima
Authors: Dammert Örjan
Published: July 2002

For a general smooth one-dimensional potential well with N minima (containing N−1 barrier humps of arbitrary shape), a phase-integral quantization condition (including σ terms) is obtained, from which the energy eigenvalues can be calculated with an accuracy relative to the chosen order of the Fröman phase-integral approximation. The quantization equation is derived from a condition for total transmission through a general system of potential barriers of arbitrary shape, by means of a limiting procedure, in which the two outermost barriers are made to grow large in a particular way. Finally, the quantization condition is expressed by means of a concise recursion formula. As an illustration, the special case N=2 is treated in some detail. For N=2, the quantization condition turns out to be identical to a previously well-known phase-integral quantization condition for the double-minimum potential, the accuracy of which has been examined in several papers by other authors. Their results concerning a symmetric double-minimum model potential show that when the σ term is included, the third- and fifth-order energy eigenvalues are accurate enough for all practical purposes, even close to the barrier maximum, but if the σ term is neglected, the quantization condition breaks down near the barrier maximum, this breakdown becoming more dramatic with increasing order. A table displaying phase-integral eigenvalues for the model potential as compared with exact (numerical) quantum mechanical results is given here; the figures are quoted from the papers mentioned. Applying the results to the special case N=3, it will be given explicitly the quantization condition for an arbitrary triple-minimum potential.
Spin-plane defects and emergence of Planck’s constant in gravity
Authors: Magnon Anne M. R.
Published: July 2002

The possibility of building Planck’s constant into space-time geometry in presence of topological excitations is considered. This is accomplished on the ground that such space-times display a spin two-plane and spin vector which incorporate the presence of space-time torsion and dislocations via global topological defects. This, in turn, is related to the nonvanishing of specific cohomological classes labeled by an integer and carrying topological charges that can be related to a multiple of the Planck length, thus inducing quantization of the spin. Such a geometrization of ℏ could shed light on quantum gravity and be compared to the geometrization of c (the other fundamental constant) by special relativity.
Quantum algebra deforming maps, Clebsch–Gordan coefficients, coproducts, R and U matrices
Authors: Curtright T. L. , Ghandour G. I. , Zachos C. K.
Published: July 2002

Quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts. Map-induced coproducts are connected to the usual ones by similarity transformations U that may be expressed either in terms of Clebsch–Gordan coefficients, or in a universal operator form. The product of two such U matrices yields the R matrix for a fixed value of the spectral parameter, which bears on the Yang–Baxterization of U as well as R. All this is explicitly illustrated for the tensor product 1/2⊗j of SU(2)q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a two-parameter manifold. Some observations are made on the general structure of U and R matrices, and of coproduct manifolds, based on the solutions of hierarchies of partial difference equations. Applications of deforming maps and U matrices to the physics of spin-chains are outlined.
The Dirac equation in external fields: Variable separation in Cartesian coordinates
Authors: Shishkin German V. , Cabos William D.
Published: July 2002

The method of separation of variables in the Dirac equation proposed in an earlier work by one of the present authors [J. Math. Phys. 30, 2132 (1989)] is developed for the complete set of interactions of the Dirac particle. The essence of the method consists of the separation of the first-order matrix differential operators that define the dependence of the Dirac bispinor on the related variables, but commutation of such operators with or between the operator of the equation is not assumed. This approach, which is perfectly justified in the presence of gravitational [Theor. Math. Phys. 70, 204 (1987)] or vector fields [J. Math. Phys. 30, 2132 (1989)], permits one to find all the possibilities of separation of variables in the Dirac equation in the case of the most general set of external fields. The complete set of interactions of the Dirac particle is determined by the symmetry group of equations, namely, viz. the SU(4) group. The interactions are scalar, vector, tensor, pseudovector and pseudoscalar. The analysis in this article is limited to Cartesian coordinates. The corresponding results for the general curvilinear coordinates will be presented in a future paper.
Symmetry and quantization: Higher-order polarization and anomalies
Authors: Aldaya V. , Navarro-Salas J. , Bisquert J. , Loll R.
Published: July 2002

The concept of (full) polarization subalgebra in a Group Approach to Quantization on a Lie group G̃ as a generalization of the analogous concept in geometric or standard quantization is discussed. The lack of full polarization subalgebras is considered as an anomaly of the corresponding system and related to its more conventional definition. A generalization of the subalgebra of (full) polarization is then provided, made out of higher-order differential operators in the enveloping algebra of G̃. Higher-order polarizations can also be used to quantize nonanomalous theories in different ‘‘representations.’’ Numerous examples are analyzed, including the finite-dimensional dynamics associated with the Schrödinger group, which presents an anomaly, and an infinite-dimensional anomalous system associated with the Virasoro group. In the last example, the operators in the higher-order polarization are in one-to-one correspondence with the null vectors in the Verma module approach.
On the Coulomb singular kernel of Lippmann–Schwinger-type equation
Authors: Latypov D. M. , Mukhamedzhanov A. M.
Published: July 2002

A Lippmann–Schwinger-type equation with Coulomb singular kernel is considered. It is shown that all its solutions are singular on the energy shell, k2/2μ=E. In order for this equation to have a solution with a singularity of the type (E−k2/2μ+iε)iα, α is shown to be equal to the Coulomb parameter η. The Coulomb singular kernel in the given class of functions is found to split into a δ function and a kernel which smoothes the singularity.
Algebraic structures of the quantum projective [sl(2,C)-invariant] field theory: The commutative version of Elie Cartan’s exterior differential calculus
Authors: Juriev D.
Published: July 2002

The commutative exterior differential calculus, appearing in the quantum projective field theory, is described.
Bi-Hamiltonian structure of super KP hierarchy
Authors: Yu Feng
Published: July 2002

The bi-Hamiltonian structure of the super KP hierarchy is obtained based on the even super KP operator Λ=θ2+∑i=−2∞Uiθ−i−1, as a supersymmetric extension of the ordinary KP bi-Hamiltonian structure. It is expected to give rise to a universal super W-algebra incorporating all known extended superconformal WN algebras by reduction. Also constructed is the super BKP hierarchy by imposing a set of anti-self-dual constraints on the super KP hierarchy.
Racah–Wigner calculus for the super-rotation algebra. I
Authors: Minnaert Pierre , Mozrzymas Marek
Published: July 2002

The symmetry properties and the pseudo-orthogonality relations of the super-rotation Clebsh–Gordan coefficients for the tensor product of two irreducible representations of the super-rotation algebra are derived. The symmetric super-rotation 3-j symbol and the symmetric and invariant super-rotation 6-j symbol are defined, their basic properties are described, and their relations to the usual 3-j and 6-j symbols are given.
The Stokes constants associated with the differential equation d2ψ/dz2−4A2z
n−2(z
n+c)ψ=0
Authors: Dzieciol Aleksander
Published: July 2002

The solutions of the differential equation d2ψ/dz2−4A2zn−2(zn+c)ψ=0 are expressed in terms of convenient integral representations of the Laplace type leading to asymptotic expressions that are rewritten as linear combinations of phase-integral functions. Formulas for the Stokes constants in the Fröman–Fröman phase-integral method are then derived. For the first- and the third-order approximations, explicit formulas are given, and the limiting case of large absolute values of Ac in particular is considered. A discussion of numerical results is also given.
Dirac equation in external fields: Separation of variables in curvilinear coordinates
Authors: Shishkin German V. , Cabos William D.
Published: July 2002

The algebraic method of separation of variables in the Dirac equation proposed by one of the present authors [Gravitation and Electromagnetism (U.P., Minsk, 1989) Issue 4, p. 156 (in Russian)] is developed for the case of the most general interaction of the Dirac particle in an external field, taking into account scalar, vector, tensor, pseudovector, pseudoscalar, and gravitation connections. The present work, which concludes this series of papers entitled ‘‘Dirac equation in external fields’’ [J. Math. Phys. 32, 3184 (1991); 33, XXX (1992)] is dedicated to the investigation of the problem in the case of general orthogonal curvilinear coordinates, and allows the introduction of gravitation through the spinor connection and Lame’s functions. Special consideration is given to the cases, when the generalized Lame’s functions do not separate the variables multiplicatively (e.g., elliptic cylindrical, parabolic cylindrical, and oblate and prolate spheroidal coordinates). All the previous results and numerous results of other authors are particular cases of this investigation.
Relativistic tautochrone
Authors: Kamath S. G.
Published: July 2002

The path joining two points A and B, which a particle falling from rest in a uniform gravitational field must adopt, so that the time of transit from A to B is independent of the location of A is called the tautochrone. In the nonrelativistic case, the path is known to be a cycloid, a standard method associated with the derivation of this result being, for example, the method of Laplace transforms. For the relativistic case that is studied herein, the methods of fractional calculus are shown to be more useful in the derivation of the exact relativistic tautochrone. This latter derivation is then checked out by the Laplace transform approach for the relativistic problem, from the point of view of consistency. Using the same method, the relativistic tautochrone associated with a charged particle of charge q and mass m falling from rest in an uniform electric field is also worked out. The tautochrone turns out to be an incomplete elliptic function of the second kind E(δ,r). As an application, the power radiated by the charged particle as it accelerates along the curve is then computed; it is found to be proportional to (1 − v2/c2)−2, with v(c) the velocity of the particle (light). Finally, an appendix highlights the utility of fractional calculus vis-á-vis the approach of Abel for the relativistic tautochrone.
Topological aspects of a fermion, chiral anomaly, and Berry phase
Authors: Banerjee Dipti , Bandyopadhyay Pratul
Published: July 2002

The relationship of the Berry phase with the topological aspects of a fermion and chiral anomaly is studied here. It is observed that this phase reflects the effect of quantum geometry when the quantization procedure of a Fermi field is considered from a classical system and may be considered as the manifestation of the multiply connected nature of quantum space-time.
Reduced Wigner coefficients for U
q[gl(n)]
Authors: Gould M. D.
Published: July 2002

This is the second paper in a series concerned with the Racah–Wigner calculus for the quantum group Uq[gl(n)]. In this approach all reduced Wigner coefficients are determined algebraically by the eigenvalues of certain generalized Casimir invariants. Here the method is applied to evaluate all fundamental Wigner coefficients and reduced Wigner coefficients for Uq[gl(n)].
Minimal coupling of electromagnetic fields in Riemann–Cartan space-times for perfect fluids with spin density
Authors: Smalley Larry L. , Krisch Jean P.
Published: July 2002

The electromagnetic field is minimally coupled to gravity in a Riemann–Cartan space-time containing a charged magnetized spinning fluid. It is required that the overall Lagrangian of the gravitational field, spinning matter, and the electromagnetic field be invariant under a gauge transformation of the vector potential. The theory preserves both charge conservation and particle number conservation. The electromagnetic field, via the vector potential, now interacts directly with the spin energy momentum. The spin transport equation, in addition to the usual Fermi–Walker transport term, contains a contribution due to the torque of the electromagnetic field acting on a magnetic dipole. In the absence of electromagnetism, the field equations reduce to those of the usual self-consistent Lagrangian formalism for a perfect fluid with spin density.
Bessel function expressions for Dirac fields
Authors: Cardoso J. G.
Published: July 2002

Penrose’s null initial data techniques are used to obtain a solution to the electromagnetism-free Dirac equation, involving Bessel functions. Each Dirac field is split up into an infinite number of elementary contributions whose densities satisfy Dirac-like distributional field density equations on the interior of the future cone of an origin of real Minkowski space. The initial data hypersurface for all the elementary fields is conveniently chosen to be the future null cone of the origin. It is observed that the field density equations are invariant under proper Lorentz transformations. The entire fields are then expressed as explicit scaling invariant convolutions taken over the initial data cone.
A Complete set of solutions to the massive Dirac equation in static open Robertson–Walker space
Authors: Kovalyov M. , Légaré M.
Published: July 2002

A sketch of the proof of completeness of a set of nonzero mass solutions to the Dirac equation in static open Robertson–Walker space is given.
In search of evolution equations in quantum and statistical field theory
Authors: Hańćkowiak J.
Published: July 2002

Using an enlarged space of n-point functions F and fictitious time s linear problems are reformulated in quantum and statistical field theory in such a way that non-negative sesquilinear forms play a crucial role in the description of system. First- and second-order ‘‘evolution’’ type equations are considered and their connection with weak and strong convergence is shown. Supersymmetrical ‘‘evolution’’ of systems is presented without increasing the number of fields. The five-dimensional field theory was found to be useful in developing the formalism.
Novel symmetry of non-Einsteinian gravity in two dimensions
Authors: Grosse Harald , Kummer Wolfgang , Prešnajder Peter , Schwarz Dominik J.
Published: July 2002

The integrability of R2-gravity with torsion in two dimensions is traced to an ultralocal dynamical symmetry of constraints and momenta in Hamiltonian phase space. It may be interpreted as a quadratically deformed iso(2,1)-algebra with the deformation consisting of the Casimir operators of the undeformed algebra. The locally conserved quantity encountered in the explicit solution is identified as an element of the center of this algebra. Specific contractions of the algebra are related to specific limits of the explicit solutions of this model.
The universal propagator for affine [or SU(1,1)] coherent states
Authors: Klauder John R. , Tomé Wolfgang A.
Published: July 2002

It is shown that it is possible to introduce a universal propagator for a general Hamiltonian appropriate to a single affine degree of freedom. This universal propagator is a single function, independent of any particular choice of fiducial vector, which, nonetheless, propagates the coherent state Hilbert space representatives correctly. Furthermore, the universal propagator for several examples is explicitly constructed.
Covariant phase space formulation of parametrized field theories
Authors: Torre C. G.
Published: July 2002

Parametrized field theories, which are generally covariant versions of ordinary field theories, are studied from the point of view of the covariant phase space: the space of solutions of the field equations equipped with a canonical (pre)symplectic structure. Motivated by issues arising in general relativity, we focus on phase space representations of the space-time diffeomorphism group, construction of observables, and the relationship between the canonical and covariant phase spaces.
White-noise calculation of annihilation and creation operators in stochastic mechanics
Authors: Fukumi Toshio
Published: July 2002

A second quantization procedure is formulated in the framework of Nelson’s stochastic mechanics. It is shown that the annihilation operator is given by an integration over a Gaussian white noise Ḃ(s) with the kernel exp(iω(t−s)) and the creation operator is given by an integration over a time reversed Gaussian white noise Ḃ*(s) with the kernel exp(−iω(t−s)). The eigenfunctions on which the operators act are shown to be Gaussian random variables whose collection is dense in a Hilbert space. Employing the above operators, we can construct {N}-representation in the stochastic mechanics.
Rigorous derivation of Planck’s law in the thermodynamic limit
Authors: Hertle Jochen , Honegger Reinhard
Published: July 2002

The limiting Gibbs state and dynamics for free thermal photons was investigated in a previous work by means of an operator algebraic approach. For the one-photon Hamiltonian in a local region, the square root of the Laplacian with Dirichlet boundary conditions was used. In the present work, the energy density distribution for thermal photons (Planck’s law) in a nearly arbitrary physical cavity is derived in the thermodynamic limit. For the cavity, only the segment property and a zero-set boundary is assumed, which is the weakest possible requirement at its border. For the thermodynamic limit, the cavity is monotonously dilated from a fixed interior point. In this sense, the work is also a contribution to Weyl’s problem, the asymptotic distribution of the eigenvalues of the wave equation in the infinite space approximation. The calculation uses comparisons of functions of the Laplacians defined in different regions of the Euclidean space by means of the ordering of positivity preserving operators.
Reflection coefficient beyond all orders for singular problems. III. Essential singularities on the real axis
Authors: Hu Jishan
Published: July 2002

In this paper uses the generalized Bremmer series to calculate the nontrivial behavior of the reflection coefficient for singular problems, which have isolated critical points and essential singularities on the real axis. The generalized Pokrovskii–Khalatnikov is not applicable in this case.
Representations of commutations relations for p-adic systems of infinitely many degrees of freedom
Authors: Zelenov E. I.
Published: July 2002

The standard C*-formalism of representations of commutations relations for systems of infinitely many degrees of freedom over a real numbers field is extended to the case of a p-adic numbers field. It is proved that any positive function on the Weyl algebra over a p-adic symplectic space is regular and restriction of the generating function of a state of the Weyl algebra on any nongenerate finite-dimensional subspace of p-adic symplectic space is continuous. Some nontrivial class of nonequivalent representations of commutations relations is described and analogies of Fock representations and representations with number operators are constructed. The criteria for the existence of free-field unitary quantization of symplectic transformations is given.
Finiteness of the discrete spectrum of (N≥3) N-body Schrödinger operators, which have some determinate subsystems that are virtual at the bottom of the continuum
Authors: Ahia Francis
Published: July 2002

A geometric proof is given in the spirit of Sigal [Lectures on the Quantum Mechanical Many-Body Problem, Toronto, Spring, 1987] for Vugal’ter’s and Zhislin’s theorem in Transactions of the Moscow Mathematical Society, 1987, pp. 97–114. This is a different proof.
Schrödinger–Fuerth quantum diffusion theory: Generalized complex diffusion equation
Authors: Kostin M. D.
Published: July 2002

According to Schrödinger–Fuerth quantum diffusion theory the equations of quantum theory can be derived by applying the Fuerth transformation to the equations of diffusion theory. By introducing a complex flux whose real and imaginary parts are the x and y components of the flux, a complex form of the two-dimensional diffusion equation is obtained and the equation is then generalized to three dimensions. Application of the Fuerth transformation to the generalized complex diffusion equation yields a quantum equation, which is shown to be equivalent to the Dirac equation.
On the instability of the n=1 Einstein–Yang–Mills black holes and mathematically related systems
Authors: Wald Robert M.
Published: July 2002

The usual approach to analyze the linear stability of a static solution of some system of equations consists of searching for linearized solutions which satisfy suitable boundary conditions spatially and which grow exponentially in time. In the case of the n=1 Einstein–Yang–Mills (EYM) black hole, an interesting situation occurs. There exists a perturbation which grows exponentially in time−and spatially decreases to zero at the horizon−but nevertheless is physically singular on the horizon. Thus, this unstable mode is unacceptable as initial data, and the question arises as to whether the n=1 EYM black hole is stable. We analyze this issue here in the more general case of a scalar field φ satisfying the wave equation ∂2φ/∂t2 = (DaDa − V)φ on a manifold R×M, where Da is the derivative operator associated with a complete Riemannian metric on M and V is a bounded function on M whose derivatives also are bounded. We prove that if the operator A = −DaDa + V fails to be a strictly positive operator on the Hilbert space L2(M), then there exists smooth initial data of compact support in M which give rise to a solution which grows unboundedly with time. This implies that the n=1 EYM black hole and other mathematically similar systems are unstable despite the nonexistence of physically acceptable exponentially growing modes. Rigorous criteria for linear stability are also obtained.
An application of first integrals method in magnetic diffusion problem
Authors: Icha Andrzej
Published: July 2002

In this paper a rigorous approach to the magnetic diffusion problem is presented under the assumption that the random fluid velocity field is independent of x and completely determined by a prescribed probability measure on the space of symmetric velocity field u(t)=u(−t). By adopting the first integrals method developed by Visik and Fursikov [Mat. Sb. 92, 347 (1973)], the space characteristic functional associated with the random induction equation has been expressed by a functional power series. The explicit formula for this functional has been obtained, without deriving and solving a suitable functional equation for the [u, B] field.
Matrix elements and Wigner coefficients for U
q[gl(n)]
Authors: Gould M. D. , Links J. , Bracken A. J.
Published: July 2002

In this series of papers the fundamental Wigner coefficients and the matrix elements of the generators for the quantum group Uq[gl(n)] are derived. The Uq[gl(n)]:Uq[gl(n−1)] reduced Wigner coefficients and reduced matrix elements are determined algebraically as eigenvalues of certain Uq[gl(n−1)] invariants in the quantum group Uq[gl(n)]. The matrix elements of the elementary quantum group generators, in the Gel’fand–Tsetlin basis, are derived below, while all fundamental Wigner coefficients are derived in the second paper of the series.
Vertex operators and generating functional for one loop scattering amplitudes
Authors: Naito Seichi
Published: July 2002

With the help of a q-number conformal mapping whose transformation function depends on a covariantly quantized string coordinate component X+(z), a generating functional for all physical vertex operators is constructed. Using these obtained vertex operators, a generating functional for one-loop scattering amplitudes among arbitrarily excited physical states is derived.
Canonical structure in SO(4)q and relations to E(3)q and SO(3,1)q
Authors: Chakrabarti A.
Published: July 2002

Starting with the SU(2)q⊗SU(2)q basis and using q-CG coefficients the closest possible q deformation of the classical canonical Gelfand–Zetlin formalism is constructed explicitly for SO(4)q. Simple introduction of q brackets for each factor of the classical matrix elements, valid for SU(n)q, is shown to need significant and unexpected modifications for SO(n)q even when n=4. Contraction to E(3)q and continuation to SO(3,1)q are studied, making explicit the possibilities and the problems that arise. Periodic representations for q, a root of unity, are commented upon briefly.
The nonsingular spiked harmonic oscillator
Authors: Estévez-Bretón Erika Sabrina , Estévez-Bretón Gentil A.
Published: July 2002

Equation (14) of the paper by Aguilera-Navarro and Guardiola on the nonsingular spiked harmonic oscillator [J. Math. Phys. 32, 2135 (1991)] appears not to be applicable when α=2. By carrying out simple transformations on this equation it is shown that it can be evaluated in closed form in the limiting case α→2.
On an old article of Tzitzeica and the inverse scattering method
Authors: Boldin A. Yu. , Safin S. S. , Sharipov R. A.
Published: July 2002

Tzitzeica’s class of surfaces in R3 is considered in connection with the inverse scattering method for the associated equation uxy=eu−e−2u. The Bäcklund transformation for this equation is derived from Tzitzeica’s Darboux transformation for it.
Hierarchy of lower Korteweg–de Vries equations and supersymmetry structure of Miura transformations
Authors: Andreev V. A. , Shmakova M. V.
Published: July 2002

The infinite hierarchy of lower Korteweg–de Vries (KdV) equations which corresponds to negative degrees of Olver’s operator is constructed. Basic forms of these equations are found. The supersymmetry structure of the Miura’s transformations is considered.
Compilation of two-point and four-point graphs in field theory in noninteger dimensions
Authors: Holovatch Yu. , Krokhmal’s’kii T.
Published: July 2002

Two-point and four-point graphs appearing in a three-loop approximation in the field theory renormalization group scheme are calculated in general dimensions. Combining Feynman parameterization and direct integration loop integrals are represented in the form of the expressions, depending on the space dimension d as a parameter. The expressions obtained enables one to consider renormalization group functions directly at noninteger d.
Qualitative analysis of diagonal Bianchi type V imperfect fluid cosmological models
Authors: Coley A. A. , Hoogen R. J. van den
Published: July 2002

The Einstein field equations for diagonal Bianchi type V imperfect fluid cosmological models with both viscosity and heat conduction are set up as an autonomous system of differential equations using dimensionless variables and a set of dimensionless equations of state. Models with and without a cosmological constant, λ, are investigated using the techniques from dynamical systems theory. It is shown that all models that satisfy the weak energy conditions isotropize. The introduction of viscosity (in particular) allows for a variety of different qualitative behaviors (including, for example, models with a negative deceleration parameter). Exact solutions that correspond to the singular points of the dynamical system are found. It is shown that the past asymptotic states are represented by self-similar cosmological models and, if λ=0, the future asymptotic states are also, in general, represented by self-similar cosmological models; in the exceptional cases the late time asymptotic state is represented by a de Sitter model with constant expansion, as is the case for solutions with λ ≠ 0.
A composite Fourier-wavelet transform and square-integrable group representations
Authors: Ali S. Twareque , Denisov L. V.
Published: July 2002

A formula for a composite Fourier-wavelet transform is suggested, from which the noncommutative Fourier and wavelet transforms can be obtained as particular examples, as well as several other Fourier-type transforms. It is shown how this composite transform can be used to extend certain results, originally obtained for square-integrable group representations, to more general measures, arising in quantum probability theory.
Feynman integrals and Fredholm determinants
Authors: Rezende Jorge
Published: July 2002

In this article simple properties of Feynman integrals such as the translation invariance, Fubini theorem, and Cameron–Martin formula are used in order to derive a product formula for Fredholm determinants. This formula enables us to compute explicitly some of such determinants as well as the index of the related Fredholm operators. Some examples are done which have applications in the semiclassical approximation (ℏ → 0).
Multiplane gravitational lensing. I. Morse theory and image counting
Authors: Petters A. O.
Published: July 2002

The image counting problem for gravitational lensing by general matter deflectors distributed over finitely many lens planes is considered. Counting formulas and lower bounds are found via Morse theory for the number of images of a point source not on a caustic. Images are counted within a compact region D not necessarily assumed to properly contain the deflector space. In addition, it is shown that Morse theory is applicable because multiplane time-delay maps Ty generically satisfy the Morse boundary conditions relative to D. All results obtained depend only on the topological properties induced in the lens planes by the deflector potentials and the behavior of grad Ty at boundary points of D.
On the new approach to variable separation in the time-dependent Schrödinger equation with two space dimensions
Authors: Zhdanov R. Z. , Revenko I. V. , Fushchych W. I.
Published: July 2002

We suggest an effective approach to separation of variables in the Schrödinger equation with two space variables. Using it we classify inequivalent potentials V(x1,x2) such that the corresponding Schrödinger equations admit separation of variables. Besides that, we carry out separation of variables in the Schrödinger equation with the anisotropic harmonic oscillator potential V=k1x21+k2x22 and obtain a complete list of coordinate systems providing its separability. Most of these coordinate systems depend essentially on the form of the potential and do not provide separation of variables in the free Schrödinger equation (V=0).
Collapsing regions and black hole formation
Authors: Burnett Gregory A.
Published: July 2002

Up to a conjecture in Riemannian geometry, we significantly strengthen a recent theorem of Eardley by proving that a compact region in an initial data surface that is collapsing sufficiently fast in comparison to its surface-to-volume ratio must contain a future trapped region. In addition to establishing this stronger result, the geometrical argument used does not require any asymptotic or energy conditions on the initial data. It follows that if such a region can be found in an asymptotically flat Cauchy surface of a spacetime satisfying the null-convergence condition, the spacetime must contain a black hole with the future trapped region therein. Further, up to another conjecture, we prove a strengthened version of our theorem by arguing that if a certain function (defined on the collection of compact subsets of the initial data surface that are themselves three-dimensional manifolds with boundary) is not strictly positive, then the initial data surface must contain a future trapped region. As a byproduct of this work, we offer a slightly generalized notion of a future trapped region as well as a new proof that future trapped regions lie within the black hole region.
Topological sectors and measures on moduli space in quantum Yang–Mills on a Riemann surface
Authors: Fine Dana Stanley
Published: July 2002

Previous path integral treatments of Yang–Mills on a Riemann surface automatically sum over principal fiber bundles of all possible topological types in computing quantum expectations. This paper extends the path integral formulation to treat separately each topological sector. The formulation is sufficiently explicit to calculate Wilson line expectations exactly. Further, it suggests two new measures on the moduli space of flat connections, one of which proves to agree with the small-volume limit of the Yang–Mills measure.
The doublet representation of non-Hilbert eigenstates of the Hamiltonian
Authors: Castagnino M. , Domenech G. , Levinas M. , Umérez N.
Published: July 2002

We find the minimal mathematical structure to represent quantum eigenstates with complex eigenvalues with no need of analytic continuation. These eigenvectors build doublets in non-Hilbert spaces. We construct exact solutions for the Friedrichs model that continuously join the ones of the free Hamiltonian. We extend the Wigner operator to these non-Hilbert spaces and enlarge the concept of normalized vectors via the definition of the doublets. Making use of these doublets, we describe systems whose states have initial conditions out of Hilbert space.
Bi-Hamiltonian structures of the coupled AKNS hierarchy and the coupled Yajima–Oikawa hierarchy
Authors: Liu Q. P.
Published: July 2002

The Hamiltonian theory for the two-component AKNS hierarchy and Yajima–Oikawa hierarchy is considered from the viewpoint of reduction. We show that the second Hamiltonian structures of the former is a Dirac reduction of the sl(3) current algebra, while the latter is related to the classical W(3)4 algebra.
Proper time and path integral representations for the commutation function
Authors: Gavrilov S. P. , Gitman D. M.
Published: July 2002

On the example of the quantized spinor field, interacting with arbitrary external electromagnetic field, the commutation function is studied. It is shown that a proper time representation is available in any dimensions. Using it, all the light cone singularities of the function are found explicitly, generalizing the Fock formula in four dimensions, and a path integral representation is constructed.
The Einstein action for algebras of matrix valued functions—Toy models
Authors: Hajac Piotr M.
Published: July 2002

Two toy models are considered within the framework of noncommutative differential geometry. In the first one, the Einstein action of the Levi–Civita connection is computed for the algebra of matrix valued functions on a torus. It is shown that, assuming some constraints on the metric, this action splits into a classical-like, a quantum-like and a mixed term. In the second model, an analogue of the Palatini method of variation is applied to obtain critical points of the Einstein action functional for M4(R). It is pointed out that a solution to the Palatini variational problem is not necessarily a Levi–Civita connection. In this model, no additional assumptions regarding metrics are made.
Differential calculi on quantum Minkowski space
Authors: Gonera Cezary , Kosiński Piotr , Maślanka Pawel
Published: July 2002

The differential calculus on n-dimensional (n≥3) quantum Minkowski space covariant with respect to left action of κ-Poincaré group is constructed and its uniqueness is shown.
Solving nonlinear recursions
Authors: Rabinovich S. , Berkolaiko G. , Havlin S.
Published: July 2002

A general method to map a polynomial recursion on a matrix linear one is suggested. The solution of the recursion is represented as a product of a matrix multiplied by the vector of initial values. This matrix is product of transfer matrices whose elements depend only on the polynomial and not on the initial conditions. The method is valid for systems of polynomial recursions and for polynomial recursions of arbitrary order. The only restriction on these recurrent relations is that the highest-order term can be written in explicit form as a function of the lower-order terms (existence of a normal form). A continuous analog of this method is described as well.
Towards the canonical tensor operators of u
q(3). I. The maximal null space case
Authors: Ališauskas Sigitas
Published: July 2002

Generalizing the SU(3) canonical tensor operator concept (Biedenharn and Louck) to the quantum algebra uq(3), the Wigner–Clebsch–Gordan coefficients of uq(3) with repeating irreducible representations are considered. Extremal projectors of the quantum algebra uq(3) in terms of the ordered generator polynomials are used for evaluation of the bilinear combinations of the uq(3) canonical isoscalar factors. Explicit expressions of the uq(3) isofactors, corresponding to the maximal null space case of the uq(3) unit canonical tensor operators, and their normalization factors (denominator functions) are presented. The transposition and conjugation phase factors for the SU(3) and uq(3) canonical isofactors are correlated with phases and zeros of boundary isofactors. Invariance of the canonical isofactors (or absence of such invariance) under interchange of the tensor operator and the initial or final state parameters is correlated with the existence and invariance (or numerical degeneracy) of the usual splitting (distinctive) conditions. Some oversights of previous publications are disclosed.
(T*ℬ)q, q-analog of model space and the Clebsch–Gordan coefficients generating matrices
Authors: Bytsko A. G. , Faddeev L. D.
Published: July 2002

We study relations between the deformed cotangent bundle (T*ℬ)q for the Borel subgroup ℬ of a given simple Lie group G, the quantum Lie algebra 𝒥q associated with the corresponding quantum group Gq and the matrices generating Clebsch–Gordan coefficients for 𝒥q. We reveal the connection of these objects to quantum analog of the model space ℳ and q-tensor operators.
Novel generalization of three-dimensional Yang–Mills theory
Authors: Anco Stephen C.
Published: July 2002

A class of new non-Abelian gauge theories for vector fields on three manifolds is presented. The theories describe a generalization of three-dimensional Yang–Mills theory featuring a novel nonlinear gauge symmetry and field equations for Lie-algebra-valued vector potential fields. The nonlinear form of the gauge symmetry and field equations relies on the vector cross-product and vector curl operator available only in three dimensions, and makes use of an auxiliary Lie bracket together with the Lie bracket used in Yang–Mills theory. A gauge covariant formulation of the new theories is given which utilizes the covariant derivative and curvature from the geometrical formulation of Yang–Mills theory. Further features of the new theories are discussed.
Linear canonical transformations in quantum mechanics
Authors: Bordner Andrew J.
Published: July 2002

We find explicit unitary operators that implement linear canonical transformations of the quantum mechanical operators for a system with N degrees of freedom. We then relate the operators effecting this transformation to the previous formulation of quantum canonical transformations in terms of an effective generating function introduced by Ghandour.
Nonconducting electromagnetic media with rotational invariance: Transition operators and Green’s functions
Authors: Tip A.
Published: July 2002

For an accurate description of multiple scattering of electromagnetic waves, such as in random dielectrics or dielectric lattices, the full off-shell single scatterer transition (T-) operator is required. Here we study this quantity for the case of a non-conducting medium with rotational invariant permeabilities. We start with the scattering problem where rotational invariance is not yet assumed and find a class of mutually different T-operators which all have the same on-shell restriction. Next we extend the usual method of expressing scattered fields for rotational invariant systems in terms of the solutions of two scalar wave equations to the resolvent (Green’s function) associated with the vector wave equation. We find that it can be expressed in terms of the resolvents of two scalar operators. Finally we turn to the Mie case (a dielectric sphere in vacuum) for which we obtain explicit expressions for the corresponding Green’s functions and the general off-shell T-matrix elements.