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Oct 1980

Volume 21, Issue 10, pp. 2475-2598


Topological solution of ordinary and partial finite difference equations

Adel F. Antippa and Nguyen Ky Toan

J. Math. Phys. 21, 2475 (1980); http://dx.doi.org/10.1063/1.524352 (6 pages) | Cited 3 times

Online Publication Date: 21 July 2008

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Using the discrete path formalism, we obtain a topological solution for ordinary, as well as partial, linear inhomogeneous finite difference equations with variable coefficients and arbitrarily specified boundary conditions. The solution is homomorphic to a set of discrete paths constructed from a set of vectors determined by the level differences of the equation.
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02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
02.30.Ks Delay and functional equations

Finite nonabelian subgroups of SU(n) with analytic expressions for the irreducible representations and the Clebsch–Gordan coefficients

U. Abresch, A. Bovier, O. Lechtenfeld, M. Lüling, V. Rittenberg, and G. Weymans

J. Math. Phys. 21, 2481 (1980); http://dx.doi.org/10.1063/1.524353 (6 pages) | Cited 5 times

Online Publication Date: 21 July 2008

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We present two sequences of finite nonabelian groups which are semidirect products of three Zn groups. Although these groups are not simply reducible (in the tensor product of two irreducible representations an irreducible representation is obtained more than once) we give analytic expressions for the irreducible representations and the Clebsch–Gordan coefficients.
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02.20.Bb General structures of groups

Equivalence of induced representations

P. M. van den Broek and R. Dirl

J. Math. Phys. 21, 2487 (1980); http://dx.doi.org/10.1063/1.524354 (4 pages)

Online Publication Date: 21 July 2008

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Equivalence of induced representations for finite groups is considered in order to determine those equivalence classes of space group representations which are linked by complex conjugation.
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02.20.Qs General properties, structure, and representation of Lie groups

Spinor fields invariant under space–time transformations

J. Beckers, J. Harnad, and P. Jasselette

J. Math. Phys. 21, 2491 (1980); http://dx.doi.org/10.1063/1.524355 (9 pages) | Cited 12 times

Online Publication Date: 21 July 2008

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Spinor fields invariant under the subgroups of the Poincaré group or under the maximal subgroups of the conformal group of space–time are analyzed. It is shown that only certain Poincaré subgroups, all of dimension less than or equal to six, can leave two component spinor fields invariant, with rather severe restrictions on the fields. Tables listing all such invariant fields for subgroups of dimension greater than or equal to four are given. Construction of Dirac spinors and connections between invariant spinors and tensors are discussed: In particular it is shown that from any two‐component spinor invariant under a Poincaré subgroup a real skew‐symmetric tensor invariant under the same group may be constructed.
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02.20.Qs General properties, structure, and representation of Lie groups
02.20.Sv Lie algebras of Lie groups
11.30.Cp Lorentz and Poincaré invariance

Oscillators submitted to squared Gaussian processes

Christian Soize

J. Math. Phys. 21, 2500 (1980); http://dx.doi.org/10.1063/1.524356 (8 pages) | Cited 1 time

Online Publication Date: 21 July 2008

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The paper is a study of oscillators governed by equations of the type amath(t)+bmath(t)+cX(t)=E(t), where a, b, c are given constants and where E(t) is for example the square of a Gaussian stationary process. A constructive and numerical method, using explicit expressions of Fourier transforms, are developed in order to compute the density of the distribution function (d.f.) of X(t) and of the joint distribution of X(t) and math(t). Hence the upcrossing rates of a given level and an approximation of the d.f. of maxt∊TX(t) can be computed. A numerical example is given.
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02.30.Hq Ordinary differential equations
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion

Factorization of operators I. Miura transformations

Allan P. Fordy and John Gibbons

J. Math. Phys. 21, 2508 (1980); http://dx.doi.org/10.1063/1.524357 (3 pages) | Cited 39 times

Online Publication Date: 21 July 2008

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The method of factorization of operators, which has been used to derive the Miura transformation of the KdV equation, is here extended to some third‐order scattering operators, and transformations between several fifth‐order nonlinear evolution equations are derived. Further applications are discussed.
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02.30.Jr Partial differential equations

A modified Bars–Durgut equation with polynomial eigenfunctions

E. Onofri

J. Math. Phys. 21, 2511 (1980); http://dx.doi.org/10.1063/1.524358 (10 pages) | Cited 1 time

Online Publication Date: 21 July 2008

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A singular integral equation in two degrees of freedom is defined. Its structure is similar to Bars–Durgut equation for baryons in two dimensions. It admits polynomial eigenfunctions and its spectrum can be studied exactly. A comparison with numerical data available for the baryon equation shows strong similarities. The asymptotic behavior of the eigenvalues for high quantum numbers is studied in the semiclassical approximation and it is found to be in good agreement with the exact spectrum. A peculiar feature of this model is the presence of a transition from a region of periodic classical orbits with constant frequency (straight Regge trajectories in the spectrum) to a regime of aperiodic orbits (nearly parabolic trajectories).
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02.30.Rz Integral equations
02.30.-f Function theory, analysis
11.10.St Bound and unstable states; Bethe-Salpeter equations
11.55.Jy Regge formalism

Variational principles for particles and fields in Heisenberg matrix mechanics

Abraham Klein, Ching Teh Li, and Moyez Vassanji

J. Math. Phys. 21, 2521 (1980); http://dx.doi.org/10.1063/1.524359 (7 pages) | Cited 8 times

Online Publication Date: 21 July 2008

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For many years we have advocated a form of quantum mechanics based on the application of sum rule methods (completeness) to the equations of motion and to the commutation relations, i.e., to Heisenberg matrix mechanics. Sporadically we have discussed or alluded to a variational foundation for this method. In this paper we present a series of variational principles applicable to a range of systems from one‐dimensional quantum mechanics to quantum fields. The common thread is that the stationary quantity is the trace of the Hamiltonian over Hilbert space (or over a subspace of interest in an approximation) expressed as a functional of matrix elements of the elementary operators of the theory. These parameters are constrained by the kinematical relations of the theory introduced by the method of Lagrange multipliers. For the field theories, variational principles in which matrix elements of the density operators are chosen as fundamental are also developed. A qualitative discussion of applications is presented.
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03.65.Ca Formalism

Dynamical symmetry and magnetic charge

Jonathan F. Schonfeld

J. Math. Phys. 21, 2528 (1980); http://dx.doi.org/10.1063/1.524360 (6 pages) | Cited 13 times

Online Publication Date: 21 July 2008

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By adding to the force between an electric and a magnetic point charge a central force arising from a specially chosen potential, one can construct a system known to have the same SO (3,1) and/or SO (4) dynamical symmetry algebra as the Kepler system. We derive projective changes of variables under which the classical orbits of any such system are put in one‐to‐one correspondence with SO (3,1)‐ and/or SO (4)‐invariant sets of curves on similarly invariant surfaces. This extends results hitherto established only for the Kepler system. This is surprising in that there is a sense in which the phase space of such a magnetic system is a truncation of the Kepler phase space and so one might have expected such global properties not to generalize. Our transformations apparently do not permit transcription of the corresponding Schrödinger equation into a manifestly SO (3,1)‐ and/or SO (4)‐symmetric form, in contrast to the pure Kepler case. Such magnetic systems play roles in the theory of quantum fields in Taub–NUT space‐times, and in the theory of quantum‐mechanical fluctuations about extended magnetic monopoles in supersymmetric gauge theories. In passing, we use the properties of the magnetic systems to formulate a very short and direct proof that the classical orbits of the Kepler system are conic sections.
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03.65.Fd Algebraic methods
45.05.+x General theory of classical mechanics of discrete systems

A factorization of M4: Construction of the principal bundle of orthogonal frames over M4 from O(3,3) spinors

Patrick L. Nash

J. Math. Phys. 21, 2534 (1980); http://dx.doi.org/10.1063/1.524361 (4 pages) | Cited 5 times

Online Publication Date: 21 July 2008

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The principal bundle of orthogonal frames over M4 is explicitly constructed from certain pairs of Ō(3,3)bar)= spinors that transform as (associated) twistors under the action of the covering group of the Poincaré group. In particular, flat space–time is constructed from these associated twistors, and is thus shown to be an object derivable from geometric structures more basic than the vectors of M4. Associated twistors describe massive elementary particles. The position in M4 of such a particle can be explicitly defined in terms of the components of these twistors. The usual momentum and angular momentum variables which coordinatize the classical phase space of this elementary relativistic system of nonzero mass and arbitrary spin may also be realized in terms of this pair of associated twistors. This realization is not equivalent to descriptions of massive particles using twistors which have previously appeared in the literature.
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04.20.Cv Fundamental problems and general formalism

Asymptotic behavior of gravitational fields in a type II coordinate system

Garry Ludwig

J. Math. Phys. 21, 2538 (1980); http://dx.doi.org/10.1063/1.524345 (5 pages) | Cited 1 time

Online Publication Date: 21 July 2008

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With the aid of Penrose’s conformal technique the asymptotic behavior of the components of the metric tensor, the Weyl tensor, the Ricci tensor and the spin‐coefficients is calculated for a large class of space‐times that includes the NUT (Newman–Unti–Tamburino) solution as well as all asymptotically flat space‐times. The calculations are done in a coordinate system associated not with null hypersurfaces but with an asymptotically shearfree twisting null congruence. For vacuum the results presented here reduce to those of Aronson and Newman to the order given in their paper.
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04.20.Jb Exact solutions

Space‐times with geodesic, shearfree, twistfree, nonexpanding rays

Garry Ludwig

J. Math. Phys. 21, 2543 (1980); http://dx.doi.org/10.1063/1.524346 (6 pages) | Cited 4 times

Online Publication Date: 21 July 2008

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The Kundt class of metrics containing geodesic rays with vanishing divergence, shear and curl is obtained for more general Ricci tensors using the standard Newman–Penrose formalism. These solutions are then rederived using Penrose’s conformal technique, thereby clarifying their asymptotic behavior.
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04.20.Jb Exact solutions

Lattice Green’s functions for cubic lattices

M. A. Rashid

J. Math. Phys. 21, 2549 (1980); http://dx.doi.org/10.1063/1.524347 (4 pages) | Cited 5 times

Online Publication Date: 21 July 2008

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The Green’s functions for a cubic lattice given by G(E)=1/π3 ∫0π∫0π∫0π (dx dydz)/[E−ω  (x,y,z)], where (i) ω(x,y,z)=(a1cosx+a2cosy)(1+cosz)+a3cosz, (ii) ω(x,y,z)=a1cosx(1+cosy+cosz+cosycosz) +a2cosy+a3cosz+a23cosycosz are evaluated exactly and expressed as products of two 2F1’s each of which represents a complete elliptic integral of the first kind. The expressions for the Green’s functions manifest the expected symmetries.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

Covariant objects and invariant equations on fiber bundles

Richard Kerner

J. Math. Phys. 21, 2553 (1980); http://dx.doi.org/10.1063/1.524348 (7 pages) | Cited 5 times

Online Publication Date: 21 July 2008

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Let P(M4,G) be a principal fiber bundle over the Minkowskian space–time M4 with the structural group G. The group G is supposed to be a compact and semisimple Lie group. Let A be a connection form on P(M4,G) and F=DA its curvature form. Let gG be the Cartan–Killing metric on G, and gM4 the Minkowskian metric on M4. Let us define dπ : TP→TM4, the differential of the canonical projection from P onto M4. Then we can define a scalar product for any two vectors from P(M4,G): gP(X,Y)=gG(A(X),A(Y))+gM4Q@qL (dπ(X),dπ(Y)). In this metric the horizontal and vertical subspaces of the connection A are orthogonal to each other. Next, we construct the Clifford algebra corresponding to the metric gP. The metric gP can be always diagonalized locally to give diag((3+N)+,1−), where N is the dimension of G. The lowest faithful representation of this algebra, which we call C(3+N,1) is of the dimension K=2[(N+5)/2]. This K‐dimensional vector space is called the space of spinors over P(M4,G). We study the decomposition of these spinors into multiplets of Lorentz spinors. We also define the generalized Dirac equation for such a spinor, construct an explicit representation in the case of G=SU(2), and give the formulae for the mass splitting. Finally, the invariant interaction with vector fields over P(M4,G) and scalar multiplets is discussed, together with the physical implications of the coupled equations.
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11.15.-q Gauge field theories
11.30.Ly Other internal and higher symmetries

S–matrix for interacting A–fields

Tchavdar D. Palev

J. Math. Phys. 21, 2560 (1980); http://dx.doi.org/10.1063/1.524349 (2 pages) | Cited 3 times

Online Publication Date: 21 July 2008

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In the framework of the Lagrangian field theory a new statistics for charged tensor fields is considered. An interaction Lagrangian is constructed such that the S–matrix is unitary, covariant and causal.
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11.10.Ef Lagrangian and Hamiltonian approach
11.55.-m S-matrix theory; analytic structure of amplitudes
02.10.De Algebraic structures and number theory

Unitarity of supergravity and Z2 or Z2×Z2 or Z2×Z2×Z2 gradings of gauge and ghost fields

P. van Nieuwenhuizen

J. Math. Phys. 21, 2562 (1980); http://dx.doi.org/10.1063/1.524350 (5 pages) | Cited 2 times

Online Publication Date: 21 July 2008

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Unitarity can be proven from the usual Z2 grading of gauge and ghost fields, or from a Z2×Z2 grading, geometrically derived by Ne’eman and Thierry‐Mieg, or from a Z2×Z2×Z2 grading derived here. The claim that only the Z2×Z2 grading leads to unitarity is incorrect. The opposite is shown to hold: signs due to different gradings are physically unobservable. We show how the Z2×Z2 grading follows from the Z2 grading by taking a product space.
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11.15.-q Gauge field theories
04.50.-h Higher-dimensional gravity and other theories of gravity
04.60.-m Quantum gravity

Space spinors

Paul Sommers

J. Math. Phys. 21, 2567 (1980); http://dx.doi.org/10.1063/1.524351 (5 pages) | Cited 24 times

Online Publication Date: 21 July 2008

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Just as Maxwell’s electromagnetic field equations govern the evolution of electric and magnetic spatial vectors if some choice of time function has been made, so also the neutrino equation and Dirac equation may be understood as governing the evolutions of certain spatial quantities. In this space‐plus‐time view of the spinor field equations, it is accurate and natural to regard a two‐component are written in 3‐plus‐1 form for both the spinor fields and the corresponding null vector fields. A spatial null vector is of the form M=E+iB, with E⋅E−B⋅B=0=E⋅B, so it is also of the correct algebraic form for describing a null electromagnetic field. The time derivative of a squared neutrino field Ma, however, is ‐i curl Ma+〈M〉cDaMc, compared with simply ‐i curl Ma for a source‐free Maxwell field. Here 〈M〉c is the real spatial unit vector in the neutrino propagation direction E×B, and Da is the spatial covariant derivative.
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03.65.Pm Relativistic wave equations

The Dirac inverse spectral transform: Kinks and boomerons

Jérôme JP. Leon

J. Math. Phys. 21, 2572 (1980); http://dx.doi.org/10.1063/1.524362 (7 pages) | Cited 8 times

Online Publication Date: 21 July 2008

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The inverse spectral transform (IST) is derived when using the eigenvalue problem for the one-dimensional Dirac operator: (D)=iσ3(d/dx)+i(r 0 0-q), σ3=(01-10), where the potentials q and r have nonzero asymptotic values. The method used is of AKNS type. It is shown that the nonlinear evolution equations (NEE) obtained are of differential type at any order (and not of integro-differential type). Some particular solutions are studied, and it is shown that their special behavior is a direct consequence of the nonzero boundary condition on (D).
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03.65.Pm Relativistic wave equations
02.30.Jr Partial differential equations

Parametrizations of unitary matrices and related coset spaces

A. J. Macfarlane

J. Math. Phys. 21, 2579 (1980); http://dx.doi.org/10.1063/1.524363 (4 pages) | Cited 1 time

Online Publication Date: 21 July 2008

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Explicit forms of U3 and U4 matrices as functions of a single adjoint vector are displayed. Parametrizations of the coset spaces U(N+r)/(UN×Ur) are discussed, most explicitly for r=1 and 2, and related, for N=3 and 4, to the results for U3 and U4 matrices.
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11.30.Pb Supersymmetry
02.20.Qs General properties, structure, and representation of Lie groups

Field fluctuations in a two‐phase random medium

Mark J. Beran

J. Math. Phys. 21, 2583 (1980); http://dx.doi.org/10.1063/1.524364 (3 pages) | Cited 9 times

Online Publication Date: 21 July 2008

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We consider here the problem of determining the mean square fluctuations in a statistically homogeneous isotropic two‐phase dielectric random medium. An expression is derived for a weighted sum of the mean square fluctuations in each phase in terms of the effective dielectric constant. From this expression bounds are derived for the mean square fluctuations in each phase. An assumption is then made to allow us to obtain exact expressions for the mean square fluctuations in a particular phase.
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41.20.Cv Electrostatics; Poisson and Laplace equations, boundary-value problems
41.20.Gz Magnetostatics; magnetic shielding, magnetic induction, boundary-value problems

Localized nonuniform patterns in a diffusion‐reaction model with autocatalysis and the Langmuir–Hinshelwood saturation law

L. L. Bonilla and M. G. Velarde

J. Math. Phys. 21, 2586 (1980); http://dx.doi.org/10.1063/1.524365 (4 pages) | Cited 4 times

Online Publication Date: 21 July 2008

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The WKB method is used to predict the onset of localized dissipative structures in a one‐dimensional reactor where a diffusion‐reaction process with autocatalysis and the Langmuir–Hinshelwood (Michaelis–Menten, Holling) saturation law takes place.
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82.20.Fd Collision theories; trajectory models
87.10.-e General theory and mathematical aspects

Resonance of nonaxial symmetric modes in circular microstrip disk antenna

W. C. Chew and J. A. Kong

J. Math. Phys. 21, 2590 (1980); http://dx.doi.org/10.1063/1.524366 (9 pages) | Cited 15 times

Online Publication Date: 21 July 2008

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Resonant frequencies of the nonaxial symmetric modes in a microstrip disk are computed using two approaches: Galerkin’s method and a perturbative approach. The perturbative approach is good when the substrate of the microstrip disk is thin compared to its radius and when the dielectric constant of the substrate is high. Galerkin’s method can be used to compute the resonant frequency to high accuracy but the perturbative approach is more efficient for thin substrate and large dielectric constant. In applying Galerkin’s method, the problem is first cast into a vector dual integral equation using vector Hankel transform (VHT). Using VHT, it is also shown that the magnetic‐wall model is only good when the substrate is of zero thickness. Using zero‐order current distribution on the disk, we also derive the radiation field and radiation pattern. Also, by taking into account the radiation loss, the resonant frequencies are complex. We find discrepancies when we compare our results for the resonant frequency shifts with that obtained by quasistatic approach.
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84.40.Ba Antennas: theory, components and accessories
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