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Apr 1979

Volume 20, Issue 4, pp. 535-756

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The Diophantine moment problem and the analytic structure in the activity of the ferromagnetic Ising model

M. Barnsley, D. Bessis, and P. Moussa

J. Math. Phys. 20, 535 (1979); http://dx.doi.org/10.1063/1.524121 (12 pages) | Cited 12 times

Online Publication Date: 29 July 2008

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We show that the intensity of magnetization I (z,x) where z=e−2βH and x=e−2βJ, for the ferromagnetic Ising model in arbitrary dimension, reduces, for rational values of x, to a Diophantine moment problem I (z) =∑∞0nkzk, where nk=∫Λ0σ (λ) λkdλ, σ (λ) is a positive measure, n0=1/2, and nk is integer for k≠0. The fact that the nk are positive integers puts very stringent constraints on the measure σ (λ). One of the simplest results we obtain is that for Λ<4, σ (λ) is necessarily a finite sum of Dirac δ functions whose support is of the form 4 cos2(pπ/m), p=0,1,2,...,m−1, with m a finite integer. For Λ=4, which correspond to the one‐dimensional Ising model, we have the result that either I (z) is a rational fraction belonging to the previous class Λ<4, or I (z) = (1/2)(1−4z)−1/2 which corresponds precisely to the exact answer for dimension 1. For Λ≳4, which is associated with Ising models in dimension d⩾2 we show that all cases are reducible to Λ=6, by a quadratic transformation which transforms integers into integers and positive measures into positive measures. The fixed point of this type of transformation is analyzed in great detail and is shown to provide a devil’s staircase measure. Various other results are also discussed as well as conjectures.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
75.10.Hk Classical spin models

Direct determination of the Langlands decompositions for the parabolic subalgebras of noncompact semisimple real Lie algebras

J. F. Cornwell

J. Math. Phys. 20, 547 (1979); http://dx.doi.org/10.1063/1.524122 (9 pages) | Cited 13 times

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A direct method for the determination of the Langlands decompositions for the parabolic subalgebras of any noncompact semisimple real Lie algebra is described in detail. The method is based on the canonical form of the Lie algebra. The physically important Lie algebras so(3,2) and so(4,2) are treated as illustrative examples.
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02.20.Sv Lie algebras of Lie groups

Geometrical theory of contractions of groups and representations

Didier Arnal and Jean‐Claude Cortet

J. Math. Phys. 20, 556 (1979); http://dx.doi.org/10.1063/1.524123 (8 pages) | Cited 6 times

Online Publication Date: 29 July 2008

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The contractions of Lie groups and Lie algebras and their representations are studied geometrically. We prove they can be defined by deformations in Poisson algebras of symplectic manifolds on which the groups act. These deformations are given by Dirac constraints which induce on C∞ functions on the deformed manifold an associative twisted product, characterizing the contracted group or its representations. We treat the contractions of SO(n) to E(n) and apply this theory to thermodynamical limits in spin systems.
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02.20.Qs General properties, structure, and representation of Lie groups

Formulation of the linearized Vlasov‐fluid model for a sharp‐boundary screw pinch

H. Ralph Lewis and Leaf Turner

J. Math. Phys. 20, 564 (1979); http://dx.doi.org/10.1063/1.524124 (8 pages) | Cited 1 time

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A theoretical formulation is derived for analyzing linearized equations appropriate to a straight, cylindrical, sharp‐boundary screw pinch within the framework of the Vlasov‐fluid model. Surrounding the plasma is a cylindrical conducting wall, and there is a nonconducting vacuum between the plasma and the wall. By introducing a perturbation‐dependent transformation of the phase space and linearizing about a zeroth‐order state which depends on the perturbation, the linearized equations of Freidberg’s Vlasov‐fluid model are put into a form which would be correct for a hypothetical problem in which the plasma boundary is a rigid cylinder. The effects of the impulsive electric field at the actual perturbed boundary are taken into account in the zeroth‐order state. The boundary conditions at the perturbed plasma boundary are continuity of the normal component of B and vanishing of the normal component of the net current density.
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52.30.-q Plasma dynamics and flow
52.55.Ez Theta pinch

Semiclassical quantization of nonseparable systems

Abraham Klein and Ching‐teh Li

J. Math. Phys. 20, 572 (1979); http://dx.doi.org/10.1063/1.524125 (7 pages) | Cited 12 times

Online Publication Date: 29 July 2008

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The problem of semiclassical quantization of nonseparable systems with a finite number of degrees of freedom is studied within the framework of Heisenberg matrix mechanics, in extension of previous work on one‐dimensional systems. The relationship between the quantum theory and multiply‐periodic classical motions is derived anew. A suitably averaged Lagrangian provides a variational basis not only for the Fourier components of the semiclassical equations of motion, but also for the general definition of action variables. A Legendre transformation to the Hamiltonian verifies that these have been properly chosen and therefore provide a basis for the quantization of nonseparable systems. The problem of connection formulas is discussed by a method integral to the present approach. The action variables are shown to be adiabatic invariants of the classical system. An elementary application of the method is given. The methods of this paper are applicable to nondegenerate systems only.
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03.65.Ca Formalism

Gribov degeneracies: Coulomb gauge conditions and initial value constraints

Vincent Moncrief

J. Math. Phys. 20, 579 (1979); http://dx.doi.org/10.1063/1.524126 (7 pages) | Cited 7 times

Online Publication Date: 29 July 2008

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We discuss, using suitable function spaces, several features of the Gribov degeneracies of non‐Abelian gauge theory. We show that the set of degenerate transverse potentials can be expected to fill entire neighborhoods in the space of transverse potentials. Specifically we show that if a transverse potential Ā1 sufficiently near Ā=0 has a Gribov copy Ā0 then in fact there is a whole neighborhood of Ā0 (in the transverse subspace) filled with Gribov copies of transverse potentials near Ā1. This means that degenerate potentials can be expected to have nonvanishing measure in path integral quantization. We also show how the breakdown of the canonical technique for solving the initial value constraint equations can be circumvented by using a covariant, noncanonical decomposition of the space of electric fields. We prove that the constraint subset of phase space is in fact a submanifold and establish a potentially useful orthogonal decomposition of its tangent space at any point.
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11.15.-q Gauge field theories

Quartic trace identity for exceptional Lie algebras

Susumu Okubo

J. Math. Phys. 20, 586 (1979); http://dx.doi.org/10.1063/1.524127 (8 pages) | Cited 21 times

Online Publication Date: 29 July 2008

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Let X be a representation matrix of generic element x of a simple Lie algebra in generic irreducible representation {λ} of the Lie algebra. Then, for all exceptional Lie algebras as well as A1 and A2, we can prove the validity of a quartic trace identity Tr(X4) =K (λ)[Tr(X2)]2, where the constant K (λ) depends only upon the irreducible representation {λ}, and its explicit form is calculated. Some applications of second and fourth order indices have also been discussed.
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02.20.Sv Lie algebras of Lie groups

Spectral and scattering theory for the adiabatic oscillator and related potentials

Matania Ben‐Artzi and Allen Devinatz

J. Math. Phys. 20, 594 (1979); http://dx.doi.org/10.1063/1.524128 (14 pages) | Cited 10 times

Online Publication Date: 29 July 2008

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See Also: Erratum

Show Abstract
We consider the Schrödinger operator H=−Δ+V (r) on Rn, where V (r) =a sin(brα)/rβ+VS(r), VS(r) being a short range potential and α≳0, β≳0. Under suitable restrictions on α, β, but always including α=β=1, we show that the absolutely continuous spectrum of H is the essential spectrum of H, which is [0,∞), and the absolutely continuous part of H is unitarily equivalent to −Δ. We use these results to show the existence and completeness of the Møller wave operators. Our results are obtained by establishing the asymptotic behavior of solutions of the equation Hu=zu for complex values of z.
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11.80.-m Relativistic scattering theory
02.30.-f Function theory, analysis
03.65.Nk Scattering theory

A note on the iteration of the Chandrasekhar nonlinear H‐equation

R. L. Bowden

J. Math. Phys. 20, 608 (1979); http://dx.doi.org/10.1063/1.524129 (3 pages) | Cited 3 times

Online Publication Date: 29 July 2008

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An iteration scheme to solve the Chandrasekhar H equation in the form H (μ) ={1−μ F 10[Ψ (s) H (s)]/(s +μ) ds}−1 is shown to converge monotonically and uniformly.
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95.30.Jx Radiative transfer; scattering
02.30.Rz Integral equations

On the quantization of spin systems and Fermi systems

Ph. Combe, R. Rodriguez, M. Sirugue‐Collin, and M. Sirugue

J. Math. Phys. 20, 611 (1979); http://dx.doi.org/10.1063/1.524130 (6 pages) | Cited 3 times

Online Publication Date: 29 July 2008

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It is shown that spin operators and Fermi operators can be interpreted as the Weyl quantization of some functions on a ’’classical phase space’’ which is a compact group. Moreover the transition from quantum spin to Fermi operators is an isomorphism of the ’’classical phase space’’ preserving the Haar measure.
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03.65.Ca Formalism

An analytical theory of pulse wave propagation in turbulent media

K. Furutsu

J. Math. Phys. 20, 617 (1979); http://dx.doi.org/10.1063/1.524131 (12 pages) | Cited 11 times

Online Publication Date: 29 July 2008

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The theory of pulse wave propagation in turbulent media is developed starting from the space–time transport equation with the forward‐scattering approximation. The solutions are obtained by a fully analytical method based on the eigenfunction expansion, and the averaged intensity of plane wave pulse is presented by two different expressions for both the Gaussian and Kolmogorov turbulence spectra. These two expressions are given in the series, and the convergence of each series is good when the convergence of the other series is poor; in the case of the Gaussian turbulence spectrum, one of these expressions precisely agrees with the previous one obtained by Sreenivasiah et al. (1976). In connection with the pulse wave width, the pulse moments are evaluated in detail. The resolvent function is fully used to find the eigenvalues and eigenfunctions.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation

Lowering and raising operators of IU(n) and IO(n) and their normalization constants

M. K. F. Wong and Hsin‐Yang Yeh

J. Math. Phys. 20, 629 (1979); http://dx.doi.org/10.1063/1.524105 (5 pages) | Cited 2 times

Online Publication Date: 29 July 2008

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Lowering and raising operators for the vector space U(n) ⊇IU(n) and O(n) ⊇IO(n) have been obtained, and their normalization constants evaluated. For U(n) ⊇IU(n), we obtain two forms, one according to Nagel and Moshinsky, and the other according to Bincer. For O(n) ⊇IO(n),we obtain the shift operators according to Bincer.
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02.20.Qs General properties, structure, and representation of Lie groups

Quasisteady primitive equations with associated upper boundary conditions

Paul Gordon

J. Math. Phys. 20, 634 (1979); http://dx.doi.org/10.1063/1.524106 (25 pages) | Cited 2 times

Online Publication Date: 29 July 2008

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This paper presents another approach to the problem of modeling large scale atmospheric flow. The major thrust of the method is to search for quasi‐steady‐state phenomena. This leads to sets of diagnostic and predictive equations that differ from those presently in use. Another important feature of the analysis is the introduction of a slowly floating upper boundary. In addition to simplifying the question of boundary conditions at the upper boundary, the floating top requires a highly significant change in the set of diagnostic variables. Two possible upper boundary conditions are derived in conjunction with the floating top. The first assumes continuous flow at the upper boundary, while the second assumes a compression‐wave type discontinuity. Two specific criteria are formulated for checking the validity of the quasi‐steady‐state model. One is a scale assumption, between the physical scale and the time scale. The other is the requirement that the solution of the diagnostic equations be the steady‐state limit of the original time‐dependent equations. Various examples are given in order to attempt to clarify the techniques and philosophy of this approach. In addition, a specific test case is solved numerically with three models: The fixed top quasi‐steady‐state model, the floating top quasi‐steady‐state model, and a hydrostatic model. At the same time various upper boundary conditions are tested and compared. The results of the investigation indicate several significant advantages in favor of the floating top quasi‐steady‐state model.
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47.10.-g General theory in fluid dynamics
92.60.-e Properties and dynamics of the atmosphere; meteorology

Clebsch–Gordan coefficients: General theory

R. Dirl

J. Math. Phys. 20, 659 (1979); http://dx.doi.org/10.1063/1.524107 (5 pages) | Cited 32 times

Online Publication Date: 29 July 2008

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A general method is given for obtaining Clebsch–Gordan coefficients for finite groups, by considering the columns of the Clebsch–Gordan matrices as G‐adapted vectors and by identifying the multiplicity index as special column indices of the Kronecker product. The matrix representations are assumed to be projective ones, however not necessarily belonging to equivalent factor systems.
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02.20.Qs General properties, structure, and representation of Lie groups

Multiplicities for space group representations

R. Dirl

J. Math. Phys. 20, 664 (1979); http://dx.doi.org/10.1063/1.524108 (7 pages) | Cited 14 times

Online Publication Date: 29 July 2008

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The multiplicity formula for nonsymmorphic space group representations is reinvestigated by using explicitly projective representations for the little cogroups Pq↘≃Gq↘/T. Thereby useful identities and relations concerning the wave vector selection rules are derived for various cases which may occur for the elements of the Brillouin zone. These relations allow for nearly all cases a closed expression for the multiplicity without reference to a special space group.
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02.20.Qs General properties, structure, and representation of Lie groups

Clebsch–Gordan coefficients for space groups

R. Dirl

J. Math. Phys. 20, 671 (1979); http://dx.doi.org/10.1063/1.524109 (8 pages) | Cited 14 times

Online Publication Date: 29 July 2008

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A general method for finding Clebsch–Gordan coefficients is used to calculate them for nonsymmorphic space groups. This method is based on the fact that the columns of the Clebsch–Gordan matrices can be seen as G‐adapted vectors and that the multiplicity index can be traced back to special column indices of the Kronecker product. Using this method we obtain simple defining equations for the multiplicity index and for nearly all cases without reference to a special space group by a simple calculation the corresponding Clebsch–Gordan matrices.
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02.20.Qs General properties, structure, and representation of Lie groups

Clebsch–Gordan coefficients for Pn3n

R. Dirl

J. Math. Phys. 20, 679 (1979); http://dx.doi.org/10.1063/1.524110 (6 pages) | Cited 8 times

Online Publication Date: 29 July 2008

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A general method for calculating Clebsch–Gordan coefficients is applied to determine such coefficients for the nonsymmorphic space group Pn3n.
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02.20.Qs General properties, structure, and representation of Lie groups

Resonances in one‐dimensional Stark effect and continued fractions

S. Graffi, V. Grecchi, S. Levoni, and M. Maioli

J. Math. Phys. 20, 685 (1979); http://dx.doi.org/10.1063/1.524111 (6 pages) | Cited 12 times

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The Stieltjes type continued fraction (i.e., any diagonal Padé approximants sequence) of the perturbation series for the resonances of the so‐called one‐dimensional Stark effect converges to the resonances.
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02.30.Mv Approximations and expansions

Linear representations of any dimensional Lorentz group and computation formulas for their matrix elements

Takayoshi Maekawa

J. Math. Phys. 20, 691 (1979); http://dx.doi.org/10.1063/1.524112 (21 pages) | Cited 4 times

Online Publication Date: 29 July 2008

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The representation matrix elements of SO(n,1) are discussed in a space spanned by the representation matrix elements of the maximal compact subgroup SO(n). A multiplier of the representation corresponding to the boost of SO(n,1) is completely determined by requiring the commutation relations of SO(n,1) for the differential operators of the multiplier representation and of the parameter group of SO(n). It is shown that the bases of the space, the representation matrix elements of SO(n), are classified by the group chains of the first and the second parameter groups of SO(n), whose differential operators commute with each other, and the characteristic numbers of SO(n,1) are the same as those of the first parameter group of SO(n−1) and a complex number appearing in the multiplier. By using the scalar product defined in the space, the matrix elements for the differential operators and the computation formulas for the representation corresponding to the boost of SO(n,1) are given for all unitary representations of SO(n,1) and useful formulas containing the d matrix elements of SO(n) are obtained. By making use of these results, even for the nonunitary representation of SO(n,1) the matrix elements for the differential operators and the computation formula for the representation corresponding to the boost are obtained by defining the matrix elements with respect to the bases of the space. It is also pointed out that the unitary representations (the complementary series) corresponding to some value of the parameter, which appear in the classification using only the matrix elements of the generators, should not be included in our classification table because of divergence of the normalization integral. The continuation to SO(n+1) and the contraction to ISO(n) from the principal series are discussed.
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02.20.Rt Discrete subgroups of Lie groups

Generalized Lie algebras

M. Scheunert

J. Math. Phys. 20, 712 (1979); http://dx.doi.org/10.1063/1.524113 (9 pages) | Cited 67 times

Online Publication Date: 29 July 2008

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The generalized Lie algebras, which have recently been introduced under the name of color (super) algebras, are investigated. The generalized Poincaré–Birkhoff–Witt and Ado theorems hold true. We discuss the so‐called commutation factors which enter into the defining identities of these algebras. Moreover, we establish a close relationship between the generalized Lie algebras and ordinary Lie (super) algebras.
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02.20.Sv Lie algebras of Lie groups

Relations among generalized Korteweg–deVries systems

W. Symes

J. Math. Phys. 20, 721 (1979); http://dx.doi.org/10.1063/1.524114 (5 pages) | Cited 10 times

Online Publication Date: 29 July 2008

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This report presents certain relations among the completely integrable Hamiltonian systems introduced by Gel’fand and Dikii. These relations generalize a formula of A. Lenard linking the higher‐order Korteweg–deVries equations, of which the Gel’fand–Dikii systems are a generalization. The general form of the relations, which connect the various isospectral deformations of linear differential operators, is described, and two examples are given explicitly.
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02.30.Hq Ordinary differential equations

Null electromagnetic fields in the generalized Einstein–Maxwell field theory

Gregory Walter Horndeski

J. Math. Phys. 20, 726 (1979); http://dx.doi.org/10.1063/1.524115 (7 pages) | Cited 2 times

Online Publication Date: 29 July 2008

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In this paper solutions to the source‐free generalized Einstein–Maxwell field equations with a null electromagnetic field are investigated. It is argued that the principal null congruence of the null electromagnetic field need not be geodesic, shear‐free or a repeated principal null congruence of the gravitational field. However, if the principal null congruence of the null electromagnetic field is geodesic and shear‐free, then it is shown that it must be hypersurface orthogonal, expansion‐free, and a repeated principal null congruence of the gravitational field. The local form of all such solutions of Petrov type III or N is presented.
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04.40.-b Self-gravitating systems; continuous media and classical fields in curved spacetime

The classical limit of quantum dissipative generators

J. V. Pulè and A. Verbeure

J. Math. Phys. 20, 733 (1979); http://dx.doi.org/10.1063/1.524116 (3 pages) | Cited 3 times

Online Publication Date: 29 July 2008

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For finite quantum systems the classical limit of the general dissipative generator of a semigroup of completely positive maps is obtained, yielding a generalized Fokker–Planck generator.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.50.Ga Markov processes

First order estimates of the error in approximate calculations of scattering phase shifts

J. W. Darewych and J. Sokoloff

J. Math. Phys. 20, 736 (1979); http://dx.doi.org/10.1063/1.524117 (4 pages) | Cited 2 times

Online Publication Date: 29 July 2008

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We discuss the question of error estimation in approximate calculations of scattering phase shifts. The Kato integral identity between the exact and approximate solutions is used as the starting point to determine an upper bound to the absolute value of the difference between the exact and approximate result. This bound involves the maximum value of the modulus of the exact wavefunction as a factor. For the potential scattering case it is shown that this maximum occurs in the asymptotic region, if the potential is monotonically decreasing (with decreasing r). For more general potentials simple calculable bounds to the maximum of the wavefunction are derived, for energies which are everywhere higher than the potential. The results are illustrated for scattering by an (attractive) exponential potential and are compared to bounds obtained previously by Bardsley, Gerjuoy, and Sukumar, who have used the Lippman–Schwinger equation to determine bounds to the maximum modulus of the wavefunction.
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11.80.-m Relativistic scattering theory
11.80.Et Partial-wave analysis

On the static Einstein–Maxwell field equations

A. Das

J. Math. Phys. 20, 740 (1979); http://dx.doi.org/10.1063/1.524118 (4 pages) | Cited 4 times

Online Publication Date: 29 July 2008

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The static Einstein–Maxwell field equations are investigated in the presence of both electric and magnetic fields. The sources or bodies are assumed to be of finite size and to not affect the connectivity of the associated space. Furthermore, electromagnetic and metric fields are assumed to have reasonable differentiabilities. It is then proved that the electric and magnetic field vectors are constant multiples of one another. Moreover, the static Einstein–Maxwell equations reduce to the static magnetovac case. If, furthermore, the variational derivation of the Einstein–Maxwell equations is assumed, then both the total electric and magnetic charge of each body must vanish. As a physical consequence it is pointed out that if a suspended magnet be electrically charged then it must experience a purely general relativistic torque.
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04.20.Jb Exact solutions
04.20.Fy Canonical formalism, Lagrangians, and variational principles
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