JMP Nameplate
  • Volume/Page
  • Keyword
  • DOI
  • Citation
  • Advanced
   
 
 
 

Flickr Twitter UniPHY Group iResearch App Facebook

Search Issue | RSS Feeds RSS
Previous Issue

Dec 1987

Volume 28, Issue 12, pp. 2807-3005

Page 1 of 2 Pages Next Page | Jump to Page

Representations and invariant equations of E(3)

Mayer Humi

J. Math. Phys. 28, 2807 (1987); http://dx.doi.org/10.1063/1.527730 (5 pages)

Full Text: | Download PDF

Show Abstract
Using methods analogous to those introduced by Gel’fand et al. [Representations of the rotation and Lorentz Groups and Their Applications (Pergamon, New York, 1963)] for the Lorentz group the matrix elements for the representations of the Lie algebra of the Euclidean group in three dimensions E(3) are explicitly derived. These results are then used to construct invariant equations with respect to this group and to show, in particular, that the nonrelativistic analog to the Dirac equation is not unique.
Show PACS
02.20.Qs General properties, structure, and representation of Lie groups
02.20.Sv Lie algebras of Lie groups
45.05.+x General theory of classical mechanics of discrete systems
02.40.Dr Euclidean and projective geometries

Weight‐2 zeros of 3j coefficients and the Pell equation

J. D. Louck and P. R. Stein

J. Math. Phys. 28, 2812 (1987); http://dx.doi.org/10.1063/1.527731 (12 pages) | Cited 4 times

Full Text: | Download PDF

Show Abstract
All weight‐2 zeros of the Wigner 3j coefficients may be obtained from the quadratic Diophantine equation known as Pell’s equation. These zeros may then be classified by the orbits of a discrete, infinite‐order subgroup of the Lorentz group SO(1,1). This is carried out by transforming the ‘‘polynomial part’’ of a weight‐2 3j coefficient to Pellian form and obtaining the fundamental zeros numerically. The relation of this polynomial to a family of binary quadratic forms is also given, together with a discussion of the invariance group.
Show PACS
02.20.Qs General properties, structure, and representation of Lie groups
02.20.Rt Discrete subgroups of Lie groups
02.10.De Algebraic structures and number theory
02.10.-v Logic, set theory, and algebra

Construction of space‐time by gauge translations

P. K. Smrz

J. Math. Phys. 28, 2824 (1987); http://dx.doi.org/10.1063/1.527680 (5 pages) | Cited 11 times

Full Text: | Download PDF

Show Abstract
The following procedure is described: Starting with a connection in a principal fiber bundle P(M,G), where G is either the Poincaré group or one of the de Sitter groups, a connection in the bundle of linear frames of a submanifold N of M is constructed by using the translational components of the original connection for frame identification. The dimension of N is gauge dependent, and the flat four‐dimensional Minkowski space‐time may appear of dimension less than 4 when certain gauges are used. It is shown that in the case of a de Sitter group, the minimum dimension to which the flat four‐dimensional space‐time can be reduced is 1, while the number is 0 for the Poincaré group. The gauge transformation that achieves the maximum dimension reduction in the de Sitter case is constant and leads to infinite strings as a result. Variable continuous gauge transformations that can reduce the dimension over a finite region of the base manifold are also considered.
Show PACS
02.40.Ky Riemannian geometries
02.40.Hw Classical differential geometry
02.40.Ma Global differential geometry
04.20.Cv Fundamental problems and general formalism

Quantization: Towards a comparison between methods

G. M. Tuynman

J. Math. Phys. 28, 2829 (1987); http://dx.doi.org/10.1063/1.527681 (12 pages) | Cited 8 times

Full Text: | Download PDF

Show Abstract
In this paper it is shown that the procedure of geometric quantiztion applied to Kähler manifolds gives the following result: the Hilbert space H consists, roughly speaking, of holomorphic functions on the phase space M and to each classical observable f (i.e., a real function on M) is associated an operator f on H as follows: first multiply by f+ 1/4 ℏΔdRf (ΔdR being the Laplace–de Rham operator on the Kähler manifold M) and then take the holomorphic part [see G. M. Tuynman, J. Math. Phys. 27, 573 (1987)]. This result is correct on compact Kähler manifolds and correct modulo a boundary term ∫Mdα on noncompact Kähler manifolds. In this way these results can be compared with the quantization procedure of Berezin [Math. USSR Izv. 8, 1109 (1974); 9, 341 (1975); Commun. Math. Phys. 40, 153 (1975)], which is strongly related to quantization by ∗‐products [e.g., see C. Moreno and P. Ortega‐Navarro; Amn. Inst. H. Poincaré Sec. A: 38, 215 (1983); Lett. Math. Phys. 7, 181 (1983); C. Moreno, Lett. Math. Phys. 11, 361 (1986); 12, 217 (1986)]. It is shown that on irreducible Hermitian spaces [see S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces (Academic, Orlando, FL, 1978] the contravariant symbols (in the sense of Berezin) of the operators f as above are given by the functions f+ 1/4 ℏΔdRf. The difference with the quantization result of Berezin is discussed and a change in the geometric quantization scheme is proposed.
Show PACS
02.40.Ky Riemannian geometries
03.65.Ta Foundations of quantum mechanics; measurement theory
03.70.+k Theory of quantized fields

A class of integrable potentials

Tanaji Sen

J. Math. Phys. 28, 2841 (1987); http://dx.doi.org/10.1063/1.527682 (10 pages) | Cited 7 times

Full Text: | Download PDF

Show Abstract
A class of time independent two‐dimensional integrable potentials, all possessing an invariant of the same general form, is constructed. One of these potentials is superintegrable, its invariants realize the symmetry algebra sO(3) for negative energies, e(2) for zero energy, and sO(2,1) for positive energies. A transformation of coupling constants reveals that in parabolic coordinates this potential is the harmonic oscillator acted on by constant forces. This and another potential in the class may be considered as successive extensions of the Kepler potential. The analytic properties of these integrable systems in the complex time plane are also discussed.
Show PACS
45.05.+x General theory of classical mechanics of discrete systems

Second‐order equation fields and the inverse problem of Lagrangian dynamics

G. Thompson

J. Math. Phys. 28, 2851 (1987); http://dx.doi.org/10.1063/1.527683 (7 pages) | Cited 2 times

Full Text: | Download PDF

Show Abstract
The transformation properties of determined, autonomous systems of second‐order ordinary differential equations, identified as vector fields on the tangent bundle of the space of dependent variables, are derived and studied. The inverse problem of Lagrangian dynamics is studied from this transformation viewpoint as well as the problem of alternative Lagrangians. In particular, regular Lagrangians which are analytic as functions of the first derivatives are considered. Finally, the inverse problem for second‐order systems corresponding to the geodesic flow of a symmetric linear connection is investigated.
Show PACS
45.05.+x General theory of classical mechanics of discrete systems
02.40.Vh Global analysis and analysis on manifolds
11.10.Ef Lagrangian and Hamiltonian approach
02.40.Re Algebraic topology

The inverse scattering problem for the soft ellipsoid

George Dassios

J. Math. Phys. 28, 2858 (1987); http://dx.doi.org/10.1063/1.527684 (5 pages) | Cited 8 times

Full Text: | Download PDF

Show Abstract
A soft triaxial ellipsoid, of unknown semiaxes and orientation, is excited into secondary radiation by a plane acoustic wave of a fixed low frequency. It is proved that one measurement of the leading low‐frequency coefficient and exactly six measurements of the second low‐frequency coefficient of the real part of the forward or the backward scattering amplitude are enough to specify completely both the semiaxes, as well as the orientation of the ellipsoid. Therefore, only the first two low‐frequency coefficients of the real part of the scattering amplitude are needed in order to solve the inverse scattering problem for the soft ellipsoid. For the case of spheroids, the number of measurements is restricted to one for the first and three for the second coefficient. Finally, the sphere is specified by a single measurement of the leading coefficient. The special cases where the orientation or the semiaxes are known are also discussed.
Show PACS
41.20.Jb Electromagnetic wave propagation; radiowave propagation
43.20.Bi Mathematical theory of wave propagation

Integrability of restricted multiple three‐wave interactions

Frank Verheest

J. Math. Phys. 28, 2863 (1987); http://dx.doi.org/10.1063/1.527685 (3 pages) | Cited 5 times

Full Text: | Download PDF

Show Abstract
Using a Hamiltonian framework with complex canonical variables allows for the determination of irreducible forms, which serve as building blocks for polynomial invariants. All the independent invariants in involution are thus obtained for the restricted multiple three‐wave interactions, where all triads are coupled through a common pump (or daughter) wave, in the case of equal coupling strengths in all triads. The mixed, common pump/daughter wave case is not integrable.
Show PACS
41.20.Jb Electromagnetic wave propagation; radiowave propagation
45.05.+x General theory of classical mechanics of discrete systems

Quantum observables: Compatibility versus commutativity and maximal information

Paul Busch, Thomas P. Schonbek, and Franklin E. Schroeck

J. Math. Phys. 28, 2866 (1987); http://dx.doi.org/10.1063/1.527686 (7 pages) | Cited 6 times

Full Text: | Download PDF

Show Abstract
Two different approaches to a characterization of the degree of (in)compatibility of quantum observables are investigated. First, recent examples of the (partial) commutativity of spectral measures of incompatible observables are proved to be generic. The analysis is extended to the case of compatible or incompatible unsharp, or stochastic observables, leading to a general criterion for commutativity of position and momentum effects. Further, a recently proposed information theoretic quantification of the (in)compatibility of noncommuting observables is generalized, and the relation between ‘‘maximal information,’’ ‘‘minimal uncertainty,’’ partial commutativity, and strict correlation is further clarified. Both approaches are illustrated in a number of examples.
Show PACS
03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Ca Formalism

The problem of moments in the phase‐space formulation of quantum mechanics

Francis J. Narcowich

J. Math. Phys. 28, 2873 (1987); http://dx.doi.org/10.1063/1.527687 (10 pages) | Cited 13 times

Full Text: | Download PDF

Show Abstract
Long ago, Moyal [Proc. Cambridge Philos. Soc. 45, 99 (1949)] formulated a moment problem in the context of the Wigner–Weyl phase‐space formulation of quantum mechanics. The problem amounts to giving necessary and sufficient conditions for a sequence of numbers to be moments of a Wigner function. In this paper, that problem is solved, and so is a truncated version of it.
Show PACS
03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Db Functional analytical methods
03.65.Sq Semiclassical theories and applications
02.30.Sa Functional analysis

Composite systems in quaternionic quantum mechanics

C. G. Nash and G. C. Joshi

J. Math. Phys. 28, 2883 (1987); http://dx.doi.org/10.1063/1.527688 (3 pages) | Cited 9 times

Full Text: | Download PDF

Show Abstract
The natural composition of systems in quaternionic quantum mechanics is examined via their lattices of propositions and it is shown that the criticisms that have been made of such a composition are unconvincing.
Show PACS
03.65.Ta Foundations of quantum mechanics; measurement theory

Component states of a composite quaternionic system

C. G. Nash and G. C. Joshi

J. Math. Phys. 28, 2886 (1987); http://dx.doi.org/10.1063/1.527689 (5 pages) | Cited 8 times

Full Text: | Download PDF

Show Abstract
The problem of finding component states given a composite state is examined for quaternionic quantum mechanics. It is shown that under very loose conditions the component state is forced to be complex.
Show PACS
03.65.Ta Foundations of quantum mechanics; measurement theory

Absolutely continuous spectra of quasiperiodic Schrödinger operators

Luigi Chierchia

J. Math. Phys. 28, 2891 (1987); http://dx.doi.org/10.1063/1.527690 (8 pages) | Cited 7 times

Full Text: | Download PDF

Show Abstract
Several aspects of the general and constructive spectral theory of quasiperiodic Schrödinger operators in one dimension are discussed. An explicit formula for the absolutely continuous (a.c.) spectral densities that yields an immediate proof of the fact that the Kolmogorov–Arnold–Moser (KAM) spectrum constructed by Dinaburg, Sinai, and Rüssmann [Funkt. Anal. Prilozen. 9, 8 (1975); Ann. Acad. Sci. 357, 90 (1980)] is a subset of the a.c. spectrum is provided. Some quasiperiodicity properties of the Deift–Simon a.c. eigenfunctions are proved, namely, that the normalized phase of such eigenfunctions is a quasiperiodic distribution. In the constructive part the Dinaburg–Sinai–Rüssmann theory is extended to quasiperiodic perturbations of periodic Schrödinger operators using a KAM Hamiltonian formalism based on a new treatment of perturbations of harmonic oscillators. Particular attention is devoted to the dependence upon the eigenvalue parameter and a complete control of KAM objects is achieved using the notion of Whitney smoothness.
Show PACS
03.65.Db Functional analytical methods
02.30.Sa Functional analysis
02.30.Tb Operator theory

On the construction of perfect Morse functions on compact manifolds of coherent states

S. Berceanu and A. Gheorghe

J. Math. Phys. 28, 2899 (1987); http://dx.doi.org/10.1063/1.527691 (9 pages) | Cited 3 times

Full Text: | Download PDF

Show Abstract
Perfect Morse functions on the manifold of coherent states are effectively constructed. The case of a compact, connected, simply connected Lie group of symmetry, having the same rank as the stationary group of the manifold of coherent states, such that the manifold of coherent states is a Kählerian C‐space, is considered. It is proved that the set of perfect Morse functions is dense in the set of energy functions for linear Hamiltonians in the elements of the Cartan algebra of the Lie algebra of the representation of the group considered. It is proved that the maximum number of orthogonal vectors on a coherent vector manifold is equal to the Euler–Poincaré characteristic of the manifold.
Show PACS
03.65.Fd Algebraic methods
03.65.Db Functional analytical methods
02.20.Qs General properties, structure, and representation of Lie groups
02.40.Ky Riemannian geometries

Exponential time‐evolution operator for the time‐dependent harmonic oscillator

Francisco M. Fernández

J. Math. Phys. 28, 2908 (1987); http://dx.doi.org/10.1063/1.527819 (1 page) | Cited 13 times

Full Text: | Download PDF

Show Abstract
The time‐evolution operator for the time‐dependent harmonic oscillator H= (1)/(2) {α(t)p2 +β(t)q2} is exactly obtained as the exponential of an anti‐Hermitian operator. The method is based on the equations of motion for the coordinate and momentum operators in the Heisenberg representation. The problem is reduced to solving the classical equations of motion.
Show PACS
03.65.Fd Algebraic methods

Quadratic zeros of Racah 6j coefficients: A geometrical approach

J. J. Labarthe

J. Math. Phys. 28, 2909 (1987); http://dx.doi.org/10.1063/1.527692 (5 pages) | Cited 1 time

Full Text: | Download PDF

Show Abstract
It is shown that the projective symmetries of the polynomial Φ of quadratic 6j coefficients form the symmetrical group S6. Nonlinear rational symmetries of Φ are found. Partial parametrizations of the zeros of Φ are presented.
Show PACS
03.65.Fd Algebraic methods
02.10.De Algebraic structures and number theory
02.20.Qs General properties, structure, and representation of Lie groups
02.20.Sv Lie algebras of Lie groups

On discrete Schrödinger equations and their two‐component wave equation equivalents

Alfred M. Bruckstein and Thomas Kailath

J. Math. Phys. 28, 2914 (1987); http://dx.doi.org/10.1063/1.527693 (11 pages) | Cited 5 times

Full Text: | Download PDF

Show Abstract
An approach to inverse scattering problems for discrete Schrödinger equations, which are discrete three‐term recursions, is presented by systematically transforming them into discrete two‐component wave‐propagation equations. The wave‐propagation equations permit the immediate application of certain computationally efficient and physically insightful ‘‘layer‐peeling’’ algorithms for inverse scattering. The mapping of three‐term recursions to two‐component evolution equations is one to many, because the relation between the ‘‘potential’’ sequence parametrizing Schrödinger equations and the ‘‘reflection coefficient’’ sequence determining local wave interaction is a nonlinear difference equation. This mapping is examined in some detail and it is used to study both direct and inverse scattering problems associated with discrete Schrödinger equations.
Show PACS
03.65.Nk Scattering theory
03.65.Ge Solutions of wave equations: bound states

Time‐dependent canonical formalism of thermally dissipative fields and renormalization scheme

I. Hardman, H. Umezawa, and Y. Yamanaka

J. Math. Phys. 28, 2925 (1987); http://dx.doi.org/10.1063/1.527694 (14 pages) | Cited 16 times

Full Text: | Download PDF

Show Abstract
The canonical formalism of thermally dissipative semifree fields in the time‐dependent situation is presented. The use of thermal covariant derivatives simplifies the formulation considerably. With this formalism one can unambiguously obtain the interaction Hamiltonian under any thermal situation which together with the free propagator enables perturbative calculations to be performed. The ‘‘on‐shell’’ renormalization condition in the time‐dependent case is also discussed. The model of a system with a thermal reservoir illustrates how the present formalism works in time‐dependent situations.
Show PACS
03.70.+k Theory of quantized fields
05.70.Ln Nonequilibrium and irreversible thermodynamics
05.30.-d Quantum statistical mechanics
11.10.-z Field theory

There is no isolated p‐p wave

P. F. Yip

J. Math. Phys. 28, 2939 (1987); http://dx.doi.org/10.1063/1.527695 (3 pages)

Full Text: | Download PDF

Show Abstract
A theorem is presented that basically states that there are no nontrivial well‐behaved, spatially asymptotically flat space‐times which are p‐p waves at infinity.
Show PACS
04.20.Cv Fundamental problems and general formalism

Some solutions of Einstein’s equations with shock waves

A. M. Anile, G. Moschetti, and O. I. Bogoyavlenski

J. Math. Phys. 28, 2942 (1987); http://dx.doi.org/10.1063/1.527696 (7 pages) | Cited 5 times

Full Text: | Download PDF

Show Abstract
The Einstein field equations in the presence of a polytropic fluid performing self‐similar motion are reduced to a dynamical system. Qualitative properties of the dynamical system are investigated in the case when the fluid motion is with shock waves.
Show PACS
04.20.Jb Exact solutions
95.30.Sf Relativity and gravitation
98.80.Cq Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.)

General exact solutions of Einstein equations for static perfect fluids with spherical symmetry

Sonia Berger, Roberto Hojman, and Jorge Santamarina

J. Math. Phys. 28, 2949 (1987); http://dx.doi.org/10.1063/1.527697 (2 pages) | Cited 18 times

Full Text: | Download PDF

Show Abstract
The gravitational field equations for a spherical symmetric perfect fluid are completely solved. The general analytical solution obtained depends on an arbitrary function of the radial coordinate. As illustrations of the proposed procedure the exterior and interior Schwarzschild solutions are regained.
Show PACS
04.20.Jb Exact solutions
04.40.-b Self-gravitating systems; continuous media and classical fields in curved spacetime

Colliding gravitational plane waves with noncollinear polarization. II

Frederick J. Ernst, Alberto García D., and Isidore Hauser

J. Math. Phys. 28, 2951 (1987); http://dx.doi.org/10.1063/1.527698 (10 pages) | Cited 20 times

Full Text: | Download PDF

Show Abstract
A simple criterion for colliding gravitational plane waves is developed. This colliding wave condition is preserved by a new realization of the Geroch group augmented by a Kramer–Neugebauer involution. A three‐parameter generalization of a two‐parameter family of solutions with noncollinear polarization discovered recently by Ferrari, Ibañez, and Bruni is presented, and two additional solutions are derived that demonstrate that much larger families are likely to be constructed in the near future.
Show PACS
04.30.-w Gravitational waves
04.20.Jb Exact solutions

The generalized Weyl correspondence and time‐dependent stochastic processes

M. Gadella, L. M. Nieto, J. M. Noriega, and E. Santos

J. Math. Phys. 28, 2961 (1987); http://dx.doi.org/10.1063/1.527699 (12 pages) | Cited 1 time

Full Text: | Download PDF

Show Abstract
The definition and some general properties of the generalized Weyl correspondence between stochastic processes and operator‐valued real functions on a Hilbert space plus a trace class operator are given. The relation between the derivatives of an operator‐valued function and the derivative of the corresponding stochastic process are studied. When the operator‐valued function is the position (or momentum) in the Heisenberg picture, a condition for the positivity of the joint distribution functions of the corresponding process is given, provided that the evolution Hamiltonian be quadratic in the position and momentum. Finally, the case of an arbitrary Hamiltonian evolution for the position operator is studied and the two‐dimensional density functions of the process is related to the Wigner function associated to some state ρ, and a necessary condition for the positivity of the densities is given.
Show PACS
05.30.-d Quantum statistical mechanics
03.65.Db Functional analytical methods
03.65.Fd Algebraic methods
02.50.Ey Stochastic processes

Effective potential determining one‐dimensional Slater sum in independent‐particle theory

N. H. March

J. Math. Phys. 28, 2973 (1987); http://dx.doi.org/10.1063/1.527700 (2 pages) | Cited 1 time

Full Text: | Download PDF

Show Abstract
Hilton, March, and Curtis have earlier focused on the utility of the effective potential U(r,β) in determining the Slater sum Z(r,β)=Z0(β)exp(−βU(r,β)). Here an explicit, though highly nonlinear, partial differential equation is derived for determining the effective potential U(x,β) in one‐dimensional problems. Direct solution of this equation by power series expansion in β leads readily to Husimi’s results obtained from off‐diagonal density matrix calculations. Perturbation theory in the potential is also developed, and thereby it is shown that an infinite subseries of the Husimi expansion is readily summed, and that a scaling property is exhibited.
Show PACS
05.30.-d Quantum statistical mechanics
02.30.Mv Approximations and expansions
02.30.Em Potential theory
02.30.Jr Partial differential equations

Electron density and Slater sum of a semi‐infinite electron gas in d dimensions

N. H. March

J. Math. Phys. 28, 2975 (1987); http://dx.doi.org/10.1063/1.527701 (2 pages) | Cited 2 times

Full Text: | Download PDF

Show Abstract
By taking the electron densities of semi‐infinite electron gases in one and in three dimensions, and forming the Slater sum by the Laplace transform, it is shown that the Slater sum is the classical partition function in d dimensions, times a function independent of dimensionality. The electron density is thereby calculated for general dimensionality, as is the kinetic energy density. As a by‐product, the dimensionality dependence of Friedel oscillations emerges in general form.
Show PACS
05.30.-d Quantum statistical mechanics
02.30.Mv Approximations and expansions
02.30.Em Potential theory
02.30.Jr Partial differential equations
Page 1 of 2 Pages Next Page | Jump to Page
Close

Google Calendar

ADVERTISEMENT

Your Recent History Recent History Help

Recently Viewed

  1. Erratum: Comment on an aspect of a paper by G. Thompson [J. Math. Phys. 27, 153 (1986)]
  2. Nonlinear equations with superposition formulas and the exceptional group G2. I. Complex and real forms of g2 and their maximal subalgebras
  3. Erratum: Imparting to a Bianchi type II space‐time [J. Math. Phys. 27, 417 (1986)]
  4. Statistical filter theory and the O’Doherty–Anstey effect: Dependence on offset
  5. Erratum: On a labeling for point group harmonics. I and II [J. Math. Phys. 26, 2413, 2441 (1985)]
Go to AIP Home
Copyright © American Institute of Physics
close