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Dec 1989

Volume 30, Issue 12, pp. 2735-3008

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Deformations of the Galilean algebra

José M. Figueroa‐O’Farrill

J. Math. Phys. 30, 2735 (1989); http://dx.doi.org/10.1063/1.528506 (5 pages) | Cited 5 times

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All the infinitesimal deformations of the Galilean algebra with and without central extension are computed, as well as their integrability properties. Among the four‐parameter family of infinitesimal deformations of the unextended algebra is found the Newton algebras, the Euclidean algebra E(4), the Poincaré algebra, the de Sitter algebras, and SO(5). For the centrally extended algebra there is found, in particular, an infinitesimal deformation containing a Poincaré subalgebra (although the embedding is not the natural one), and centrally extended versions of the Newton algebras.
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02.10.-v Logic, set theory, and algebra

Multivariable Meixner, Krawtchouk, and Meixner–Pollaczek polynomials

M. V. Tratnik

J. Math. Phys. 30, 2740 (1989); http://dx.doi.org/10.1063/1.528507 (10 pages) | Cited 10 times

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A multivariable biorthogonal generalization of the Meixner, Krawtchouk, and Meixner–Pollaczek polynomials is presented. It is shown that these are orthogonal with respect to subspaces of lower degree and biorthogonal within a given subspace. The weight function associated with the Krawtchouk polynomials is the multivariate binomial distribution.
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02.10.De Algebraic structures and number theory

Algebras connected with the Zn elliptic solution of the Yang–Baxter equation

Hou Bo‐yu and Wei Hua

J. Math. Phys. 30, 2750 (1989); http://dx.doi.org/10.1063/1.528508 (6 pages) | Cited 11 times

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The quantum and classical algebras connected with the Zn‐symmetric elliptic solution of the Yang–Baxter equation are derived; their structure constants and the relations between the quantum algebra and the classical one are investigated in detail. Moreover, the trigonometric limit of these algebras is worked out.
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02.20.-a Group theory
05.20.-y Classical statistical mechanics

The four sets of additive quantum numbers of SU(3)

J. Patera

J. Math. Phys. 30, 2756 (1989); http://dx.doi.org/10.1063/1.528509 (7 pages) | Cited 9 times

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The four maximal sets of additive quantum numbers and related fine gradings of the Lie algebra sl(3,C) are described in detail. The quantum numbers are determined by a grading of the Lie algebra, fine gradings providing maximal sets of them. Two sets of additive quantum numbers are equivalent precisely if the corresponding gradings are equivalent under a transformation from the automorphism group of the Lie algebra.
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02.20.-a Group theory
03.65.-w Quantum mechanics

New gradings of sl(3,C) representations

J. Patera, R. T. Sharp, and J. Van Der Jeugt

J. Math. Phys. 30, 2763 (1989); http://dx.doi.org/10.1063/1.528510 (7 pages) | Cited 6 times

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Two new gradings of the Lie algebra sl(3,C) are the refined sl(2,C) and o(3) gradings. The grading in each case utilizes the sl(2,C), or o(3), weight together with a new additive modular label. A complete set of sl(3,C) representation basis states labeled by each to the two sets of additive quantum numbers is found. The newlabeling operator in both cases fails to commute with the cubic sl(3,C) Casimir operator, and hence mixes states of the contragredient representations (p,q) and (q,p)
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02.20.Qs General properties, structure, and representation of Lie groups

Lie algebras associated with scalar second‐order ordinary differential equations

F. M. Mahomed and P. G. L. Leach

J. Math. Phys. 30, 2770 (1989); http://dx.doi.org/10.1063/1.528511 (8 pages) | Cited 25 times

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Second‐order ordinary differential equations are classified according to their Lie algebra of point symmetries. The existence of these symmetries provides a way to solve the equations or to transform them to simpler forms. Canonical forms of generators for equations with three‐point symmetries are established. It is further shown that an equation cannot have exactly r ∊{4,5,6,7} point symmetries. Representative(s) of equivalence class(es) of equations possessing s ∊{1,2,3,8} point symmetry generator(s) are then obtained.
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02.20.Sv Lie algebras of Lie groups
02.30.Hq Ordinary differential equations

A geometric approach to quantum vortices

Vittorio Penna and Mauro Spera

J. Math. Phys. 30, 2778 (1989); http://dx.doi.org/10.1063/1.528512 (7 pages) | Cited 12 times

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In this paper a geometrical description is given of the theory of quantum vortices first developed by Rasetti and Regge [Physica A 80, 217 (1975)] relying on the symplectic techniques of Marsden and Weinstein [J. Phys. D 7, 305 (1983)], and Kirillov–Kostant–Souriau geometric quantization. The RR‐current algebra is interpreted as the natural Hamiltonian algebra associated to a certain coadjoint orbit of the group G=SDiff(R3), the KKS prequantization condition of which is related to the Feynman–Onsager relation. This orbit is also shown to possess a G‐invariant Kaehler structure, whence, in principle, it is possible to quantize it in a natural way.
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02.40.-k Geometry, differential geometry, and topology
47.10.-g General theory in fluid dynamics
05.30.-d Quantum statistical mechanics

New aspects of the path integrational treatment of the Coulomb potential

D. P. L. Castrigiano and F. Stärk

J. Math. Phys. 30, 2785 (1989); http://dx.doi.org/10.1063/1.528513 (4 pages) | Cited 15 times

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The well‐known treatment of the path integral for the Coulomb potential, by means of the Kustaanheimo–Stiefel transformation and a time transformation, is made more transparent by reducing the problem to the equality of two measures on the space of paths. For this equality two proofs are given: an elementary computational one and a short one recurring to general features of stochastic processes. It is shown that the time transformation is a special case of the well‐known time change for a continuous local martingale, by which the process is changed to a Brownian motion and which is determined by its quadratic variation.
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02.50.-r Probability theory, stochastic processes, and statistics
02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)

On asymptotic expansions of twisted products

Ricardo Estrada, José M. Gracia‐Bondía, and Joseph C. Várilly

J. Math. Phys. 30, 2789 (1989); http://dx.doi.org/10.1063/1.528514 (8 pages) | Cited 27 times

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The series development of the quantum‐mechanical twisted product is studied. The series is shown to make sense as a moment asymptotic expansion of the integral formula for the twisted product, either pointwise or in the distributional sense depending on the nature of the factors. A condition is given that ensures convergence and is stronger than previously known results. Possible applications are examined.
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02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)
03.65.Sq Semiclassical theories and applications

Angular reduction in multiparticle matrix elements

D. R. Lehman and W. C. Parke

J. Math. Phys. 30, 2797 (1989); http://dx.doi.org/10.1063/1.528515 (10 pages) | Cited 1 time

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A general method for reduction of coupled spherical harmonic products is presented. When the total angular coupling is zero, the reduction leads to an explicitly real expression in the scalar products of the unit vector arguments of the spherical harmonics. For nonscalar couplings, the reduction gives Cartesian tensor forms for the spherical harmonic products; tensors built from the physical vectors in the original expression. The reduction for arbitrary couplings is given in closed form, making it amenable to symbolic manipulation on a computer. The final expressions do not depend on a special choice of coordinate axes, nor do they contain azimuthal quantum number summations, or do they have complex tensor terms for couplings to a scalar; consequently, they are easily interpretable from the properties of the physical vectors they contain.
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02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)
03.65.Fd Algebraic methods
21.45.-v Few-body systems
23.20.Js Multipole matrix elements

Canonical formalism for path‐dependent Lagrangians. Coupling constant expansions

X. Jaén, R. Jáuregui, J. Llosa, and A. Molina

J. Math. Phys. 30, 2807 (1989); http://dx.doi.org/10.1063/1.528516 (8 pages) | Cited 2 times

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A canonical formalism obtained for path‐dependent Lagrangians is applied to Fokker‐type Lagrangians. The results are specialized for coupling constant expansions and later on are applied to relativistic systems of particles interacting through symmetric scalar and vector mesodynamics and electrodynamics.
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45.05.+x General theory of classical mechanics of discrete systems
03.50.-z Classical field theories
03.30.+p Special relativity

Application of perturbation theory to the damped sextic oscillator

Sunita Srivastava and Vishwamittar

J. Math. Phys. 30, 2815 (1989); http://dx.doi.org/10.1063/1.528464 (4 pages) | Cited 1 time

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Perturbation theory for the anharmonic oscillator with large damping has been used to solve the equation of motion for the damped sextic oscillator. The results so obtained are compared with the values found through numerical integration of the equation of motion.
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45.05.+x General theory of classical mechanics of discrete systems
02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)

Application of spectral deformation to the Vlasov–Poisson system. II. Mathematical results

Peter D. Hislop and John David Crawford

J. Math. Phys. 30, 2819 (1989); http://dx.doi.org/10.1063/1.528465 (19 pages) | Cited 9 times

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This paper presents a mathematical description of the linearized Vlasov–Poisson operator Lk acting on a family of Banach spaces Xp, related to L p (R), and the application of the method of spectral deformation to this model. It is shown that a type‐A analytic family of operators Lk(θ), θ∊C, Lk(0) =Lk can be associated with Lk. By means of this family, the Landau damped modes of the plasma are identified as the spectral resonances of Lk. Existence and uniqueness of solutions to the initial‐value problem for the evolution equation ∂ν g =Lk(θ)g is proven. An expansion of any solution to the initial‐value problem (with sufficiently smooth initial data) is obtained in terms of the eigenfunctions of Lk(θ) and a spectral integral over the essential spectrum. This is applied to derive an expansion for solutions to the Vlasov equation in which the Landau damped portions of the distribution function are manifestly exhibited. A self‐contained discussion of the spectral deformation method and an extension of it to certain closed operators on Banach spaces is also given.
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46.05.+b General theory of continuum mechanics of solids
41.20.Jb Electromagnetic wave propagation; radiowave propagation
52.35.Fp Electrostatic waves and oscillations (e.g., ion-acoustic waves)

Solution of the Dirac equation in Kasner’s space‐time

Sushil K. Srivastava

J. Math. Phys. 30, 2838 (1989); http://dx.doi.org/10.1063/1.528466 (7 pages) | Cited 11 times

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The Dirac equation for the spin‐ 1/2 field in Kasner’s space‐time is discussed and various possible solutions are obtained. For further interpretation of the theory a current vector is derived and Gordon decomposition of the current for massive fields is discussed.
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03.65.-w Quantum mechanics
04.20.-q Classical general relativity

Approximate relativistic quantum Hamiltonians for N interacting particle systems

E. Ruiz

J. Math. Phys. 30, 2845 (1989); http://dx.doi.org/10.1063/1.528467 (9 pages)

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Time evolution of relativistic particle systems can be accomplished by means of a Schrödinger wave equation, provided the Hamiltonian of the N particle system satisfies some commutator equations involving the other generators of a suitable representation of the Poincaré group on the initial data space of the Schrödinger equation. This set of operator equations is solved in the N free particle case. Further, the structure of N interacting particle Hamiltonians is worked out to include second‐order relativistic corrections, i.e., (1/c2)‐terms. It is shown that Breit and Barker–O’Connell Hamiltonians (nonzero spin particles) and the Bažański Hamiltonian (zero‐spin particles) are particular cases of this more general Hamiltonian.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Pm Relativistic wave equations

Quantum mechanics on p‐adic fields

Ph. Ruelle, E. Thiran, D. Verstegen, and J. Weyers

J. Math. Phys. 30, 2854 (1989); http://dx.doi.org/10.1063/1.528468 (21 pages) | Cited 15 times

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A formulation of quantum mechanics on p‐adic number fields is presented. Quantum amplitudes are taken as complex functions of p‐adic variables and it is shown how the Weyl approach to quantum mechanics can be generalized to the p‐adic case. The p‐adic analogs of simple one‐dimensional systems (free particle, compact and noncompact oscillators) are defined by a ‘‘group of motion,’’ which is an Abelian subgroup of SL (2,Qp). In each case the evolution operator is a unitary representation of the appropriate group. Its spectrum is given by characters and its eigenstates are calculated.
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03.65.Ca Formalism
02.20.-a Group theory

Dirac operators with a spherically symmetric δ‐shell interaction

J. Dittrich, P. Exner, and P. Šeba

J. Math. Phys. 30, 2875 (1989); http://dx.doi.org/10.1063/1.528469 (8 pages) | Cited 16 times

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Dirac operators with a contact interaction supported by a sphere are studied restricting attention to the operators that are rotationally and space‐reflection symmetric. The partial wave operators are constructed using the self‐adjoint extension theory, a particular attention being paid to those among them that can be interpreted as δ‐function shells of scalar and vector nature. The class of interactions for which the sphere becomes impenetrable is specified and spectral properties of the obtained Hamiltonians are discussed.
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03.65.Db Functional analytical methods
03.65.Pm Relativistic wave equations

Coherent and thermal coherent state

A. Mann, M. Revzen, K. Nakamura, H. Umezawa, and Y. Yamanaka

J. Math. Phys. 30, 2883 (1989); http://dx.doi.org/10.1063/1.528470 (8 pages) | Cited 8 times

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The characterization of coherent states as the quantum states that split into two uncorrelated beams is considered. The characterization leads to the study of coherent states at finite temperature—thermal coherent states (TCS’s). These TCS’s are defined within the formalism of thermo field dynamics (TFD). TFD allows a generalization of the uncertainty relation that accounts for both thermal and quantum fluctuations. The TCS is shown to be a minimal state for the generalized uncertainty relation.
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03.65.Db Functional analytical methods
42.50.-p Quantum optics

o(4,2)×o(4,2) group theoretical expression of the interelectronic Coulomb potential

E. de Prunelé

J. Math. Phys. 30, 2891 (1989); http://dx.doi.org/10.1063/1.528471 (4 pages) | Cited 6 times

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The interelectronic Coulomb interaction between two electrons is expressed in terms of the o(4,2) generators of each electron. The formulation allows an expansion in terms of scaled hydrogenic (also called Sturmian) states of each electron with respect to a common center if the total orbital angular momentum is different from zero. The formulation is exact in the limit where a dimensionless parameter goes to infinity. Numerical evaluation of matrix elements of the interelectronic potential between hydrogenic configurations illustrates the convergence with respect to this parameter.
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03.65.Fd Algebraic methods
31.15.-p Calculations and mathematical techniques in atomic and molecular physics
02.20.Sv Lie algebras of Lie groups

Gravitational analogs of the Aharonov–Bohm effect

V. B. Bezerra

J. Math. Phys. 30, 2895 (1989); http://dx.doi.org/10.1063/1.528472 (5 pages) | Cited 24 times

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A quantum scalar particle is considered in the following background gravitational fields due to (a) a tubular matter source with axial interior magnetic field and vanishing exterior magnetic field; (b) slowly moving mass currents (weak approximation); and (c) a spinning cosmic string. It is shown that in the flat space‐time around these sources, the energy spectrum and wave function of the particle depend on the amount of matter and magnetic field (tubular matter source case), on the velocity of the moving mass currents, and on the angular momentum in the spinning cosmic string case. These represent gravitational analogs of the Aharonov–Bohm effect in electrodynamics and are due to global (topological) features of the background space‐times under consideration.
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03.65.Ge Solutions of wave equations: bound states
04.60.-m Quantum gravity

Scattering of a wave packet by an interval of random medium

William G. Faris and Woody J. Tsay

J. Math. Phys. 30, 2900 (1989); http://dx.doi.org/10.1063/1.528473 (4 pages) | Cited 2 times

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This paper deals with the scattering theory for the one‐dimensional discrete Schrödinger equation with a random potential having large support. The main result is that a fluctuation deep within the scattering region has a very small effect on the scattering of wave packets; the region of random potential is effectively opaque.
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03.65.Nk Scattering theory
72.15.Rn Localization effects (Anderson or weak localization)
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.50.-r Probability theory, stochastic processes, and statistics

Weyl‐ordered fermions and path integrals

Giuliano M. Gavazzi

J. Math. Phys. 30, 2904 (1989); http://dx.doi.org/10.1063/1.528474 (3 pages)

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It is shown that a correspondence exists between the Weyl‐ordered Hamiltonian and the mid‐point prescription in the discrete path integral for fermions. It is then proven that the Feynman rules obtained from the discrete and continuous path integral are equivalent.
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03.70.+k Theory of quantized fields
03.65.-w Quantum mechanics

Reduction of the Einstein equations in 2+1 dimensions to a Hamiltonian system over Teichmüller space

Vincent Moncrief

J. Math. Phys. 30, 2907 (1989); http://dx.doi.org/10.1063/1.528475 (8 pages) | Cited 76 times

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In this paper the ADM (Arnowitt, Deser, and Misner) reduction of Einstein’s equations for three‐dimensional ‘‘space‐times’’ defined on manifolds of the form Σ×R, where Σ is a compact orientable surface, is discussed. When the genus g of Σ is greater than unity it is shown how the Einstein constraint equations can be solved and certain coordinate conditions imposed so as to reduce the dynamics to that of a (time‐dependent) Hamiltonian system defined on the 12g−12‐dimensional cotangent bundle, T∗T(Σ), of the Teichmüller space, T(Σ), of Σ. The Hamiltonian is only implicitly defined (in terms of the solution of an associated Lichnerowicz equation), but its existence, uniqueness, and smoothness are established by standard analytical methods. Similar results are obtained for the case of genus g=1, where, in fact, the Hamiltonian can be computed explicitly and Hamilton’s equations integrated exactly (as was found previously by Martinec). The results are relevant to the problem of the reduction of the 3+1‐dimensional Einstein equations (formulated on circle bundles over Σ×R and with a spacelike Killing field tangent to the fibers of the chosen bundle) and to the recent discussion by Witten of the possible exact solvability of the ‘‘topological dynamics’’ associated with Einstein’s equations in 2+1 dimensions.
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04.20.Cv Fundamental problems and general formalism
03.65.-w Quantum mechanics

Static ‘‘semi‐plane‐symmetric’’ metrics yielded by plane‐symmetric electromagnetic fields

Jian‐zeng Li and Can‐bin Liang

J. Math. Phys. 30, 2915 (1989); http://dx.doi.org/10.1063/1.528476 (3 pages) | Cited 1 time

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The task of seeking a general static solution to the Einstein–Maxwell equations representing ‘‘semi‐plane‐symmetric’’ metrics yielded by plane‐symmetric electromagnetic fields is reduced to solving a single ordinary differential equation. A special solution is given, showing that there does exist some electrovac metric that does not share some of the symmetries of the electromagnetic fields.
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04.20.Jb Exact solutions

Thermodynamic perfect fluid. Its Rainich theory

Bartolomé Coll and Joan Josep Ferrando

J. Math. Phys. 30, 2918 (1989); http://dx.doi.org/10.1063/1.528477 (5 pages) | Cited 15 times

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The conditions for a relativistic perfect fluid to admit a thermodynamic scheme are considered, and the necessary and sufficient requirements for a metric to define a thermodynamic perfect fluid space‐time are given.
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04.20.-q Classical general relativity
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