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Dec 1991

Volume 32, Issue 12, pp. 3231-3552

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Generalized graphs and unitary irrational central charge in the superconformal master equation

M. B. Halpern and N. A. Obers

J. Math. Phys. 32, 3231 (1991); http://dx.doi.org/10.1063/1.529483 (10 pages) | Cited 2 times

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For each magic basis of Lie g, it is known that the Virasoro master equation on affine g contains a generalized graph theory of conformal level‐families. In this paper, it is found that the superconformal master equation on affine g×SO(dim g) similarly contains a generalized graph theory of superconformal level‐families for each magic basis of g. The superconformal level‐families satisfy linear equations on the generalized graphs, and the first exact unitary irrational solutions of the superconformal master equation are obtained on the sine‐area graphs of g=SU(n), including the simplest unitary irrational central charges c=6nx/(nx+8 sin2(rsπ/n)) yet observed in the program.
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02.20.-a Group theory
11.10.-z Field theory
02.20.Qs General properties, structure, and representation of Lie groups

The realizations of quantum groups of An−1 and Cn types in q‐deformed oscillator systems at classical and quantum levels

Zhe Chang, Jian‐Xiong Wang, and Hong Yan

J. Math. Phys. 32, 3241 (1991); http://dx.doi.org/10.1063/1.529484 (5 pages) | Cited 7 times

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In the classical q‐deformed oscillators system, the Poisson bracket realizations of the quantum enveloping algebras of An−1 and Cn types are given in the symplectic space (V,Ω) (without deformation). When the oscillators system is canonically quantized, the Lie bracket realizations of the quantum enveloping algebras of An−1 and Cn types are obtained. The Hopf structures of these quantum enveloping algebras are supplied.
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02.20.-a Group theory
02.40.-k Geometry, differential geometry, and topology
45.05.+x General theory of classical mechanics of discrete systems
03.65.Fd Algebraic methods

Examples of braided groups and braided matrices

Shahn Majid

J. Math. Phys. 32, 3246 (1991); http://dx.doi.org/10.1063/1.529485 (8 pages) | Cited 69 times

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Matrix braided groups are developed as an analog of the ‘‘coordinate functions’’ on a group or supergroup. The ±1 in the super case is replaced by braid statistics. There are braided group analogs of all the classical simple Lie groups as well as braided matrix groups and braided matrices B(R) for every regular solution R of the quantum Yang–Baxter equations. A direct verification of B(R) is provided and some of the simplest examples are computed in detail.
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02.20.-a Group theory
11.30.Pb Supersymmetry

Symmetry scattering

R. F. Wehrhahn, Yu. F. Smirnov, and A. M. Shirokov

J. Math. Phys. 32, 3254 (1991); http://dx.doi.org/10.1063/1.529486 (7 pages) | Cited 5 times

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Scattering connected with the symmetry of a system is treated. Symmetries related to Riemannian symmetric spaces of the noncompact type are studied. Several examples are discussed.
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02.20.-a Group theory
11.80.-m Relativistic scattering theory

Universal R matrices and invariants of quantum supergroups

R. B. Zhang and M. D. Gould

J. Math. Phys. 32, 3261 (1991); http://dx.doi.org/10.1063/1.529487 (7 pages) | Cited 16 times

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A general method is developed for constructing invariants of quantum supergroups using universal R matrices. Applied to Uq(gl(2/1)), this method yields the invariants of this quantum supergroup in explicit form. The eigenvalues of these invariants in arbitrary irreducible highest weight representations are computed.
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02.20.-a Group theory

Dynkin‐like diagrams and representations of the strange superalgebra P(n)

L. Frappat, A. Sciarrino, and P. Sorba

J. Math. Phys. 32, 3268 (1991); http://dx.doi.org/10.1063/1.529488 (10 pages) | Cited 2 times

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Using an extension P(n) of the noncontragredient superalgebra P(n), on which a nondegenerate Killing form can be constructed, simple root systems (SRSs) are introduced. To each SRS is associated a Dynkin‐like diagram. Then the maximal regular sub‐(super)algebras are obtained. Moreover, a procedure to construct highest weight irreducible representations of P(n) is presented.
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02.20.Sv Lie algebras of Lie groups
02.20.-a Group theory

Transcendentally small reflection of waves for problems with/without turning points near infinity: A new uniform approach

Harry Gingold and Jishan Hu

J. Math. Phys. 32, 3278 (1991); http://dx.doi.org/10.1063/1.529489 (7 pages) | Cited 1 time

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In this paper the generalized Liouville–Green approximation is used to study the wave reflection with a turning point at infinity. The method provided here unifies the work by many authors in finding the nontrivial behavior of the reflection coefficient for high‐energy particles above barrier in the semiclassical limit.
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02.30.Hq Ordinary differential equations
41.20.Jb Electromagnetic wave propagation; radiowave propagation

Hyperbolicity of invariant wave equations

Teymour Darkhosh

J. Math. Phys. 32, 3285 (1991); http://dx.doi.org/10.1063/1.529490 (3 pages)

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The method of characteristics is used to construct the fundamental solution to a system of second‐order partial differential equations. Conditions that the coefficients have to satisfy to preserve the hyperbolicity are determined and the influence of lower‐order terms is indicated. As an example, the Proca wave equation in an external field is analyzed.
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02.30.Jr Partial differential equations
03.65.Pm Relativistic wave equations

Local symplectic operators and structures related to them

I. Ya Dorfman and O. I. Mokhov

J. Math. Phys. 32, 3288 (1991); http://dx.doi.org/10.1063/1.529491 (9 pages) | Cited 3 times

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Matrices with entries being differential operators, that endow the phase space of an evolution system with a (pre)symplectic structure are considered. Special types of such structures are explicitly described. Links with integrability, geometry of loop spaces, and Bäcklund transformations are traces.
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02.30.Jr Partial differential equations
02.20.Tw Infinite-dimensional Lie groups

Steady currents in composite conductors, eigenfunction expansions, and exactly solvable linear integral equations

A. A. Inayat‐Hussain

J. Math. Phys. 32, 3297 (1991); http://dx.doi.org/10.1063/1.529492 (15 pages)

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A novel connection is uncovered between the simple physics of steady current flow in a composite conductor and the theory of integral equations. With a judicious choice of eigenfunction expansions, exploitation of the physical continuity of current flow across a chosen interface in a composite conductor is shown to yield an infinite class of integral equations with exact closed‐form solutions. The mathematical derivation of this class is based on the elementary (but also new) notion of formally equating two different eigenfunction expansions of a given arbitrary function. The new class contains as special cases the celebrated Abel integral equation of classical mechanics and the Kramers–Kronig relations of electromagnetic scattering. But it also contains new integral equations (with exact solutions), some with the Cauchy‐singularity 1/(x−y) in their kernels, and a new summation equation. These new equations are in themselves intriguing and their exact solutions do not appear to be derivable by the known methods for solving integral equations. An application of the new class of integral equations is given in the context of a particular composite conductor, which consists of a semi‐infinite strip imbedded in an otherwise homogeneous whole space conductor (containing a uniform current flow parallel to the strip).
The coefficient in the eigenfunction expansion of the potential in the strip satisfies a one‐dimensional singular integral equation with a Cauchy‐singularity. This singularity is regularized by the application of an integral equation and its exact solution from the new class, resulting in an integral equation with a smooth kernel. This equation together with the eigenfunction expansion provides an elegant representation for the potential in the strip. (The only known exact solutions are for the cases of elliptic‐cylinder and ellipsoid geometries in two and three dimensions, respectively.) The new class of integral equations yields the first examples of singular kernels which possess a bilinear expansion in terms of two different complete sets of eigenfunctions, with only the diagonal terms (i.e., those terms in which the summation indices or integration variables are equal) in the expansion being nonzero. Such an expansion for square‐integrable kernels (as opposed to singular kernels) is well known in the Hilbert–Schmidt theory of Hermitian operators and in Schmidt’s extension to the non‐Hermitian case, and it forms the basis for a method of solving Fredholm integral equations. None of these theories, however, yields the bilinear expansions for the singular kernels of our new class.
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02.30.Rz Integral equations
02.60.-x Numerical approximation and analysis

Error bounds for maximum entropy approximate solutions to Fredholm integral equations

S. Kopeć

J. Math. Phys. 32, 3312 (1991); http://dx.doi.org/10.1063/1.529493 (3 pages) | Cited 1 time

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A new method of calculating a posteriori error bounds for maximum entropy approximate solutions to Fredholm integral equations is introduced. A numerical example confirming the efficacy of the method is also presented.
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02.60.-x Numerical approximation and analysis

Curvature and torsion of the world curve in the action of the relativistic particle

V. V. Nesterenko

J. Math. Phys. 32, 3315 (1991); http://dx.doi.org/10.1063/1.529494 (6 pages) | Cited 10 times

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A generalization of the relativistic particle action is considered. It contains, in addition to the length of the world trajectory, the integrals along the world curve of its curvature and torsion. The generalized Hamiltonian formalism for this model in the D‐dimensional space‐time is constructed. A complete set of the constraints in the phase space is obtained and their division into the first‐class and the second‐class constraints is accomplished. On this basis the canonical quantization of the model is fulfilled. For D=3 the mass spectrum is obtained in the sector without tachyonic states, the mass of the state being dependent on its spin. It is shown that in the framework of this model when D=3 we have the possibility to describe the states with integral, half‐odd‐integral and continuous spins. Interaction with an external Abelian gauge field is introduced in the geometrical way.
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03.30.+p Special relativity
11.90.+t Other topics in general theory of fields and particles (restricted to new topics in section 11)
14.80.-j Other particles (including hypothetical)

Nonlinear resonant scattering and plasma instability: an integrable model

C. Claude, A. Latifi, and J. Leon

J. Math. Phys. 32, 3321 (1991); http://dx.doi.org/10.1063/1.529443 (10 pages) | Cited 32 times

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A detailed study of a system of coupled waves is given for which an initial‐boundary value problem is solved by means of the spectral transform theory. This system represents the nonlinear interaction of an electrostatic high‐frequency wave with the ion acoustic wave in a two component homogeneous plasma. As a result it is understood the plasma instability as (i) a continuous secular transfer of energy from the laser beam to the acoustic wave, (ii) the evolution toward the formation of local singularities of the electrostatic wave (collapsing), (iii) a mutual trapping of the acoustic wave and the scattered Langmuir wave.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation
52.38.Bv Rayleigh scattering; stimulated Brillouin and Raman scattering

Painlevé analysis and Lie group symmetries of the regularized long‐wave equation

David K. Rollins

J. Math. Phys. 32, 3331 (1991); http://dx.doi.org/10.1063/1.529444 (2 pages) | Cited 2 times

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The Painlevé formulation is given for the regularized long‐wave equation, which has been used to describe shallow water waves and drift waves in a plasma. The Lie group symmetries of the equation is then studied and it is shown that the only symmetries are due to the three conserved quantities.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation
47.35.-i Hydrodynamic waves
52.35.Kt Drift waves
02.20.Qs General properties, structure, and representation of Lie groups

Kac–Moody and new infinite‐dimensional Lie algebras

Susumu Okubo

J. Math. Phys. 32, 3333 (1991); http://dx.doi.org/10.1063/1.529445 (6 pages) | Cited 1 time

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It will be shown that general polynomials of the Kac–Moody field can generate either a new infinite‐dimensional Lie algebra or W3 ‐type algebra, at least for the classical field case. For the case of the infinite dimensional Lie algebra, it leads to an infinite number of conserved classical quantities which generate hierarchy equations analogous to that of the KdV system. The chiral boson equation emerges as the lowest‐order equation of the hierarchy.
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03.50.Kk Other special classical field theories
02.20.-a Group theory

On the geometric quantization of Poisson manifolds

Izu Vaisman

J. Math. Phys. 32, 3339 (1991); http://dx.doi.org/10.1063/1.529446 (7 pages) | Cited 11 times

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In a paper by Huebschmann [J. Reine Angew. Math. 408, 57 (1990)], the geometric quantization of Poisson manifolds appears as a particular case of the quantization of Poisson algebras. Here, this quantization is presented straightforwardly. The results include a geometric prequantization integrality condition and its discussion in particular cases such as Dirac brackets, an adaptation of the notion of a polarization and a construction of a quantum Hilbert space, and a computational example. In the last section of the paper the general prequantization representations in the sense of Urwin [Adv. Math. 50, 126 (1983)] are described for the Poisson and Jacobi manifolds.
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03.65.Ca Formalism
02.40.-k Geometry, differential geometry, and topology
02.20.Sv Lie algebras of Lie groups

Quantum dynamics on (super)groups: Constraints and the ‘‘BRST’’ supergroup

V. Aldaya, R. Loll, and J. Navarro‐Salas

J. Math. Phys. 32, 3346 (1991); http://dx.doi.org/10.1063/1.529447 (15 pages) | Cited 1 time

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The smallest supergroup B (K) containing among its generators those of a constraint group K, the BRST charge, and the corresponding ghost and antighost operators is constructed. The supergroup B (K) can be enlarged to include the ordinary dynamical variables of the unconstrained physical system. In a group approach to quantization B (K) generalizes the ordinary U(1) phase invariance of wave functions. In particular this mechanism reproduces the essential features of the BRST quantization. When K is diff S1 the critical values of string theory are those for which the space of polarized functions on the enlarged BRST supergroup is not irreducible.
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03.65.Fd Algebraic methods
11.30.Pb Supersymmetry
11.25.-w Strings and branes

Constants of the motion and diagonalization of the Hamiltonian for some scattering processes

R. K. Colegrave and P. Croxson

J. Math. Phys. 32, 3361 (1991); http://dx.doi.org/10.1063/1.529448 (8 pages) | Cited 3 times

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Complex invariants (for SHO modes) or pairs of real invariants (for RHO modes) are obtained for the frequency converter, the parametric amplifier, and for a three‐mode system that describes Raman or Brillouin scattering. All possible constants of the motion and the diagonalization of the Hamiltonian follow as a consequence. Under the assumption made in this paper, the Hamiltonian has also been considered by Moshinsky and Winternitz, who have established the existence of its reduction to diagonal form. However, it is not easy to select the correct transformation (from the totality comprising the symplectic group) to reduce a given H. This has been done explicitly showing, among other results, that the constants of the motion are similar (often identical) in the stable (frequency converter) domain and in the unstable (amplifier) domain.
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03.65.Fd Algebraic methods

Group theory of the Smorodinsky–Winternitz system

N. W. Evans

J. Math. Phys. 32, 3369 (1991); http://dx.doi.org/10.1063/1.529449 (7 pages) | Cited 89 times

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The three degrees of freedom Smorodinsky–Winternitz system is a degenerate or super‐integrable Hamiltonian that possesses five functionally independent globally defined and single‐valued integrals of the motion in both classical and quantum mechanics. This is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the Hamiltonian under a group of symmetry transformations isomorphic to the three‐dimensional unitary unimodular group, SU(3). In turn, this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other, the so‐called dynamical group. All the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group. So, in common with the hydrogen atom and the harmonic oscillator, the quantum mechanics of the Smorodinsky–Winternitz system may be completely solved within the framework of group theory alone.
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03.65.Fd Algebraic methods
11.30.-j Symmetry and conservation laws
02.20.-a Group theory

A study of a 90° vortex–vortex scattering process

J. Burzlaff and P. McCarthy

J. Math. Phys. 32, 3376 (1991); http://dx.doi.org/10.1063/1.529450 (5 pages) | Cited 3 times

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Following Ruback, the evidence for scattering at right angle of two vortices in a head‐on collision is discussed. The evidence is given in terms of the approximate solutions of the equations of motion. This makes it possible to extend the analysis to the case of a small net repulsive force between the corresponding static vortex configurations. The ordinary differential equations, which result from the ansatz for the approximate solutions, are solved by Taylor series at the origin and by asymptotic series at infinity.
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03.65.Ge Solutions of wave equations: bound states
11.10.Lm Nonlinear or nonlocal theories and models
11.15.-q Gauge field theories

Path integrals and supercoherent states

M. Chaichian, D. Ellinas, and P. Prešnajder

J. Math. Phys. 32, 3381 (1991); http://dx.doi.org/10.1063/1.529451 (11 pages) | Cited 12 times

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For the real supergroup Osp(1‖2;R), with both its compact and noncompact versions, supercoherent states are introduced with a method close to the one by Perelomov for the even subgroups SU(2) or SU(1,1). These states labeled by a complex c number and Grassmann variable minimize the uncertainty of the quadratic Casimir operator of the group. A path integral formalism is developed for the transition amplitude between supercoherent states for a Hamiltonian linear in the generators of the superalgebra, which leads to a super‐Riccati equation. Finally, in the classical limit the canonical equations of motion are derived which involve a generalized super Poisson bracket.
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03.65.Ge Solutions of wave equations: bound states
02.20.Sv Lie algebras of Lie groups

Upper and lower bounds of the ground state energy of anharmonic oscillators using renormalized inner projection

F. Vinette and J. Čížek

J. Math. Phys. 32, 3392 (1991); http://dx.doi.org/10.1063/1.529452 (13 pages) | Cited 59 times

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Renormalization is a method often used to approximate the eigenvalues of a Hamiltonian that cannot be solved exactly. It consists of splitting the Hamiltonian into a solvable part and a remainder which is then minimized. The inner projection technique, first introduced by Löwdin in the sixties, was developed to bracket the eigenvalues between lower and upper bounds. Combining renormalization and Löwdin’s inner projection yielded the so‐called ‘‘renormalized inner projection technique.’’ In this study, this method will be applied to the quartic, sextic, and octic anharmonic oscillators. Lower and upper energy bounds are obtained for finite values of the coupling constant as well as for the infinite case. The relation between the renormalized inner projection and perturbation theory will also be discussed. Another feature of this study is the importance of symbolic computation in allowing us to manipulate expressions with unevaluated parameters and to perform calculations in rational arithmetics or high decimal precision. Thus Löwdin’s rational approximants can be expressed explicitly as rational fractions in terms of the coupling constant and values for the limit constant can be obtained with amazing high accuracy, namely, 62, 33, and 21 decimal places for the quartic, sextic, and octic oscillator, respectively.
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03.65.Ge Solutions of wave equations: bound states
02.30.Mv Approximations and expansions

Semiclassical Jaynes–Cummings model, Painlevé test, and exact solutions

W.‐H. Steeb, N. Euler, and P. Mulser

J. Math. Phys. 32, 3405 (1991); http://dx.doi.org/10.1063/1.529453 (2 pages)

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The semiclassical model equations for a quantum system of N two‐level atoms interacting with a single‐mode field in a resonant cavity are discussed. For this system of ordinary differential equations the Painlevé test is performed and first integrals are found. It is also shown that exact solutions can be found with the help of the Painlevé test.
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03.65.Sq Semiclassical theories and applications
05.45.-a Nonlinear dynamics and chaos
42.60.Da Resonators, cavities, amplifiers, arrays, and rings

A note on the universal behavior of Schwinger terms in higher dimensional conformal field theories

Wenxin Jiang

J. Math. Phys. 32, 3407 (1991); http://dx.doi.org/10.1063/1.529502 (2 pages)

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It is shown that the structures of the c‐number parts of the Schwinger terms for the current algebra and stress tensor algebra are universal for conformal field theories in higher than two dimensions. New results on the structure of these Schwinger terms are presented. Some related general conclusions will be obtained from these results.
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03.70.+k Theory of quantized fields
02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)

The anomalous Ward identities in gauge and gravitational theories

Wenxin Jiang

J. Math. Phys. 32, 3409 (1991); http://dx.doi.org/10.1063/1.529454 (3 pages) | Cited 1 time

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In this paper a simple derivation of the anomalous Ward identity in the Aa0=0 gauge of Yang–Mills theory is presented. The same method is also applied to the gravitational case and the anomalous Ward identities in both the quasiconformal gauge and the spatial gauge are obtained. The relation between the Schwinger terms and the consistent anomalies is discussed as an application of these Ward identities.
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03.70.+k Theory of quantized fields
02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)
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