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Dec 1988

Volume 29, Issue 12, pp. 2519-2699

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Projector bases and algebraic spinors

G. Bergdolt

J. Math. Phys. 29, 2519 (1988); http://dx.doi.org/10.1063/1.528092 (4 pages) | Cited 1 time

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In the case of complex Clifford algebras a basis is constructed whose elements satisfy projector relations. The relations are sufficient conditions for the elements to span minimal ideals and hence to define algebraic spinors.
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02.10.Hh Rings and algebras
02.20.Qs General properties, structure, and representation of Lie groups
02.20.Sv Lie algebras of Lie groups

SU(1,1), its connections with SU(2), and the vector model

E. de Prunelé

J. Math. Phys. 29, 2523 (1988); http://dx.doi.org/10.1063/1.528093 (10 pages) | Cited 10 times

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A relation between representation functions (RF’s) of positive discrete unitary irreducible representations (UIR’s) of SU(1,1) and the RF’s of the UIR’s of SU(2) is given. The classical vector model is worked out for the positive discrete UIR’s of SU(1,1). Classical domains, probability densities, and algorithms for numerical computations of SU(1,1) RF’s and Clebsch–Gordan coefficients are derived, in full analogy with the SU(2) case.
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02.20.Qs General properties, structure, and representation of Lie groups
02.70.-c Computational techniques; simulations
03.65.Fd Algebraic methods

Integral‐spin fields on (3+2)‐de Sitter space

Jean‐Pierre Gazeau and Michel Hans

J. Math. Phys. 29, 2533 (1988); http://dx.doi.org/10.1063/1.528094 (20 pages) | Cited 8 times

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Nowadays, (3+2)‐de Sitter (or anti‐de Sitter space) appears as a very attractive possibility at several levels of theoretical physics. The Wigner definition of an elementary system as associated to a unitary irreducible representation of the Poincaré group may be extended to the de Sitter group SO(3,2) [or ∼(SO(3,2))] without great difficulty. The constant curvature, as small as it can be, is a natural candidate to play the role of a regularization parameter with respect to the flat‐space limit. Massless particles in (3+2)‐de Sitter theory are composite (singletons). On the other hand, supergravity theories necessitate a (large) constant curvature. The content of this paper is group theoretical. It attempts to continue the ‘‘math la Wigner’’ program for SO(3,2), already largely broached by Fronsdal. Three recurrence formulas are presented. They permit one to build up the carrier states for representations with arbitrary integral spin. Two of them are valid for the ‘‘massive’’ representations whereas the third one is applicable to the indecomposable massless representations. In addition, other presumably indecomposable, though nonphysical, representations are studied, in relation to the existence of ‘‘generalized’’ gauge fields and divergences. The recurrence formulas also allow one to build up the invariant two‐point functions or homogeneous propagators. Hence it becomes possible to examine the problems of light‐cone propagation and ‘‘reverberation’’ into the light cone and to make the following assertion: for a certain choice of the gauge‐fixing parameters, the massless states with arbitrary spin propagate only on the light cone and whatever gauge one chooses their physical parts propagate on the light cone.
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11.30.Cp Lorentz and Poincaré invariance
11.10.Gh Renormalization
02.20.Qs General properties, structure, and representation of Lie groups
04.20.Cv Fundamental problems and general formalism

Wigner operator and Racah operator of the Lie superalgebra OSP(1,2)

Gao‐Jian Zeng and Xiou‐Hua Yuan

J. Math. Phys. 29, 2553 (1988); http://dx.doi.org/10.1063/1.528095 (10 pages) | Cited 1 time

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Starting with the Wigner–Eckart theorem about the OSP(1,2) irreducible tensor [J. Phys. A: Math Gen. 20, 5423 (1987)], the OSP(1,2) Wigner operator and Racah operator are studied, their general definitions, orthogonality properties, and coupling laws are given, and the connections between them and the corresponding operators of the SO(3) algebra are established. The results obtained in this paper are a natural development of the authors’ theory on the OSP(1,2) irreducible tensor.
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02.20.Sv Lie algebras of Lie groups
02.20.Qs General properties, structure, and representation of Lie groups
03.65.Fd Algebraic methods
11.30.Pb Supersymmetry

Analysis and solution of a nonlinear second‐order differential equation through rescaling and through a dynamical point of view

P. G. L. Leach, M. R. Feix, and S. Bouquet

J. Math. Phys. 29, 2563 (1988); http://dx.doi.org/10.1063/1.528096 (7 pages) | Cited 24 times

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The solutions of the equation math+ymath+βy3=0, where β is a free parameter, are investigated. For β= (1)/(9) the equation is linearizable through an eight‐parameter symmetry group and is completely integrable. For β≠ (1)/(9) only two symmetries subsist, but through a dynamical description the analytical asymptotic solutions and their behavior are given according to the value of β and according to the initial conditions.
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02.30.Hq Ordinary differential equations
02.20.Sv Lie algebras of Lie groups
45.05.+x General theory of classical mechanics of discrete systems

Gauge and Bäcklund transformations for the variable coefficient higher‐order modified Korteweg–de Vries equation

Yu‐kun Zheng and W. L. Chan

J. Math. Phys. 29, 2570 (1988); http://dx.doi.org/10.1063/1.528097 (6 pages) | Cited 2 times

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A family of higher‐order modified Korteweg–de Vries equations with variable coefficients (t‐ho‐mKdV) is introduced. A one‐to‐one correspondence between a real solution of these equations and a complex solution of the variable coefficient higher‐order Korteweg–de Vries (t‐ho‐KdV) equations is established through a complex Miura transformation. An auto‐Bäcklund transformation for these t‐ho‐mKdV equations is derived from that of the t‐ho‐KdV equations. The associated gauge transformations of the corresponding AKNS systems are presented. They enable one to construct a hierarchy of solutions of the t‐ho‐mKdV equations from a known hierarchy without solving the differential equations for the wave functions except the first one. A new family of higher‐order evolution equations with an auto‐Bäcklund transformation is also derived in connection with the gauge transformation of the t‐ho‐mKdV equations.
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02.30.Uu Integral transforms
02.30.Vv Operational calculus
02.30.Jr Partial differential equations

A generalization of manifolds as space‐time models

J. Gruszczak, M. Heller, and P. Multarzyński

J. Math. Phys. 29, 2576 (1988); http://dx.doi.org/10.1063/1.528098 (5 pages) | Cited 10 times

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A smooth manifold can be defined as the pair (M, C), where C is a family of functions on a set M, satisfying suitable axioms. The manifold concept can be easily generalized by dropping the axiom ensuring the manifold to be locally diffeomorphic to Rn. The resulting concept, the so‐called d‐space, turns out to be a geometrically workable structure. The existence of the pseudo‐Riemannian structure (both Riemannian and Lorentz) on d‐spaces is discussed. It is proposed to model the physical space‐time by a d‐space rather than by a manifold. In some quantum gravity situations space‐time may still be a d‐space but already not a manifold.
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02.40.Sf Manifolds and cell complexes
02.40.Ky Riemannian geometries

Theory of fluctuations and small oscillations for quantum lattice systems

D. Goderis, A. Verbeure, and P. Vets

J. Math. Phys. 29, 2581 (1988); http://dx.doi.org/10.1063/1.528099 (5 pages) | Cited 6 times

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The theory of fluctuations for a quantum lattice system is rigorously defined. The property of stability of equilibrium states against macroscopic fluctuations is formulated and small oscillations around equilibrium are studied.
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03.65.Ta Foundations of quantum mechanics; measurement theory
02.50.Ey Stochastic processes
03.65.Fd Algebraic methods
05.30.-d Quantum statistical mechanics

Evolution of SU(2) and SU(1,1) states: A further mathematical analysis

G. Dattoli, M. Richetta, and A. Torre

J. Math. Phys. 29, 2586 (1988); http://dx.doi.org/10.1063/1.528100 (3 pages) | Cited 10 times

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The problem of the evolution of SU(2) and SU(1,1) states is analyzed from a unified point of view. A generalized Rabi matrix for a time‐dependent SU(2)–SU(1,1) coherence preserving Hamiltonian is also derived.
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03.65.Fd Algebraic methods
02.20.Qs General properties, structure, and representation of Lie groups
02.20.Sv Lie algebras of Lie groups

Finite‐dimensional representations of the special linear Lie superalgebra sl(1,n). II. Nontypical representations

Tchavdar D. Palev

J. Math. Phys. 29, 2589 (1988); http://dx.doi.org/10.1063/1.528101 (10 pages) | Cited 10 times

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All nontypical irreducible representations of the special linear Lie superalgebra sl(1,n) are constructed for any n. Explicit expressions for the transformation of the basis under the action of the generators are given. The results of this paper together with those obtained in Paper I [J. Math. Phys. 28, 2280 (1987)] solve the problem of the finite‐dimensional irreducible representations of sl(1,n).
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03.65.Fd Algebraic methods
02.10.-v Logic, set theory, and algebra
11.30.Pb Supersymmetry
02.20.Qs General properties, structure, and representation of Lie groups

Infinite‐dimensional symmetry algebras and an infinite number of conserved quantities of the (2+1)‐dimensional Davey–Stewartson equation

Minoru Omote

J. Math. Phys. 29, 2599 (1988); http://dx.doi.org/10.1063/1.528102 (5 pages) | Cited 8 times

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An infinite‐dimensional symmetry algebra of the Davey–Stewartson equation is explicitly presented. It is shown that this algebra is a gauge generalization of the symmetry transformation for the Schrödinger equation, and that the Virasoro algebra appears as the subalgebra. An infinite number of conserved quantities associated with the transformations are also obtained.
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03.65.Ge Solutions of wave equations: bound states
02.20.Tw Infinite-dimensional Lie groups
02.30.Jr Partial differential equations
03.65.Fd Algebraic methods

Adiabatic switching for time‐dependent electric fields

Márcia A. G. Scialom and Rafael J. Iório

J. Math. Phys. 29, 2604 (1988); http://dx.doi.org/10.1063/1.528103 (7 pages) | Cited 1 time

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In this work the scattering theory associated with the differential equation i(∂ψ/∂t)=(−Δ+e−Ε‖t‖g(t)x1 +q(x))ψ is considered, where x=(x1,x⊥)∊R×R2, Ε≥0, ω>0, α∊R, g(t), t∊R is continuous, periodic with mean value zero over a period, and q(x) approaches to zero sufficiently fast as ‖x‖→∞. In the case Ε>0, it is shown that the usual theory is adequate; however, a limit does not exist when Ε↓0. A modified theory is developed where the limit does exist as Ε↓0. Furthermore, the concepts of bound states and scattering states for Ε≥0 are discussed.
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03.65.Ge Solutions of wave equations: bound states
03.65.Nk Scattering theory
11.80.-m Relativistic scattering theory

Padé oscillators and a new formulation of perturbation theory

M. Znojil

J. Math. Phys. 29, 2611 (1988); http://dx.doi.org/10.1063/1.528104 (7 pages) | Cited 3 times

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Being guided by the problem of bound states in potentials close to their Padé approximants, a new Rayleigh–Schrödinger‐type perturbation theory is developed. The unperturbed system is understood: here in a broader sense: its solutions are not needed, but merely the related nondiagonal unperturbed propagator R. In particular, all the chain models H0ψ=ES0ψ (H0, S0=band matrices) with arbitrary perturbations are then perturbatively solvable, with R constructed in terms of auxiliary matrix continued fraction fn. Alternatively, a ‘‘generalized unperturbed spectrum’’ mathn may be required as an input: The algebraically constructed asymptotics of the fn’s play this role in our Padé examples. Due to S≠I, the‘‘Sturmians’’ may also be constructed. In the test evaluations of the binding energies and/or couplings, the simultaneous upper and lower bounds of high precision are shown to be numerically obtainable.
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03.65.Ge Solutions of wave equations: bound states
03.65.Fd Algebraic methods

The exterior metric approach to a charged axially symmetric celestial body—the fourth‐order approximate solutions of Einstein–Maxwell equations

Zhou Qi‐huang

J. Math. Phys. 29, 2618 (1988); http://dx.doi.org/10.1063/1.528105 (4 pages)

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Starting with the general expression of a static state axisymmetric metric and using the principle of equivalence and the Maccullagh formula, the Einstein–Maxwell equations of a charged axisymmetric celestial body are obtained. Next, using the method of undetermined coefficients these equations are solved up to fourth‐order approximate. These sets of solutions are generally appropriate for all kinds of charged axisymmetric celestial bodies.
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04.20.Jb Exact solutions
95.30.Sf Relativity and gravitation
04.40.-b Self-gravitating systems; continuous media and classical fields in curved spacetime

Collision of impulsive gravitational waves followed by dust clouds

A. H. Taub

J. Math. Phys. 29, 2622 (1988); http://dx.doi.org/10.1063/1.528106 (6 pages) | Cited 8 times

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The evolution of a space‐time containing two colliding plane impulsive gravitational waves each of whose leading edges is followed by a distribution of null dust is determined. The conditions on the Ricci tensor that ensure that the evolution of such a space‐time is unambiguous are determined. These are the same as those that apply in the planar case. The equations of motion of the medium contained in the region of interaction of the dust clouds are determined. These equations determine the change in energy density of each dust cloud as the interaction proceeds and involve the functions whose specifications ensure that the evolution of the space‐time is unambiguous.
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04.30.-w Gravitational waves
04.20.Jb Exact solutions
95.30.Sf Relativity and gravitation
04.20.-q Classical general relativity

The bi‐Hamiltonian formulation of the Landau–Lifshiftz equation

E. Barouch, A. S. Fokas, and V. G. Papageorgiou

J. Math. Phys. 29, 2628 (1988); http://dx.doi.org/10.1063/1.528053 (6 pages) | Cited 11 times

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The Landau–Lifshitz (LL) equation is a universal model for integrable magnetic systems. It contains the sine–Gordon (SG), nonlinear Schrödinger (NLS), and the Heisenberg model (HM) equations as particular or limiting cases. It is well known that the NLS, SG, and HM equations possess recursion operators. A recursion operator of an equation in Hamiltonian form generates (a) a hierarchy of integrable equations, and (b) a second Hamiltonian operator and more generally a hierarchy of Poisson structures. Here the recursion operator of the LL equation is obtained algorithmically, and hence its bi‐Hamiltonian formulation is established.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
75.10.Jm Quantized spin models, including quantum spin frustration
75.10.Dg Crystal-field theory and spin Hamiltonians

Algebraic construction of partition functions for c<1 minimal conformal field theories

P. Suranyi

J. Math. Phys. 29, 2634 (1988); http://dx.doi.org/10.1063/1.528054 (7 pages)

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Partition functions with periodic and twisted boundary conditions are constructed for c<1 minimal conformal field theories using the modular transformation properties of the characters of the Virasoro algebra alone. The construction helps to clarify the connection between twisted partition functions of c<1 and periodic Gaussian conformal field theories.
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11.10.Ef Lagrangian and Hamiltonian approach
11.30.Ly Other internal and higher symmetries

Some global properties and invariance of bundle metrics in the Kaluza–Klein scheme

C. H. Oh, K. Singh, and C. H. Lai

J. Math. Phys. 29, 2641 (1988); http://dx.doi.org/10.1063/1.528055 (12 pages)

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It is shown that the transition functions that give the global structure of the fiber bundle play an important role in the construction of the metric. The invariance properties of this metric under general gauge transformations are discussed and it is found that the usual requirement of a gauge‐invariant metric leads to severe constraints on the gauge fields. To avoid them, it is shown that the metric should instead be covariant with respect to these transformations. Moreover the existence of global actions that are essential in the context of the consistency problem is also discussed. The presence of such actions is studied in both the principal and their associated bundles. In the case of a homogeneous bundle with G/H as the typical fiber, it is shown that a ‘‘spliced’’ bundle with G×N(H)/H as the structure group has to be used. The unified space is then taken as the bundle space of its associated bundle.
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11.10.Kk Field theories in dimensions other than four
11.10.Lm Nonlinear or nonlocal theories and models
11.30.Ly Other internal and higher symmetries
02.40.Ky Riemannian geometries

A simplified model for orbifold compactification

Krzysztof Zabłocki

J. Math. Phys. 29, 2653 (1988); http://dx.doi.org/10.1063/1.528056 (6 pages) | Cited 1 time

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The problem of vacuum stability in orbifold compactification is addressed. The spectrum of fermions and the effective potential in a simple model of compactification on the T2/Z2 orbifold with a topologically nontrivial gauge field background is calculated.
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11.10.Kk Field theories in dimensions other than four
11.10.Ef Lagrangian and Hamiltonian approach

Evaluation of Feynman diagrams in the logarithmic approach to quantum field theory

Carl M. Bender and H. F. Jones

J. Math. Phys. 29, 2659 (1988); http://dx.doi.org/10.1063/1.528057 (7 pages) | Cited 8 times

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The techniques necessary to compute the O(δ2) contributions to the Green’s functions of a scalar field theory with self‐interaction λ(ϕ2)1+δ in d‐dimensional space‐time are developed. The resulting expressions are evaluated explicitly for d=1, 0, and some negative even dimensions. For d=3 and 4 we calculate their leading behavior in terms of a spatial cutoff a.
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11.10.Lm Nonlinear or nonlocal theories and models
11.15.Bt General properties of perturbation theory
11.15.Tk Other nonperturbative techniques
11.10.Ef Lagrangian and Hamiltonian approach

On the Painlevé property of nonlinear field equations in 2+1 dimensions: The Davey–Stewartson system

R. A. Leo, G. Mancarella, G. Soliani, and L. Solombrino

J. Math. Phys. 29, 2666 (1988); http://dx.doi.org/10.1063/1.528058 (6 pages) | Cited 10 times

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With the purpose of clarifying some aspects of the complete integrability of nonlinear field equations, a singular‐point analysis is performed of the Davey–Stewartson system, which can be considered as an extension in 2+1 dimensions of the nonlinear Schrödinger equation. It is found that the system under consideration possesses the Painlevé property and allows a set of Bäcklund transformations obtained by truncating the series expansions of the solutions about the singularity manifold.
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11.10.Lm Nonlinear or nonlocal theories and models
03.50.-z Classical field theories
11.30.Cp Lorentz and Poincaré invariance

Periodic reduction of self‐dual Yang–Mills equations

Yoshimasa Nakamura

J. Math. Phys. 29, 2672 (1988); http://dx.doi.org/10.1063/1.528059 (3 pages) | Cited 1 time

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A class of GL(∞)‐invariant self‐dual Yang–Mills (SDYM) equations is considered. It is shown that the GL(∞) SDYM equations are reduced to the GL(n,C) SDYM hierarchy by imposing a periodic condition. This reduction procedure makes clear a relationship between our GL(∞) SDYM equations and the usual infinite matrix representation of a single GL(n,C) SDYM equation.
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11.15.-q Gauge field theories
02.20.-a Group theory

Electromagnetic scattering for a class of anisotropic layered media

Thomas M. Roberts, Harold A. Sabbagh, and L. David Sabbagh

J. Math. Phys. 29, 2675 (1988); http://dx.doi.org/10.1063/1.528060 (7 pages) | Cited 1 time

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The problem of computing fields produced by currents in the presence of unflawed layered media is reduced to quadrature. Each layer either has single‐axis‐type anisotropy or is isotropic. Integral equations are derived for fields in the presence of layered media with flaws.
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03.50.De Classical electromagnetism, Maxwell equations
74.25.N- Response to electromagnetic fields
78.20.-e Optical properties of bulk materials and thin films

A new eight‐vertex model with an infinite number of commensurate phases

Diptiman Sen

J. Math. Phys. 29, 2682 (1988); http://dx.doi.org/10.1063/1.528061 (6 pages)

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A symmetric eight‐vertex model, containing four even vertices with reak weights and four odd vertices with imaginary weights, is found to exhibit an infinite number of commensurate phases. The phase diagram is conjectured to be a complete devil’s staircase similar to that of certain one‐dimensional systems. Associated naturally with the model are two diffeomorphic one‐dimensional maps whose aymptotic trajectories are either stable cycles or intermittently chaotic, depending on the phase.
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64.60.Cn Order-disorder transformations
05.45.-a Nonlinear dynamics and chaos
05.70.Fh Phase transitions: general studies
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

A general representation for the effective dielectric constant of a composite

G. F. Dell’Antonio and V. Nesi

J. Math. Phys. 29, 2688 (1988); http://dx.doi.org/10.1063/1.528062 (7 pages) | Cited 2 times

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In its dependence on the dielectric constants ϵk of its homogeneous components, the effective dielectric constant ϵ∗ of a composite is a function analytic in a domain Ω. Some relevant results about the effective dielectric constant are collected, including the form of Ω, and then a general representation of ϵ∗ as an analytic function is given. In this representation, the dependence on the geometry is separated from the dependence on the ϵ′k’s.
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77.22.Ch Permittivity (dielectric function)
77.90.+k Other topics in dielectrics, piezoelectrics, and ferroelectrics and their properties (restricted to new topics in section 77)
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