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Dec 1990

Volume 31, Issue 12, pp. 2757-3088

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Hidden symmetry and potential group of the Maxwell fish‐eye

Alejandro Frank, François Leyvraz, and Kurt Bernardo Wolf

J. Math. Phys. 31, 2757 (1990); http://dx.doi.org/10.1063/1.528979 (12 pages) | Cited 7 times

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The Maxwell fish‐eye is an exceptional optical system that shares with the Kepler problem and the point rotor (mass point on a sphere) a hidden, higher rotation symmetry. The Hamiltonian is proportional to the Casimir invariant. The well‐known stereographic map is extended to canonical transformations between of the phase spaces of the constrained rotor and the fish‐eye. Their dynamical group is a pseudoorthogonal one that permits a succint ‘‘4π’’ wavization of the constrained system. The fish‐eye exhibits, unavoidably, chromatic dispersion. Further, a larger conformal dynamical group contains the potential group, that relates the closed, inhomogeneous fish‐eye system to similar, scaled ones. Asymptotically, it is related to free propagation in homogenous media.
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02.20.-a Group theory
42.25.Dd Wave propagation in random media

On Clebsch–Gordan coefficients and matrix elements of representations of the quantum algebra Uq(su2)

V. A. Groza, I. I. Kachurik, and A. U. Klimyk

J. Math. Phys. 31, 2769 (1990); http://dx.doi.org/10.1063/1.528980 (12 pages) | Cited 36 times

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Clebsch–Gordan coefficients and matrix elements of irreducible representations of the quantum algebra Uq(su2) were considered in several papers. In particular, a few expressions for them were derived. An approach to Clebsch–Gordan coefficients and to matrix elements of representations of Uq(su2) on the base of the theory of basic hypergeometric functions is given. This approach allows one to obtain q‐analogs of all well‐known classical expressions for Clebsch–Gordan coefficients (most of them were absent). New symmetry relations, generating functions, and recurrence formulas for Clebsch–Gordan coefficients of Uq(su2) are obtained. Unlike other papers, Clebsch–Gordan coefficients and matrix elements are considered on the base of minimal theoretical constructions (in fact, without using the notion of a C∗ algebra and of a Hopf algebra).
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02.20.-a Group theory
03.65.Fd Algebraic methods

Vector coherent state constructions of U(3) symmetric tensors and their SU(3)⊇SU(2)×U(1) Wigner coefficients

K. T. Hecht and L. C. Biedenharn

J. Math. Phys. 31, 2781 (1990); http://dx.doi.org/10.1063/1.528981 (16 pages) | Cited 8 times

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Generalized vector coherent state constructions of totally symmetric U(3) tensors are used to gain new expressions for the SU(3)⊇SU(2)×U(1) Wigner coefficients for the coupling (λ1μ1)×(λ20)→(λ3μ3). These expressions show how the extremely simple formulas of Le Blanc and Biedenharn, involving a single 9‐j coefficient, arise as special cases of a general result that involves 12‐j coefficients. A simpler general result involving only 9‐j coefficients and K‐normalization factors is derived in a way that can, in principle, be generalized to the generic coupling with multiplicity.
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02.20.-a Group theory

New inhomogeneous boson realizations and inhomogeneous differential realizations of Lie algebras

Hong‐Chen Fu and Chang‐Pu Sun

J. Math. Phys. 31, 2797 (1990); http://dx.doi.org/10.1063/1.528982 (6 pages) | Cited 5 times

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The inhomogeneous boson realizations (IHBR) and the corresponding inhomogeneous differential realizations (IHDR) of Lie algebras, which play an important role in the search of quasi‐exactly solvable problems (QESP) of quantum mechanics, are studied. All possible IHDR of semisimple Lie algebras can be obtained in this way. As examples, the IHBR and the corresponding IHDR of Lie algebras SU(2) and SU(3) are studied in detail.
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02.20.Qs General properties, structure, and representation of Lie groups
03.65.Fd Algebraic methods

Branching rules for a class of typical and atypical representations of gl(m‖n)

M. D. Gould, P. D. Jarvis, and A. J. Bracken

J. Math. Phys. 31, 2803 (1990); http://dx.doi.org/10.1063/1.528983 (8 pages) | Cited 3 times

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The irreducible representations of the Lie superalgebra gl(m‖n) with highest weights of the form Λ=(λ1,λ2,...λm‖ω) are investigated using a recently introduced induced module construction for atypical modules. The gl(m‖n)↓gl(m‖n−1) branching rules are obtained and a suitable Gel’fand–Tsetlin basis is introduced. The class of representations considered includes some multiply atypical irreducible representations of gl(m‖n) and all irreducible representations of gl(m‖1).
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02.20.Qs General properties, structure, and representation of Lie groups
11.30.Pb Supersymmetry

Local realizations of kinematical groups with a constant electromagnetic field. II. The nonrelativistic case

Javier Negro and Mariano A. del Olmo

J. Math. Phys. 31, 2811 (1990); http://dx.doi.org/10.1063/1.528984 (11 pages) | Cited 6 times

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In this paper, nonrelativistic elementary physical systems interacting with constant external electromagnetic fields are studied. The method is to construct a special kind of realizations of the Galilei group, which depend on the electromagnetic field. The linearization of this problem, which consists in obtaining these local realizations via the linear representations of another group, leads to a new representation group: the nonrelativistic Maxwell group. The study of the representations of this group and the related invariant equations completes this work.
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02.20.Qs General properties, structure, and representation of Lie groups
11.30.-j Symmetry and conservation laws

Cohomology theory and deformations of Z2‐graded lie algebras

K. C. Tripathy and M. K. Patra

J. Math. Phys. 31, 2822 (1990); http://dx.doi.org/10.1063/1.528985 (10 pages) | Cited 7 times

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The algebraic cohomology and the spectral sequences for a Z2‐graded Lie algebra are briefly reviewed. The reducibility property of a strongly semisimple Lie superalgebra is established. The role of second and third cohomologies in the deformation of a Lie superalgebra is discussed. Using spectral sequences, the second cohomology of the full BRS algebra is shown to be the ground field and the third cohomology being trivial implies that osp(1,2) is the only graded Lie algebra obtained by deformation of the full BRS algebra. A similar analysis yields the superconformal algebra as a deformation of the super Poincaré algebra. The superconformal algebra so derived contains so(4,1) as the even part, ruling out the existence of negative curvature of a de Sitter universe!
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02.20.Sv Lie algebras of Lie groups

Note on asymptotic series expansions for the derivative of the Hurwitz zeta function and related functions

S. Rudaz

J. Math. Phys. 31, 2832 (1990); http://dx.doi.org/10.1063/1.528986 (3 pages) | Cited 3 times

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Asymptotic series for the Hurwitz zeta function, its derivative, and related functions (including the Riemann zeta function of odd integer argument) are derived as an illustration of a simple, direct method of broad applicability, inspired by the calculus of finite differences.
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02.30.-f Function theory, analysis
02.70.-c Computational techniques; simulations

Integrable Hamiltonian systems related to the polynomial eigenvalue problem

Yunbo Zeng and Yishen Li

J. Math. Phys. 31, 2835 (1990); http://dx.doi.org/10.1063/1.528987 (5 pages) | Cited 23 times

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The independent integrals of motion in involution for the Hamiltonian system related to the second‐order polynomial eigenvalue problem are constructed by using relevant recursion formula. The hierarchy of Hamiltonian systems obtained from the above problem and the time part of the Lax pair are shown to be completely integrable and they are shown to commute with each other. Furthermore, their solution solves the evolution equation associated with the Lax pair.
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02.30.-f Function theory, analysis
03.65.-w Quantum mechanics

Introduction to a covariant theory of special functions of mathematical physics

J. C. Lucquiaud

J. Math. Phys. 31, 2840 (1990); http://dx.doi.org/10.1063/1.528935 (11 pages) | Cited 1 time

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Using harmonic analysis techniquies, the covariant expression of Jacobi and Hermite polynomials on an n‐dimensional space endowed with a metric g of signature (p+, q−) is given. The properties of these polynomials are studied and their relations with the hypergeometric function are given.
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02.30.Gp Special functions
02.40.-k Geometry, differential geometry, and topology
05.20.-y Classical statistical mechanics
11.10.-z Field theory

Complete integrability and analytic solutions of a KdV‐type equation

Zhi‐xiong Chen, Ben‐yu Guo (Pen‐yu, and Long‐wan Xiang

J. Math. Phys. 31, 2851 (1990); http://dx.doi.org/10.1063/1.528936 (5 pages) | Cited 5 times

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The complete integrability of the variable coefficient version of a KdV equation via the Painlevé approach is analyzed. Through the Painlevé–Bäcklund equations, its auto‐Bäcklund transformation, Lax pairs, symmetry, strong symmetry, bilinear form, and analytic solutions are obtained.
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02.30.Jr Partial differential equations
41.20.Jb Electromagnetic wave propagation; radiowave propagation
02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)

Lie algebraic methods and solutions of linear partial differential equations

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

J. Math. Phys. 31, 2856 (1990); http://dx.doi.org/10.1063/1.528937 (8 pages) | Cited 13 times

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In this paper, an algebraic method to obtain the solution of linear partial differential equations of the evolution type is discussed. The proposed method exploits the Lie differential operators and their matrix realization, to reduce the equation to an easily solvable generalized matrix form. Some applications to problems of specific interest are also discussed.
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02.30.Jr Partial differential equations
02.20.Sv Lie algebras of Lie groups
03.65.Fd Algebraic methods

A geometric approach to the path integral formalism of p‐branes

Menelaos S. Kafkalidis

J. Math. Phys. 31, 2864 (1990); http://dx.doi.org/10.1063/1.528938 (8 pages)

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The configuration space for a path integral description of a p‐brane is seen as a vector bundle over moduli spaces. The Einstein condition, applied to such vector bundles over compact Kähler manifolds, provides the required stability conditions. Consequently moduli spaces for such extended objects of higher dimensionality are constructed. Finally a Hermitian metric can be introduced in these moduli spaces.
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02.40.-k Geometry, differential geometry, and topology
11.25.-w Strings and branes

Bäcklund transformations for surfaces in Minkowski space

Bennett Palmer

J. Math. Phys. 31, 2872 (1990); http://dx.doi.org/10.1063/1.528939 (4 pages) | Cited 4 times

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A Bäcklund transformation is constructed between spacelike surfaces of constant negative curvature and timelike surfaces of constant negative curvature in three‐dimensional Minkowksi space. The transformation gives a differential geometric interpretation to a Bäcklund transformation between the elliptic sine‐Gordon equation and the elliptic sinh‐Gordon equation studied by Leibbrandt [J. Math. Phys. 19, (1978)].
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02.40.-k Geometry, differential geometry, and topology

Extremal properties of Synge’s world function and discrete geometry

Yu. A. Rylov

J. Math. Phys. 31, 2876 (1990); http://dx.doi.org/10.1063/1.528940 (15 pages) | Cited 2 times

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Properties of σ space [a set Ω of points P with a real function σ(P,P′) given on Ω] are investigated. A continuity of the set Ω is not necessary and, generally, geometry is discrete. The properties of the world function σ are investigated. At certain (extremal) world function properties the σ space is shown to be a subset of points of Euclidean space or Riemannian space. The presented approach has the peculiarity that no operation other than the function σ is given on σ space. In particular, all such operations as linear operation over vectors, constructing lines and planes, and dimension of the space are expressed through the world function σ and only through it (if it is extremal). A violation of the σ‐space extremality leads to going out beyond the frames of Riemannian geometry (lines are substituted by tubes of lines, etc.). The presented approach can be useful in quantum gravitation, string models, and other problems, where the properties of the event space at small distances are important.
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02.40.Dr Euclidean and projective geometries
02.40.Ky Riemannian geometries
02.30.-f Function theory, analysis

Quantum mechanics as an infinite‐dimensional Hamiltonian system with uncertainty structure: Part I

Renzo Cirelli, Alessandro Manià, and Livio Pizzocchero

J. Math. Phys. 31, 2891 (1990); http://dx.doi.org/10.1063/1.528941 (7 pages) | Cited 15 times

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Schrödinger quantum mechanics is formulated as an infinite‐dimensional Hamiltonian system whose phase space carries an additional structure (uncertainty structure) to account for the probabilistic character of the theory. The algebra of observables is described as an algebra of smooth functions on the quantal phase space, with a product naturally induced by the geometrical structures living on that manifold. The possibility of generalizing Schrödinger mechanics along these lines is discussed.
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02.40.Sf Manifolds and cell complexes
03.65.Ta Foundations of quantum mechanics; measurement theory

Quantum mechanics as an infinite‐dimensional Hamiltonian system with uncertainty structure: Part II

Renzo Cirelli, Alessandro Manià, and Livio Pizzocchero

J. Math. Phys. 31, 2898 (1990); http://dx.doi.org/10.1063/1.528942 (6 pages) | Cited 4 times

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Making reference to the formalism developed in Part I to formulate Schrödinger quantum mechanics, the properties of Kählerian functions in general, almost Kählerian manifolds, are studied.
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02.40.Sf Manifolds and cell complexes
03.65.Ta Foundations of quantum mechanics; measurement theory

Bäcklund transformation, conservation laws, and inverse scattering transform of a model integrodifferential equation for water waves

Y. Matsuno

J. Math. Phys. 31, 2904 (1990); http://dx.doi.org/10.1063/1.528943 (13 pages) | Cited 3 times

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The Bäcklund transformation (BT), an infinite number of conservation laws, and the inverse scattering transform (IST) of a model integrodifferential equation for water waves in fluids of finite depth [Y. Matsuno, J. Math. Phys. 29, 49(1989)] are constructed by employing the bilinear transformation method. The model equation is also shown to pass the Painlevé test. These facts prove the complete integrability of the equation. Both the deep‐ and shallow‐water limits of various results thus obtained are then investigated in detail. In addition, a new method to evaluate conserved quantities for pure N‐soliton is developed by utilizing actively the time part of the BT. It is found that the structure of conservation laws exhibits peculiar characteristics in comparison with those of usual water wave equations such as the Benjamin–Ono and the Korteweg–de Vries equations. The most important problem left open in this paper is to solve various IST equations.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation
02.30.-f Function theory, analysis
02.60.Nm Integral and integrodifferential equations
02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)

Exact plane‐wave solutions of the coupled Maxwell–Klein–Gordon equations

A. Das, T. Biech, and D. Kay

J. Math. Phys. 31, 2917 (1990); http://dx.doi.org/10.1063/1.528944 (4 pages) | Cited 1 time

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Solutions of the classical Maxwell–Klein–Gordon equations are investigated for which the Klein–Gordon field is assumed to be ψ(x)=αeipμxμ. It is shown that for this class the exponential factor can be ‘‘gauged away’’ and the resulting system of equations can be reduced to a single (complicated) nonlinear equation. Furthermore, the electromagnetic four‐potential field becomes massive ‘‘absorbing scalar particles.’’ The steady‐state (or stationary) subclass of the resulting system of equations is examined. It is proved that in absence of any magnetic field, the steady‐state system does not have a solution. In the simple case for which four‐potential components Aμ depend on one spatial coordinate, the equations are completely solved and explicitly analyzed.
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03.50.De Classical electromagnetism, Maxwell equations
03.65.Sq Semiclassical theories and applications

The functional Ito formula in quantum stochastic calculus

G. O. S. Ekhaguere

J. Math. Phys. 31, 2921 (1990); http://dx.doi.org/10.1063/1.528945 (9 pages) | Cited 1 time

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Using an Op‐∗‐algebraic approach, noncommutative analogs of the Ito formula of classical stochastic calculus within the framework of the Hudson–Parthasarathy formulation of Boson quantum stochastic calculus are proven.
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03.65.-w Quantum mechanics

A posterior Schrödinger equation for continuous nondemolition measurement

V. P. Belavkin

J. Math. Phys. 31, 2930 (1990); http://dx.doi.org/10.1063/1.528946 (5 pages) | Cited 32 times

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A continuous model for a nondemolition observation of an atom is given. An equation for the corresponding instrument is found and a stochastic dissipative Schrödinger equation for the unnormalized posterior wave function of the atom is derived. It is shown that the continuously observed isolated atom relaxes to the ground state without mixing.
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03.65.Ta Foundations of quantum mechanics; measurement theory
31.70.Hq Time-dependent phenomena: excitation and relaxation processes, and reaction rates
FREE

Corrections to the coherent state path integral: Comments upon a speculation of L. S. Schulman

T. L. Marchioro

J. Math. Phys. 31, 2935 (1990); http://dx.doi.org/10.1063/1.528947 (11 pages) | Cited 11 times

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Intuitively, the Feynman path integral corresponds to a weighted sum over classical paths, an interpretation that fails for the phase space path integral. To address the question of whether there exists a path integral expression conforming to a sum over paths in phase space, an examination of the discrete coherent state path integral (CSPI) is undertaken. Via an alternative formulation of the CSPI, it is shown that the coherent state action for a broad class Hamiltonians can be transformed from the variables (q,p) to (q,math). For these Hamiltonians, such a transformation along with the inclusion of all terms O(ϵ(zi−zi−1)) yields an expression which, for finite ϵ, can be interpreted as a sum over classical paths with Gaussian weight. The numerical evaluation of this expression through importance sampling (Monte Carlo) is demonstrated.
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03.65.Db Functional analytical methods

Exactness of the supersymmetric JWKB quantization formula

M. Crescimanno

J. Math. Phys. 31, 2946 (1990); http://dx.doi.org/10.1063/1.528948 (6 pages) | Cited 4 times

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A modification of the Fröman and Fröman [JWKB Approximation: Contributions to the Theory (North‐Holland, Amsterdam, 1965)] technique is developed in order to study the supersymmetric JWKB formula of Comtet et al. [Phys. Lett. B 150, 159 (1985)]. With this modification, in a direct and nonperturbative manner, it can be proven that this quantization condition is exact for a class of solvable models.
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03.65.Sq Semiclassical theories and applications

Semiclassical structure of trace formulas

Robert G. Littlejohn

J. Math. Phys. 31, 2952 (1990); http://dx.doi.org/10.1063/1.528949 (26 pages) | Cited 34 times

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Trace formulas provide the only general relations known connecting quantum mechanics with classical mechanics in the case that the classical motion is chaotic. In particular, they connect quantal objects such as the density of states with classical periodic orbits. In this paper, several trace formulas, including those of Gutzwiller, Balian and Bloch, Tabor, and Berry, are examined from a geometrical standpoint. New forms of the amplitude determinant in asymptotic theory are developed as tools for this examination. The meaning of caustics in these formulas is revealed in terms of intersections of Lagrangian manifolds in phase space. The periodic orbits themselves appear as caustics of an unstable kind, lying on the intersection of two Lagrangian manifolds in the appropriate phase space. New insight is obtained into the Weyl correspondence and the Wigner function, especially their caustic structures.
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03.65.Sq Semiclassical theories and applications
02.40.-k Geometry, differential geometry, and topology
05.45.-a Nonlinear dynamics and chaos

How solvable is (2+1)‐dimensional Einstein gravity?

Vincent Moncrief

J. Math. Phys. 31, 2978 (1990); http://dx.doi.org/10.1063/1.528950 (5 pages) | Cited 20 times

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In this paper, the relationship between Witten’s approach to the (2+1)‐dimensional, vacuum Einstein equations (for spatially compact space‐times) and the conventional Annowitt, Deser, and Misner (ADM) Hamiltonian approach is discussed. It is argued (especially for the space‐times with higher genus Cauchy surfaces) that neither approach is complete in itself; Witten’s because it does not provide a technique (even at the classical level) for recovering the space‐time metric and the conventional approach because it provides no mechanism for solving a seemingly intractable set of Hamilton equations. It is also argued, however, that the two formulations are instead complementary in the sense that the Wilson loops, which play a key role in Witten’s approach, provide (at least in principle) a mechanism for solving the reduced Hamilton equations and thereby completing the picture at the classical level. An example of this synthesis for the (explicitly computable) case of genus‐one hypersurfaces is provided. The more tenuous problem of whether this synthesis can be extended to the quantized Einstein equations will also be discussed. A principal open question is whether the Wilson loops, when expressed in terms of the ADM canonical variables, can be ordered in such a way as to preserve, quantum mechanically, their (classical) Poisson bracket algebra.
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04.20.-q Classical general relativity
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