Volume 20, Issue 4, April 1979
Index of content:

The Diophantine moment problem and the analytic structure in the activity of the ferromagnetic Ising model
View Description Hide DescriptionWe show that the intensity of magnetizationI (z,x) where z=e ^{−2βH } and x=e ^{−2βJ }, for the ferromagneticIsing model in arbitrary dimension, reduces, for rational values of x, to a Diophantine moment problem I (z) =∑^{∞} _{0} n _{ k } z ^{ k }, where n_{ k }=∫^{Λ} _{0}σ (λ) λ^{ k } dλ, σ (λ) is a positive measure,n _{0}=1/2, and n _{ k } is integer for k≠0. The fact that the n _{ k } 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 cos^{2}(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.

Direct determination of the Langlands decompositions for the parabolic subalgebras of noncompact semisimple real Lie algebras
View Description Hide DescriptionA 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.

Geometrical theory of contractions of groups and representations
View Description Hide DescriptionThe 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.

Formulation of the linearized Vlasov‐fluid model for a sharp‐boundary screw pinch
View Description Hide DescriptionA 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.

Semiclassical quantization of nonseparable systems
View Description Hide DescriptionThe 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.

Gribov degeneracies: Coulomb gauge conditions and initial value constraints
View Description Hide DescriptionWe 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 _{1} sufficiently near =0 has a Gribov copy _{0} then in fact there is a whole neighborhood of _{0} (in the transverse subspace) filled with Gribov copies of transverse potentials near _{1}. 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.

Quartic trace identity for exceptional Lie algebras
View Description Hide DescriptionLet 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 A _{1} and A _{2}, we can prove the validity of a quartic trace identity Tr(X ^{4}) =K (λ)[Tr(X ^{2})]^{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.

Spectral and scattering theory for the adiabatic oscillator and related potentials
View Description Hide DescriptionWe consider the Schrödinger operator H=−Δ+V (r) on R ^{ n }, where V (r) =a sin(b r ^{α})/r ^{β}+V _{ S }(r), V _{ S }(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 Mo/ller wave operators. Our results are obtained by establishing the asymptotic behavior of solutions of the equationH u=z u for complex values of z.

A note on the iteration of the Chandrasekhar nonlinear H‐equation
View Description Hide DescriptionAn iteration scheme to solve the Chandrasekhar H equation in the form H (μ) ={1−μ F ^{1} _{0}[Ψ (s) H (s)]/(s +μ) d s}^{−1} is shown to converge monotonically and uniformly.

On the quantization of spin systems and Fermi systems
View Description Hide DescriptionIt 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.

An analytical theory of pulse wave propagation in turbulent media
View Description Hide DescriptionThe 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 e t a l. (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.

Lowering and raising operators of IU(n) and IO(n) and their normalization constants
View Description Hide DescriptionLowering 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.

Quasisteady primitive equations with associated upper boundary conditions
View Description Hide DescriptionThis 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 hydrostaticmodel. 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.

Clebsch–Gordan coefficients: General theory
View Description Hide DescriptionA 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.

Multiplicities for space group representations
View Description Hide DescriptionThe multiplicity formula for nonsymmorphic space group representations is reinvestigated by using explicitly projective representations for the little cogroups P ^{q↘}≃G ^{q↘}/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.

Clebsch–Gordan coefficients for space groups
View Description Hide DescriptionA 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.

Clebsch–Gordan coefficients for P n3n
View Description Hide DescriptionA general method for calculating Clebsch–Gordan coefficients is applied to determine such coefficients for the nonsymmorphic space group P n3n.

Resonances in one‐dimensional Stark effect and continued fractions
View Description Hide DescriptionThe 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.

Linear representations of any dimensional Lorentz group and computation formulas for their matrix elements
View Description Hide DescriptionThe 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.

Generalized Lie algebras
View Description Hide DescriptionThe 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.