bluuebunny/crossref_metadata_2025 huggingface dataset:
A gauge symmetric approach to Fierz identities
Authors: Reifler Frank , Morris Randall
Published: July 2002

Seventy-five years ago Cartan invented spinors by mapping C2 onto isotropic (null) vectors in C3. In recent work this map was extended and it was shown that bispinors are isomorphic to a class of Yang–Mills vector triplets Fk =Ek +iHk which satisfy the following SU(2)×U(1) gauge invariant constraint: Fj ⋅ Fk =s2 δjk, where s2 = (1)/(3) Fk ⋅ Fk (k summed from 1 to 3). Thus bispinors have inherent SU(2) ×U(1) gauge symmetry. In this paper it is shown, using the extended Cartan map and the gauge symmetry of the constrained Yang–Mills fields, that all the Fierz identities reduce to a single equation. Moreover, this equation includes not only the 75 identities recently derived by Takahashi [Y. Takahashi, J. Math. Phys. 24, 1783 (1983)] but an additional 75 which come from interchanging gauge and vector components. It is further shown that the Fierz identities for bispinors can be generalized to any multiplet, Ψ∈C2n, consisting of 2n−1 spinors (n=1 for spinors, n=2 for bispinors, n=3 for bispinor doublets, etc.). The generalized identities can also be used to show that the 2n−1 spinor multiplets are isomorphic to multiplets of constrained Yang–Mills vector fields.
Some remarks on the nonlinear integral equation in Kirkpatrick’s theory of glass transition
Authors: Yoshimura Tetz
Published: July 2002

The nonlinear singular integral equation for ‘‘self-energy’’ Σ(k,z) arising in Kirkpatrick’s mode-coupling theory of glass transition is analyzed without suppressing the k dependence. An equation that is equivalent to Kirkpatrick’s equation and suitable for high density is set up. Applicability of Lika’s generalization of the Newton–Kantorovich successive approximation is discussed. The possibility of solutions that cannot be found by iteration is pointed out.
Phase-integral formulas for Bessel functions and their relation to already existing asymptotic formulas
Authors: Fröman Per Olof , Karlsson Finn , Yngve Staffan
Published: July 2002

The phase-integral method devised by Fröman and Fröman [N. Fröman and P. O. Fröman, JWKB Approximation, Contributions to the Theory (North-Holland, Amsterdam, 1965); Ann. Phys. (NY) 83, 103 (1974); Nuovo Cimento B 20, 121 (1974); N. Fröman, Ark. Fys. 32, 541 (1966); Ann. Phys. (NY) 61, 451 (1970)], involving a general phase-integral approximation of arbitrary order, which is generated from an unspecified base function, is used for deriving first- and higher-order phase-integral formulas for Bessel functions. For different choices of the base function one thus obtains in a systematic way different kinds of asymptotic formulas. By series expansion of these formulas one obtains already existing asymptotic formulas presented is standard handbooks. The phase-integral formulas are seen to have certain advantages that those latter formulas do not possess.
The Schwinger–DeWitt proper-time method for odd-dimensional gauge theory
Authors: Lee Taejin
Published: July 2002

The Schwinger–DeWitt proper-time method (WKB expansion) is applied to calculate the anomaly in odd-dimensional gauge theories. The parity violating part of effective action for gauge theory in odd dimensions with massless fermion is calculated explicitly and efficiently by this method. It is shown to be precisely the local Chern–Simons term.
The 38 assemblies of the general generator sets for 1421 magnetic double space groups
Authors: Kim Shoon K.
Published: July 2002

Thirty eight assemblies of general expressions of the generator sets for 1421 magnetic double space groups in terms of the 32 minimal general generator sets of the unitary space groups are provided.
Charge quantization without superheavy masses in a Kaluza–Klein description of electromagnetism
Authors: Ross D. K.
Published: July 2002

A scalar matter field coupled to general relativity and electromagnetism in a five-dimensional Kaluza–Klein model is considered. The five-dimensional space is assumed to be a fiber bundle as in the usual description of a gauge theory and not a more general manifold. Properly taking this into account allows one to use a Lagrangian density for the scalar field which includes charge quantization but not the unphysical superheavy masses found by other authors. A natural, satisfactory explanation of why charge is quantized results.
Nonequilibrium entropy in classical and quantum field theory
Authors: Kandrup Henry E.
Published: July 2002

This paper proposes a definition of nonequilibrium entropy appropriate for a bosonic classical or quantum field, viewed as a collection of oscillators with equations of motion which satisfy a Liouville theorem (as is guaranteed for a Hamiltonian system). This entropy S is constructed explicitly to provide a measure of correlations and, as such, is conserved absolutely in the absence of couplings between degrees of freedom. This means, e.g., that there can be no entropy generation for a source-free linear field in flat space, but that S need no longer be conserved in the presence of couplings induced by nonlinearities, material sources, or a nontrivial dynamical background space-time. Moreover, through the introduction of a ‘‘subdynamics,’’ it is proved that, in the presence of such couplings, the entropy will satisfy an H-theorem inequality, at least in one particular limit. Specifically, if at some initial time t0 the field is free of any correlations, it then follows rigorously that, at time t0+Δt, the entropy will be increasing: dS/dt>0. Similar arguments demonstrate that this S is the only measure of ‘‘entropy’’ consistent mathematically with the subdynamics. It is argued that this entropy possesses an intrinsic physical meaning, this meaning being especially clear in the context of a quantum theory, where a direct connection exists between entropy generation and particle creation. Reasonable conjectures regarding the more general time dependence of the entropy, which parallel closely the conventional wisdom of particle mechanics, lead to an interpretation of S which corroborates one’s naive intuition as to the behavior of an ‘‘entropy.’’
Existence and approximation of the solutions of some nonlinear problems
Authors: Vatsya S. R.
Published: July 2002

Nonlinear equations defined by a positive definite, elliptic operator and nonlinear functions are considered. It is proved that a unique solution exists for a wider class of problems than previously determined, which may be approximated by an iterative method. These results are shown to hold for an equation arising in some reaction-diffusion phenomena.
On the existence of infinitely many resonances in the scattering problem based on moment conditions and entire functions
Authors: Hwang J. S.
Published: July 2002

It has been well proved that there are infinitely many resonances in the scattering problem provided the potential V(t) satisfies the moment conditions ∫∞0‖V(t)‖tn dt =O(n(1−ε)n), ε>0, n=0,1, ... . In particular, if V(t) has a compact support then the result of Rollnik [H. Rollnik, Z. Phys. 145, 654 (1956)] and Regge [T. Regge, Nuovo Cimento, 8, 671 (1958)] is obtained and if ‖V(t)‖∼exp(−τt1+ε), τ,ε>0, we have the results of Sartori [L. Sartori, J. Math. Phys. 4, 1408 (1963)].
Quantum mechanics of a charged scalar boson with respect to an observer’s past light cone
Authors: Derrick G. H.
Published: July 2002

An observer whose instantaneous ‘‘here-now’’ has Minkowski coordinates zλ(λ=0,1,2,3) can only be aware of events within or on the past light cone with vertex at zλ. In conventional quantum mechanics his current quantum state would refer to some spacelike surface containing zλ, for example, x0=z0. This is, however, a region of space-time about which the observer can know nothing except the single event zλ, his current here-now. The aim of the present paper is to give a version of quantum mechanics in which the intrinsically unknowable ‘‘quantum state at the present time’’ is replaced by the ‘‘quantum state on the past light cone.’’ The theory is an extension and adaptation of Dirac’s point mechanics [Rev. Mod. Phys. 21, 392 (1949)].
Two comments on nonlinear Schrödinger equations
Authors: Hadeler K. P. , Lange H.
Published: July 2002

In a recent paper, Brüll and Lange [Expos. Math. 4, 279 (1986); Math. Meth. Appl. Sci. 8, 559 (1986)] have discussed a class of nonlinear Schrödinger equations with rather general nonlinearities which comprises various cases occurring in the literature. Although the ‘‘potentials’’ in these equations are quite complicated, the equations admit various invariance properties. The present paper has two aims. First several local and global conservation laws related to conservation of mass, impulse, and energy are exhibited. One of these laws seems to be new, though not surprising. Then it is shown that the equation defined by Brüll and Lange is just suitable to apply some transformations which reduce the problem of solitary waves to a relatively simple Hamiltonian system in the plane. This method of transforming the phase plane problem into normal form is, in some respects, similar to the transformations introduced by Hadeler [Proc. Math. Soc. Edinburgh, to be published; Free Boundary Problems: Theory and Applications, Montecatini Conference, 1981, edited by A.Fasano and M. Primicerio (Pitman, New York, 1983), Vol. II, pp. 664–671] for parabolic and hyperbolic reaction diffusion equations.
Double Kerr–Schild equivalence and hyperheavens
Authors: Rózga Krzysztof
Published: July 2002

Double Kerr–Schild equivalence of hyperheavens is studied within a spinorial formalism based on the fact of the existence of a congruence of self-dual null strings. In particular, it is shown that a given hyperheavenly metric structure generates all members of the corresponding equivalence class in terms of a certain spinor field and the generalized key function subject to the generalized hyperheavenly equation. The treatment is manifestly covariant: no coordinates are employed. Finally, a link with the classical results on hyperheavens (Plebański and Robinson [J. F. Plebański and I. Robinson, Phys. Rev. Lett. 37, 493 (1976)] and Finley and Plebański [J. D. Finley, III and J. F. Plebański, J. Math. Phys. 17, 2207 (1976)]) is established.
Some solutions of Einstein’s equations with shock waves
Authors: Anile A. M. , Moschetti G. , Bogoyavlenski O. I.
Published: July 2002

The Einstein field equations in the presence of a polytropic fluid performing self-similar motion are reduced to a dynamical system. Qualitative properties of the dynamical system are investigated in the case when the fluid motion is with shock waves.
Electron density and Slater sum of a semi-infinite electron gas in d dimensions
Authors: March N. H.
Published: July 2002

By taking the electron densities of semi-infinite electron gases in one and in three dimensions, and forming the Slater sum by the Laplace transform, it is shown that the Slater sum is the classical partition function in d dimensions, times a function independent of dimensionality. The electron density is thereby calculated for general dimensionality, as is the kinetic energy density. As a by-product, the dimensionality dependence of Friedel oscillations emerges in general form.
A note on the electron self-energy
Authors: Abbasabadi Alireza , Repko Wayne W.
Published: July 2002

Using dimensional regularization, the lowest-order electron self-energy function in an arbitrary covariant gauge is derived. For off-mass-shell electrons, the usual expression for the finite portion Σ f of the self-energy as the number of space-time dimensions n approaches 4 is recovered. In the case of on-mass-shell electrons, the condition (p/−m)Σ f→0 as p/→m, which is necessary to make the usual separation of the renormalization constant unambiguous, requires that n approach 4 from above, i.e., n→4+. This necessary condition on Σ f is not satisfied by the off-mass-shell expression in the limit p2→m2 due to a branch point in the self-energy operator.
Boson algebra as a symplectic Clifford algebra
Authors: Bacry H. , Boon M.
Published: July 2002

It is well known that the n-fermion algebra is a complex Clifford algebra of dimension 22n with the orthogonal group O(2n,C) as group of automorphisms. The n-boson algebra viewed similarly as a complex ‘‘symplectic Clifford’’ algebra is investigated. It is of infinite dimension and has the symplectic group Sp(2n,C) as a group of automorphisms. Special attention is given to the case n=1 and various bases. In particular, the matrix elements of the bases between harmonic oscillator states and their relation with special functions are investigated.
Stochasticity in Yang–Mills theory
Authors: Villarroel Javier
Published: July 2002

Some stochastic properties of a subsystem of Yang–Mills classical mechanics are studied which do not fulfill automatically Gauss’ law, a possibility not considered before. However, significant differences with other subsystems already studied are not found for this system.
Proper affine collineations in Robertson–Walker space-times
Authors: Collinson C. D.
Published: July 2002

It is shown that the only Robertson–Walker space-times admitting a proper affine collineation are static, hence proving the uniqueness of the collineation found recently by Maartens [J. Math. Phys. 28, 2051 (1987)].
Matrix hierarchies and vector bundles over Riemann surfaces
Authors: Helminck G. F.
Published: July 2002

To a compact Riemann surface, a point on the surface, a local coordinate around this point, and a holomorphic transfer function of degree m is associated a point in the Grassmann manifold of l2(Z)m. This leads to solutions of the nonlinear hierarchies corresponding to maximal Abelian subalgebras of End(Cm).
Long wave equations and metrizable Lie algebras
Authors: Kupershmidt B. A.
Published: July 2002

An infinite commuting hierarchy of three-Hamiltonian integrable extensions of the dispersive water waves (DWW) hierarchy is constructed for each finite-dimensional Lie algebra 𝒢 with an invariant symmetric bilinear form on it. The construction is based on special two-cocycles on a one-parameter family of differential Lie algebras attached to 𝒢.
Simple orthogonal and unitary compact quantum systems and the İnönü–Wigner contraction
Authors: Fawcett R. J. B. , Bracken A. J.
Published: July 2002

Compact quantum dynamics, based on the Lie algebra so(n), is known to be relevant to the description of the internal motion of the electron and neutrino. It is shown here that a compact dynamics can also be based on u(n). In both cases there exists a contraction to a noncompact algebra containing a Weyl–Heisenberg subalgebra and consequently, there also exists an approximation scheme defined entirely by the contraction process, in which a sequence of compact systems described in terms of finite square matrices of increasing size, approaches a noncompact system of the usual type, described in terms of operators on a Hilbert space. Particular boson realizations of so(n) and u(n) are shown to be especially convenient for the discussion of the contractions. Some simple illustrative examples of compact systems are treated algebraically and numerically in order to show how their properties approach those of the corresponding noncompact systems.
Complete orthonormal sets on the past light cone. I. Functions belonging to mass zero and helicity s
Authors: Derrick G. H.
Published: July 2002

This is the first of a series of papers preparing the mathematical framework for a past light-cone formulation for the quantum mechanics of particles of arbitrary mass and spin. The aim of past light-cone quantum theory is to define quantum states solely in terms of data accessible to an observer, i.e., information from within his current past light cone. In order to set up such a theory one needs to define on the past light cone complete orthonormal sets of functions that belong to the appropriate unitary irreducible representation of the Poincaré group. Such functions are interpreted as energy-momentum eigenfunctions. The present paper treats the discrete spin, zero mass case for all values of the helicity s=0, 1/2 , (3)/(2) ,. . . .
Multiplication formulas of orthogonal polynomials of boson field operators: Derivation based on the generalized phase-space method
Authors: Ito H. M.
Published: July 2002

The products of several orthogonal polynomials of boson field operators, the quantum mechanical version of multiple Wiener integrals, are expressed as linear combinations of the polynomials. The expression is obtained by making use of the correspondence rules of boson operators and complex numbers.
Diffeomorphisms, orientation, and pin structures in two dimensions
Authors: Da̧browski L. , Percacci R.
Published: July 2002

A set of generators for the modular group of a surface with boundary, both in the orientable and nonorientable cases is given. All inequivalent pin structures are constructed and their transformations under these generators are computed.
Notes on a cross product of vectors
Authors: Plebański J. F. , Przanowski M.
Published: July 2002

The relation between quaternionlike algebras and cross products of vectors is demonstrated. A classification of all cross products of vectors is given.
Double-solution family of self-dual SU(2) gauge fields
Authors: Zhong Zai-Zhe
Published: July 2002

The method of Witten [Phys. Rev. D 19, 718 (1979)] is extended to a double-complex form. Many new solutions of the self-dual SU(2) gauge fields are generated by the double-inverse scattering method, and the soliton solutions can form an infinite network.
Ordered exponentials and differential equations
Authors: Fried H. M.
Published: July 2002

An independent and elementary method of guessing the leading form of certain SU(2) and SU(3) ordered exponentials in the strong-coupling limit of rapidly fluctuating input is presented. For SU(N), with N≥3, the method is not unambiguous, but it can be used to check previous estimates.
Collectivity and geometry. VI. Spectra and shapes in the three-dimensional case
Authors: Chacón E. , Hess P. , Moshinsky M.
Published: July 2002

In this paper the reports on collectivity and geometry are concluded, where the microscopic description of many-body collective motions and their relation with the symplectic geometry of the n-particle system are reexamined. In the present paper it is shown that, modulo linear canonical transformations, the symplectic algebra sp(6,R) admits only three maximal subalgebras sp(2,R)⊕o(3), u(3), and cm(3), which contain the rotation algebra o(3). The objective is to discuss the spectra and shapes of ‘‘pure’’ many-body systems for which the Hamiltonian is associated with a Casimir operator up to the second degree in the generators of a given maximal subalgebra, as well as those of ‘‘transitional’’ systems, where the Hamiltonian is a function of the generators and Casimir operators of several of the maximal subalgebras.
The Schwinger functions of a rational interaction: Local existence of the Borel transform
Authors: da Costa G. A. T. F. , Gomes M.
Published: July 2002

The local existence of the Borel transform of a two-dimensional field theoretical model characterized by the rational interaction g2x6/(1+gx2) is proven.
All directing fields that are polynomial in the (n−1) velocity are geodesic
Authors: Coleman Robert Alan , Korte Herbert
Published: July 2002

It is proven that all directing fields that are polynomial in the (n−1) velocities ξα1 are, in fact, cubic in these variables; so that they are geodesic directing fields (projective structures). This result, in conjunction with previously published work, entails that (n−1) forces must be nonpolynomial in the (n−1) velocities. A large class of acceleration fields that give rise to directing fields is briefly discussed in order to illustrate the source of this nonpolynomial behavior.
Technical methods for solving bound-state equations in momentum space
Authors: Boukraa S. , Basdevant J.-L.
Published: July 2002

Some technical methods used to solve two- and three-body bound-state equations in momentum space are described. These methods are, in particular, very efficient when applied to the calculation of spectra in semirelativistic confining potential Hamiltonians, which are used in hadron spectroscopy. They have a vast range of applicability.
A class of integral transforms associated with the Majorana representation of SL(2,C)
Authors: Basu Debabrata , Kar T. K.
Published: July 2002

It is shown that the boson operators for the Majorana representation of SL(2,C) realized as hyperdifferential operators in the Bargmann Hilbert space of analytic functions yield, on exponentiation, a parametrized continuum of integral transforms. Each value of the group parameters yields an integral transform pair. The formula for the inversion of the transform is obtained simply by replacing the group element by its inverse.
Point symmetries of conditionally integrable nonlinear evolution equations
Authors: Rubin J. , Winternitz P.
Published: July 2002

The Lie point symmetries of the first two equations in the Kadomtsev–Petviashvili (KP) hierarchy, introduced by Jimbo and Miwa, are investigated. The first is the potential KP equation, the second involves four independent variables and is called the Jimbo–Miwa (JM) equation. The joint symmetry algebra for the two equations is shown to have a Kac–Moody–Virasoro structure, whereas the symmetry algebra of the JM equation alone does not. Subgroups of the joint symmetry group are used to perform symmetry reduction and to obtain invariant solutions.
Radiative transfer on curved surfaces
Authors: Tessendorf J.
Published: July 2002

After a review of appropriate concepts in local surface geometry, a formally exact solution of the radiative transfer equation is constructed, for transfer from one surface of arbitrary shape to another. The solution is obtained from repeated application of the linear interaction principle to form a path integral over paths that cross many intermediate surfaces. Invariant imbedding in general geometries is presented and found to be manifest in the path integral solution as an invariance under local coordinate transformations of the intermediate surfaces. Aspects of possible numerical implementations of this geometrical approach are discussed.
Defining equations for supergroup orbits in super Clifford modules
Authors: Bergvelt M. J.
Published: July 2002

The defining equations for group orbits of a maximal subgroup of Glp‖q in certain highest weight representations of the Lie super algebra glp‖q are discussed.
Deformations of algebraic types of the energy-momentum tensor
Authors: Sánchez A. R. Guzmán , Plebański J. F. , Przanowski M.
Published: July 2002

Arnold’s miniversal deformations of the matrix canonical forms for the traceless Ricci tensor 𝒞 are found. The diagram presenting all possible degenerations for the algebraic types of 𝒞 is given.
Schrödinger equation for convex plane polygons: A tiling method for the derivation of eigenvalues and eigenfunctions
Authors: Amar V. , Pauri M. , Scotti A.
Published: July 2002

Motivated by a recently advanced conjecture on the ergodic properties of Quantum Systems, the problem of solving the Schrödinger equation for a free particle in a plane polygonal enclosure is revisited. It will be shown that two elementary lemmas suffice to give a complete characterization of the polygons for which a solution can be found in terms of a finite superposition of plane waves, without making use of advanced group-theoretical techniques. It turns out, inter alia, that these polygons, considered as classical billiards, are all and only those which are completely integrable in the sense of Arnold’s theorem.
Nonimpulsive colliding gravitational waves with noncollinear polarizations
Authors: Li Wei , Hauser Isidore , Ernst Frederick J.
Published: July 2002

A new method of constructing solutions of the Einstein equations describing nonimpulsive colliding gravitational waves with noncollinear polarizations is introduced. Application of the procedure is illustrated by constructing a large family of such solutions of the vacuum Einstein equations. The new solutions are shown to possess Killing–Cauchy horizons when n+2δ1 is set equal to ±3 or ±1.
Stationary axisymmetric interior metric for a charged perfect fluid source
Authors: García D. Alberto
Published: July 2002

A stationary axisymmetric type-D solution of the Einstein equations coupled with an electromagnetic field and a rigidly rotating perfect fluid distribution of finite dimensions is presented. For vanishing fluid parameters, it reduces to the Carter [A] metric, which contains among others the Kerr–Newman solution. When the electromagnetic field is switched off, it reduces to the Wahlquist perfect fluid metric. The Debever–Penrose directions are geodesic, shear-free, twisting, and diverging. This solution belongs to the Wainwright class II. The expressions of the functions of other new solutions are also given.
Canonical structures on anti-self-dual four-manifolds and the diffeomorphism group
Authors: Chakravarty Sarbarish , Mason Lionel , Newman Ezra T.
Published: July 2002

An unusual and attractive system is studied that arises from the anti-self-dual (ASD) Yang–Mills equations with maximal translational symmetry and with gauge group the volume preserving diffeomorphisms of an auxiliary four-manifold ℳ. The resulting equations lead to a system consisting of a volume form together with four independent vector fields on ℳ satisfying three simple Lie bracket relations. This structure is shown to give rise to a two-sphere’s worth of closed simple two-forms which in turn lead to the standard hyper-Kahler structure of an anti-self-dual metric on ℳ. The system determines not only the ASD metric, but also a frame that is proportional to an orthonormal frame. It is shown that the freedom in the choice of frame is related to a pair of solutions of a modified Laplacian and can always be chosen so that the proportionality factor is unity. The Plebanski first and second heavenly forms for general ASD metrics are written out in terms of these structures (the first being the standard description in terms of a complex structure and Kahler scalar). One of the scalars is interpreted as the generating function for the diffeomorphisms (symplectomorphism) in line with the origin of the system as the ASD Yang–Mills equations with the volume preserving diffeomorphisms as gauge group.
Derivation of many-body potential among charged particles in the S-matrix method
Authors: Ohta Tadayuki , Kimura Toshiei
Published: July 2002

A general method of deriving a classical potential from the S-matrix element of particle scattering in the theory of quantized fields is applied to electrodynamics to the post-post-Coulombian approximation. To obtain the many-body potential, a consistent prescription is implemented in subtracting the contributions of the repetition of lower-order potential from the S-matrix elements of the higher-order diagrams. The result shows that the four-body potential between charged particles has a characteristic feature at a large distance and the two-body potential is identical with that given in the reduced Hamiltonian of Wheeler–Feynman electrodynamics. The advantage of the S-matrix method over the canonical formalism is to give the potential directly, without complicated treatment of the interaction with higher derivatives by a method of constrained dynamics.
Stationary axisymmetric charged perfect fluid type D solutions
Authors: García Alberto , Tellez Jorge
Published: July 2002

A complete class of stationary axisymmetric Wainwright class I charged perfect fluid solutions is given.
Intrinsic characterization of space-time symmetric tensors
Authors: Bona Carles , Coll Bartolomé , Morales Juan Antonio
Published: July 2002

This paper essentially deals with the classification of a symmetric tensor on a four-dimensional Lorentzian space. A method is given to find the algebraic type of such a tensor. A system of concomitants of the tensor is constructed, which allows one to know the causal character of the eigenspace corresponding to a given eigenvalue, and to obtain covariantly their eigenvectors. Some algebraic as well as differential applications are considered.
Exotic quantum double, its universal R matrix, and their representations
Authors: Sun Chang-Pu
Published: July 2002

The exotic quantum double and its universal R matrix for the quantum Yang–Baxter equation are constructed in terms of Drinfeld’s quantum double theory. As a new quasitriangular Hopf algebra, it is different from those standard quantum doubles that are the q deformations for Lie algebras or Lie superalgebras. By studying its representation theory, many-parameter representations of the exotic quantum double are obtained with an explicit example associated with Lie algebra A2. The multiparameter R matrices for the quantum Yang–Baxter equation can result from the universal R matrix of this exotic quantum double and these representations.
Asymptotic (semiclassical) equivalence for Schrödinger equations with singular potentials and for related systems of two first-order equations
Authors: Aquilanti Vincenzo , Cavalli Simonetta , Sevryuk Mikhail B.
Published: July 2002

The asymptotic equivalence of systems of two ordinary first-order linear differential equations with complex independent variable and a small parameter at the derivatives is analyzed in the case of arbitrary numbers and multiplicities of turning points and singular points. The set of all the transformation matrices realizing the equivalence is described and a recursive procedure for constructing these matrices is developed. By persistently using the determinant properties of the transformation matrices, the number of integration operations at each step of this procedure is halved compared with the algorithms known before. The theory is specialized to the case of time-independent one-dimensional Schrödinger equations with singular potentials. Some generalizations to multichannel Schrödinger equations are also presented.
On a family of supersymmetrization procedures
Authors: Beckers J. , Debergh N.
Published: July 2002

Generalized expressions are proposed for the supercharges in N=2 supersymmetric quantum mechanics and their implications are analyzed mainly in the one-dimensional context. In fact, these generalizations are exploited in three typical examples admitting the largest numbers of symmetries, i.e., the harmonic oscillator, the hydrogen atom, and their superposition, the so-called ‘‘Calogero context,’’ successively. Different matrix realizations are exploited in the harmonic oscillator case more particularly. The link between these considerations and quantum deformations must be insisted on, our deformation b parameter referring to a family of physically acceptable supersymmetrization procedures including, in particular, the standard one (when b→1) initially proposed by Witten.
Quantum phenomena in resonators with moving walls
Authors: Dodonov V. V. , Klimov A. B. , Nikonov D. E.
Published: July 2002

New solutions for the mode functions of the electromagnetic field in ideal cavity which boundary oscillates at a resonance frequency are obtained in the long-time limit. The rate of photons creation from initial vacuum state is shown to be time independent and proportional to the amplitude of oscillations and the resonance frequency. Temperature corrections are evaluated. The squeezing coefficients for the quantum states of the field generated are calculated, as well as the backward reaction of the field on the vibrating wall.
1+1 solvable relativistic field models
Authors: Degasperis A. , Tinebra F.
Published: July 2002

Nonlinear Lorentz invariant field models are constructed in such a way to be integrable by a transformation of a linear free field. The transformation is explicitly found by requiring that the U(1) conserved current keeps the same expression of the free field. This method is applied to the complex scalar and spinor fields in 1+1 dimensions. Localized nondispersive waves exist and interact with each other. A way of introducing nonlinear interactions between different fields is also given with the property of preserving integrability.
Elliptical billiards and hyperelliptic functions
Authors: Crespi Bruno , Chang Shau-Jin , Shi Kang-Jie
Published: July 2002

The geometrical properties of the elliptical billiard system are related to Poncelet’s theorem. This theorem states that if a polygon is inscribed in a conic and circumscribed about a second conic, every point of the former conic is a vertex of a polygon with the same number of sides and the same perimeter. Chang and Friedberg have extended this theorem to three and higher dimensions. They have shown that the geometrical properties of the hyperelliptic billiard system are related to the algebraic character of a Poincaré map in the phase space. The geometrical and algebraic properties of the system can be understood in terms of the analytical structure of the equations of motion. These equations form a complete system of Abelian integrals. The integrability of the physical system is reflected by the topology of the Riemann surfaces associated to these integrals. The algebraic properties are connected with the existence of addition formulas for hyperelliptic functions. The main purpose of this study is to establish such a connection, and to provide an algebraic proof of Poncelet’s theorem in three and higher dimensions.
The solution of the Ising model on the truncated icosahedron lattice of C60
Authors: Samuel Stuart
Published: July 2002

The high-temperature expansion for the partition function of the Ising model on the truncated icosahedron lattice is computed exactly. The average energy and heat capacity are presented.
Adler–Kostant–Syms theorem and supersymmetric integrable systems
Authors: Chapa Sen Dolan , Chowdhury A. Roy
Published: July 2002

The Adler, Kostant, and Syms theorem has been used to construct integrable nonlinear dynamical systems involving both bosonic and fermionic variables. Explicit forms of the equations are displayed in case of two super Lie Algebra OSPU(1,1/1) and Sl(2/2). Integrability is proved by explicit construction of higher order conserved quantities.
L-operator for Belavin’s R-matrix acting on the space of theta functions
Authors: Hasegawa Koji
Published: July 2002

For the Z/nZ symmetric elliptic R matrix, L operators which act on the space of A(1)n−1 integrable characters, or the affine Weyl group invariants, of the fixed level l, are constructed. For l=1, the matrix elements of the L operator and those of the R matrix are shown to be the same.
The induced representations of the κ-Poincaré group
Authors: Maślanka Pawel/
Published: July 2002

The induced representations of the κ-Poincaré group for the massive case and the massless case are described. It is shown that it extends many of the features of the classical case.
On the complete basis of Pauli-allowed states of three-cluster systems in the Fock–Bargmann space
Authors: Filippov G. F. , Rybkin I. Yu. , Korennov S. V. , Katō K.
Published: July 2002

In the Fock–Bargmann space, the complete harmonic-oscillator basis for the three-cluster system is constructed. The indices of the reduction U(6)⊇U(2)×U(3) are chosen as the quantum numbers. The Pauli-forbidden states are eliminated by the orthogonal transformation of basis functions. The basis states are obtained in terms of hypergeometric functions and the spherical Wigner functions. Their simple form allows one to solve the problem of calculating the matrix elements of the microscopic Hamiltonian needed for the study of three-cluster systems within the algebraic version of RGM.
SU(2) coherent states on an m-sheeted covering of the sphere
Authors: Vourdas A.
Published: July 2002

SU(2) coherent states on an m-sheeted covering of the sphere are introduced, and properties like overcompleteness and resolution of the identity are studied. Operators J+(m), J−(m), Jz(m) that obey the SU(2) algebra but shift the ‖jn〉 states by m steps are considered, and it is shown that the properties of our coherent states with respect to them are analogous to the properties of the standard SU(2) coherent states with respect to the usual angular momentum operators J+, J−, Jz. An extended SU(2) Bargmann representation on an m-sheeted sphere is introduced in which the transformations of an m-sheeted covering of the SU(2) group are implemented as extended Mobius conformal mappings. The formalism is applied in the study of Hamiltonians that contain the operators J+(m), J−(m), Jz(m) and describe processes where the state of a particle is shifted by m steps. The method is also used in the context of both the Holstein–Primakoff and Schwinger representation of SU(2).
Large deviations in a semi-infinite layered ferromagnet with mean-field interactions: Classical and quantum models
Authors: Angelescu N. , Bundaru M. , Pulé J. V.
Published: July 2002

We consider the probability distribution in a classical semi-infinite layered ferromagnet with mean-field interactions and prove that this satisfies a large deviation principle, giving the rate function explicitly. For an analogous quantum system we show that the limiting states are product states, which are completely characterized by the classical mean-field solutions.
Long-range effects in asymptotic fields and angular momentum of classical field electrodynamics
Authors: Herdegen Andrzej
Published: July 2002

Asymptotic properties of classical field electrodynamics are considered. Special attention is paid to the long-range structure of the electromagnetic field. It is shown that conserved Poincaré quantities may be expressed in terms of the asymptotic fields. Long-range variables are shown to be responsible for an angular momentum contribution which mixes Coulomb and infrared free field characteristics; otherwise angular momentum and energy-momentum separate into electromagnetic and matter fields contributions.
Path integral over the generalized coherent states
Authors: Kochetov E. A.
Published: July 2002

A path integral written in terms of the group theoretic coherent states by using the Kähler structure of the coherent state manifold with the particular emphasis on the boundary-fixing term derivation is considered herein. The path integral for a propagator of the system with Hamiltonian linear in the SU(2)/SU(1,1) generators is shown to be diagonalized by an appropriate motion in the phase space.
The fundamental invariant of the Hecke algebra Hn(q) characterizes the representations of Hn(q), S
n, SUq(N), and SU(N)
Authors: Katriel J. , Abdesselam B. , Chakrabarti A.
Published: July 2002

The irreducible representations (irreps) of the Hecke algebra Hn(q) are shown to be completely characterized by the fundamental invariant of this algebra, Cn. This fundamental invariant is related to the quadratic Casimir operator, 𝒞2, of SUq(N), and reduces to the transposition class-sum, [(2)]n, of Sn when q → 1. The projection operators constructed in terms of Cn for the various irreps of Hn(q) are well behaved in the limit q → 1, even when approaching degenerate eigenvalues of [(2)]n. In the latter case, for which the irreps of Sn are not fully characterized by the corresponding eigenvalue of the transposition class-sum, the limiting form of the projection operator constructed in terms of Cn gives rise to factors that depend on higher class-sums of Sn, which effect the desired characterization. Expanding this limiting form of the projection operator into a linear combination of class-sums of Sn, the coefficients constitute the corresponding row in the character table of Sn. The properties of the fundamental invariant are used to formulate a simple and efficient recursive procedure for the evaluation of the traces of the Hecke algebra. The closely related quadratic Casimir operator of SUq(N) plays a similar role, providing a complete characterization of the irreps of SUq(N) and—by constructing appropriate projection operators and then taking the q → 1 limit—those of SU(N) as well, even when the quadratic Casimir operator of the latter does not suffice to specify its irreps.
Manifestly covariant forms for Lorentz group unitary representations
Authors: Mukunda N. , Simon R.
Published: July 2002

We show that the unitary irreducible representations (UIRs) of the proper homogeneous Lorentz group SO(3,1) can be realized in a manifestly covariant form, using Hilbert spaces of functions on suitable geometrical configurations of vectors in space time. The concept and properties of isotropic lines, which are generators of spacelike hyperboloids, and of stability groups of lightlike vectors, are crucial to the constructions. Both principal and supplementary Series UIRs are treated.
Some trigonometric identities encountered by McCoy and Orrick
Authors: Gervois A. , Mehta M. L.
Published: July 2002

A proof is given for the trigonometric identities (1.1)–(1.5) arising in the perturbative treatment of chiral Potts model.
Batalin–Vilkovisky formalism and integration theory on manifolds
Authors: Khudaverdian O. M. , Nersessian A.
Published: July 2002

The correspondence between the BV formalism and integration theory on supermanifolds is established. An explicit formula for the density on a Lagrangian surface in a superspace provided with an odd symplectic structure and a volume form is proposed.
Localization of the photon on phase space
Authors: Brooke J. A. , Schroeck F. E.
Published: July 2002

We obtain phase space representations of the Poincaré group for zero mass particles of all helicities, including photons. A natural quantization scheme for massless particles arises, and a covariant phase space localization operator is found.
Multi-periodic coherent states and the WKB exactness
Authors: Fujii Kazuyuki , Funahashi Kunio
Published: July 2002

We construct the path integral formula in terms of a ‘‘multi-periodic’’ coherent state as an extension of the Nielsen–Rohrlich formula for spin. We make an exact calculation of the formula and show that, when a parameter corresponding to the magnitude of spin becomes large, the leading order term of the expansion coincides with the exact result. We also give an explicit correspondence between the trace formula in the multi-periodic coherent state and the one in the ‘‘generalized’’ coherent state.
Loop variables and holonomies for a class of conical space–times
Authors: Bezerra V. B. , Letelier P. S.
Published: July 2002

We compute the loop variables for a class of space–times with topological defects. In particular we compute these quantities for multiple moving cosmic strings and two plane topological defects crossed by a cosmic string, showing that these quantities are elements of the homogeneous Lorentz group. We also compute the loop variables for a multi-chiral cone and we show that in the context of Einstein theory the loop variables are elements of the inhomogeneous Lorentz group, but in the context of Einstein-Cartan theory they are elements of the homogeneous Lorentz group.
Graded contractions of symplectic Lie algebras in collective models
Authors: Tolar J. , Trávnı́ček P.
Published: July 2002

All ℤ2×ℤ2-graded contractions preserving space isotropy and the grading induced by the time reversal Θ and the discrete canonical transformation K exchanging coordinates and momenta are found for the Lie algebras C2≡sp(2,ℂ), sp(2,↛), C3≡sp(3,ℂ), and sp(3,↛). Some properties and physical aspects of these contractions are discussed, especially in connection with the algebraic formulation of nuclear collective models.
Geometrical stochastic control and quantization
Authors: Zastawniak Tomasz
Published: July 2002

A class of diffusion processes controlled by the geometry of the manifold on which they evolve is considered. The kinetic energy of such diffusions is shown to be a geometric object expressed in terms of the curvature and torsion tensors. This gives rise to an action functional leading to a variational principle, from which a nontrivial critical geometry with nonvanishing torsion emerges. The resulting criticality condition is related to the Schrödinger equation in a manner that reproduces the features of Nelson’s stochastic approach to quantum mechanics.
Three-dimensional electromagnetic inverse scattering for biisotropic dispersive media
Authors: He Sailing , Weston Vaughan H.
Published: July 2002

The time-domain inverse problem of determining the three-dimensional parameter functions for the biisotropic dispersive medium is considered. Maxwell’s equations are rewritten in terms of the tangential fields. Time domain wave-splitting of Maxwell’s equations is applied to the total field that is generated by a dipole exterior to the scattering medium. The structure of the electromagnetic fields is analyzed, and the transport equations are given. The reconstruction condition is derived for a layer-stripping approach.
INTRODUCTION
Authors: Bellissard J.
Published: July 2002

Condensed Matter Physics has been one of the most important sources of technology during the past fifty years. But until the beginning of the eighties most theoretical advances did not need the help of sophisticated mathematical methods, and various important discoveries such as Anderson’s localization, the quantum Hall effect, Kondo’s effect, quasicrystals, high Tc superconductors, and persistent currents in mesoscopic systems brought about many problems requiring the use of the most advanced theoretical and mathematical procedures. This special issue aims at focusing on a few of the problems posed in this field, summarizing the present understanding, tracing the methods used, and the relations among them.
Spectral correlations in the crossover between GUE and Poisson regularity: On the identification of scales
Authors: Guhr Thomas , Müller-Groeling Axel
Published: July 2002

Motivated by questions of present interest in nuclear and condensed matter physics we consider the superposition of a diagonal matrix with independent random entries and a GUE. The relative strength of the two contributions is determined by a parameter λ suitably defined on the unfolded scale. Using results for the spectral two-point correlator of this model obtained in the framework of the supersymmetry method we focus attention on two different regimes. For λ≪1 the correlations are given by Dawson’s integral while for λ≫1 we derive a novel analytical formula for the two-point function. In both cases the energy scales, in units of the mean level spacing, at which deviations from pure GUE behavior become noticeable, can be identified. We also derive an exact expansion of the local level density for a finite level number.
Equivalence of isotropic submanifolds and symmetry
Authors: Janeczko S.
Published: July 2002

It is shown that any two generating families of the same isotropic submanifold are equivalent. Also the singularity theory of symmetric isotropic submanifolds is developed and the basic classification theorems on prenormal forms, in particular for Z2 action, are proved.
The Meissner effect and the Ginzburg–Landau equations in the presence of an applied magnetic field
Authors: Tsutsumi Masayoshi , Kasai Hironori , Ōishi Takeshi
Published: July 2002

It is shown that a way of phenomenological description of the Meissner effect in a superconductor under an applied magnetic field is to consider the minimizing problem of the Gibbs free energy under the constraints of complete expulsion of magnetic field from the superconducting states.
On representations of Uqosp(1|2) when q is a root of unity
Authors: Chung W.-S. , Suzuki Takashi
Published: July 2002

The infinite dimensional highest weight representations of Uqosp(1|2) for the deformation parameter q being a root of unity are investigated. As in the cases of q-deformed nongraded Lie algebras, we find that every irreducible representation is isomorphic to the tensor product of a highest weight representation of sl2(R) and a finite dimensional one of Uqosp(1|2). The structure is investigated in detail.