Volume 26, Issue 7, July 1985
Index of content:

On the algebra of Dirac bispinor densities: Factorization and inversion theorems
View Description Hide DescriptionThe algebraic system formed by Dirac bispinor densities ρ_{ i }≡ψ̄Γ_{ i }ψ is discussed. The inverse problem—given a set of 16 real functions ρ_{ i }, which satisfy the bispinor algebra, find the spinor ψ to which they correspond—is solved. An expedient solution to this problem is obtained by introducing a general representation of Dirac spinors. It is shown that this form factorizes into the product of two noncommuting projection operators acting on an arbitrary constant spinor.

A generalized Molien function for field theoretical Hamiltonians
View Description Hide DescriptionA generatingfunction, or Molien function, the coefficients of which give the number of independent polynomial invariants in G, has been useful in the Landau and renormalizationgroup theories of phase transitions. Here a generalized Molien function for a field theoretical Hamiltonian (with short‐range interactions) of the most general form invariant in a group G is derived. This form is useful for more general renormalization group calculations. Its Taylor series is calculated to low order for the FΓ^{−} _{2} representation of the space group R3̄c and also for the l=1 (faithful) representation of SO(3).

A projection‐based solution to the SP(2N) state labeling problem
View Description Hide DescriptionA projection‐based solution to the symplectic group state labeling problem is presented. The approach yields a nonorthogonal Gel’fand–Tsetlin basis for the irreducible representations of Sp(2n). A method for evaluating the corresponding overlap coefficients is discussed. The action of the Sp(2n) generators, in the basis obtained, is determined and some matrix element formulas are derived. The results obtained are comparable to the matrix element formulas for O(n) and U(n).

On the denominator function for canonical SU(3) tensor operators
View Description Hide DescriptionThe definition of a canonical unit SU(3) tensor operator is given in terms of its characteristic null space as determined by group‐theoretic properties of the intertwining number. This definition is shown to imply the canonical splitting conditions used in earlier work for the explicit and unique (up to ± phases) construction of all SU(3) WCG coefficients (Wigner–Clebsch–Gordan). Using this construction, an explicit SU(3)‐invariant denominator function characterizing completely the canonically defined WCG coefficients is obtained. It is shown that this denominator function (squared) is a product of linear factors which may be obtained explicitly from the characteristic null space times a ratio of polynomials. These polynomials, denoted G ^{ t } _{ q }, are defined over three (shift) parameters and three barycentric coordinates. The properties of these polynomials (hence, of the corresponding invariant denominator function) are developed in detail: These include a derivation of their degree, symmetries, and zeros. The symmetries are those induced on the shift parameters and barycentric coordinates by the transformations of a 3×3 array under row interchange, column interchange, and transposition (the group of 72 operations leaving a 3×3 determinant invariant). Remarkably, the zeros of the general G ^{ t } _{ q }polynomial are in position and multiplicity exactly those of the SU(3) weight space associated with irreducible representation [q−1,t−1,0]. The results obtained are an essential step in the derivation of a fully explicit and comprehensible algebraic expression for all SU(3) WCG coefficients.

Algebraic formulas for some nontrivial U_{ n } 6j symbols and U_{ m n }⊃U_{ m }×U_{ n } 3j m symbols
View Description Hide DescriptionWe give tables of algebraic formulas for some nontrivial 6j symbols and 3j m symbols of the unitary groups. The tables demonstrate that the building‐up method can be used successfully to obtain the rank dependence of unitary group j and j m symbols. To emphasize the rank‐dependent nature of this calculation, we have employed the composite Young tableaux notation (or back‐to‐back notation) to label the unitary group irreps. In using this notation, the transpose conjugate symmetry of the corresponding composite Young diagram leads to a new symmetry of the unitary group 6j and 3j m symbols. The transposition of the groups U_{ m } and U_{ n } gives rise to a further symmetry of the 3j m symbols of U_{ m n }⊃U_{ m }×U_{ n }.

Group theoretical analysis of the Hartree–Fock–Bogoliubov equation. I. General theory
View Description Hide DescriptionThis paper describes a general group theoretical analysis of the temperature–Hartree‐Fock–Bogoliubov (HFB) equation and its solution. The action of the symmetry group G _{0} of the system on the HFB Hamiltonian, the HFB density matrix, and the HFB Green function are defined. It is shown that the HFB equation and its solution is classified by a subgroup G of G _{0}, which is the invariance group of the HFB Hamiltonian, the HFB density matrix, and the HFB Green function corresponding the solution. General expression of the instability of a solution and its decomposition into R‐rep (single‐valued irreducible representation over the real number field) components of the invariance group of the solution are obtained. The self‐consistent field (SCF) condition is decomposed into R‐rep components of G _{0}.

Group theoretical analysis of the Hartree–Fock–Bogoliubov equation. II.The case of the electronic system with tricrinic lattice symmetry
View Description Hide DescriptionThis paper describes a group theoretical classification of the temperature–Hartree–Fock–Bogoliubov (HFB) equation in a crystalline solid system and the electronic state of the system. It is shown that the state with a single‐order parameter (charge density wave,spin density wave, etc.) is classified into 47 classes and the BCS state coexistent with other nonsuperconducting orders, such as magnetic superconductors, is classified into 26 classes. The standard HFB Hamiltonian for each class is obtained. It is found that in each of above coexistence states (except BCS+ferromagnetism) an abnormal Cooper pair occurs.

An alternate approach to finding and using the Lie group of the Vlasov equation
View Description Hide DescriptionTwo recent methods for finding exact solutions to the Vlasov–Maxwell equations using Lie grouptheory are compared by the introduction of an ‘‘intermediate’’ approach. In the latter, the Lie group and general similarity solutions of the Vlasov equation are found through a method which treats independent and dependent variables as forms that are on an essentially equal footing. A Maxwell equation is then used to constrain the solutions further. The procedure is shown to illustrate a more general theorem that implies that the reduction of the number of variables in a set of equations through the use of canonical variables generated from the Lie group invariance of one equation in the set leads to the same solutions as are found by considering the invariance of the entire set.

Polynomial tensors for the space groups p m and p4m
View Description Hide DescriptionGenerating functions are calculated for polynomialtensors in the components of tensors of the one‐ and two‐dimensional space groups pm and p4m. For tensors whose k vectors lie at rational points (denominator q) of the Brillouin zone, the problem is imaged by the corresponding problem for an appropriate finite group, and known methods are used for its solution. For tensors with continuous k the solution is obtained by letting q become infinite.

Operator Racah algebra and Laplace‐type expansions of irreducible spherical tensors
View Description Hide DescriptionDifferential formulas for coefficients in the Laplace‐type series of an arbitrary spherical tensorf _{ L M } (r+R) are given in terms of an operator N applied to the radial part φ(r) of f _{ L M }(r). Very compact and convenient expressions for N in terms of operator Pochhammer symbols are established. A special representation of the coefficients of the Laplace‐type series, in terms of the operator Gauss function _{2} F _{1}, is given, which, in turn, provides a remarkably short proof of two earlier Sack expansions. More general gradient formulas are introduced and numerous particular cases of the Laplace‐type expansions are considered in detail.

On the polynomial solutions of ordinary differential equations of the fourth order
View Description Hide DescriptionThe density of zeros of polynomial solutions of ordinary differential equations of the fourth order with coefficients depending only on the independent variable is analyzed. The first four moments of such a density are given directly in terms of the coefficients which characterize the differential operator. Application to the nonclassical orthogonal polynomials corresponding to the names Krall–Legendre, Krall–Laguerre, and Krall–Jacobi is done. Global asymptotic properties of the zeros of these polynomials are also obtained.

Nonlinear multigroup neutron‐flux systems: Blowup, decay, and steady states
View Description Hide DescriptionThis article investigates a nonlinear system of partial differential equations describing multigroup neutron‐flux reaction‐diffusion inside a nuclear fission reactor. The neutrons are divided into nenergy groups, with fission and scattering rates dependent on temperature, which gives rise to the (n+1)th equation. Cases concerning directly coupled and down scattering, when group transfers only occur from higher to lower energy groups, are considered. Quantitative conditions on the various fission and scattering rates are found for eventual ‘‘blowup’’ and ‘‘decay’’ of neutron concentrations. Finally, a system with fission and scattering rates dependent on neutron density is also investigated. Sufficient conditions for positive nontrivial steady state are found.

Differential‐Stäckel matrices
View Description Hide DescriptionWe show that additive separation of variables for linear homogeneous equations of all orders is characterized by differential‐Stäckel matrices, generalizations of the classical Stäckel matrices used for multiplicative separation of (second‐order) Schrödinger equations and additive separation of Hamilton–Jacobi equations. We work out the principal properties of these matrices and demonstrate that even for second‐order Laplace equations additive separation may occur when multiplicative separation does not.

Multiple scales analysis of a nonlinear ordinary differential equation
View Description Hide DescriptionAsymptotic solutions of the nonlinear ordinary differential equationd ^{2}θ/d Z ^{2} +a dθ/d Z +f(θ)=0 for large a are obtained by the singular perturbation method of multiple scales analysis. They are in the form of θ(Z)=A(Z/a)+B(Z/a)exp(−a Z). Initial and boundary value problems are discussed. The special case of f(θ)=γ+cos 2θ(γ<1), encountered in shearing nematic liquid crystal soliton problems and other physical systems, is solved in detail. Previously obtained analytic solutions are recovered and justified. Our results are applicable to the unsteady shearing nematic problem.

Generalized moments applied to Fermi‐type functions
View Description Hide DescriptionGeneralized integral moments are defined for a general class of saturating functions [ f(0)=ρ_{0}, f(∞)=0]. They are useful as independent variables for describing surface properties or macroscopic dynamics of finite systems. Applied especially to functions of the Fermi type, analytic solutions are given in terms of a semiconverging, and of a numerically semiunstable expansion, respectively, suitable for numerical evaluation. Results are compared to the semidivergent expansion as given by Åberg, of which some properties are exhibited here, and with the exact numerical solutions known for this special example.

Inequalities of Schwarz and Hölder type for random operators
View Description Hide DescriptionLet A and B be random operators on a Hilbert space, and let 〈 〉 denote averages (expectations). We prove the inequality ∥〈A*B〉∥≤∥〈A*A〉∥^{1} ^{/} ^{2}∥〈B*B 〉∥^{1} ^{/} ^{2}. A generalized Hölder inequality involving traces is also proved.

Graded manifolds and vector bundles: A functorial correspondence
View Description Hide DescriptionA functorial correspondence between the category of graded manifolds and the category of vectors bundles is given. Given a graded manifold (X,A), a vector bundle G over X is given as a subset of the product X×A ^{0} where A ^{0} is the dual coalgebra of A. This bundle has an obvious coalgebra structure on each fiber. The correspondence is achieved by showing that the sheaf A is isomorphic as a sheaf of algebras to the sheaf of sections of the vector bundle G’ dual to G.

On the space‐times admitting a synchronization of constant curvature
View Description Hide DescriptionA synchronization S on the space‐time is a foliation by spacelike hypersurfaces. We study here the vector fields, tangent to S, which are Killing fields for the induced metric on every instant of S but which are not necessarily Killing fields of the whole space‐time metric; they are called S‐Killing vector fields. We analyze the multiplicity of the maximal symmetry or complete integrability case, that is the case for which the space‐times admit a synchronization S with the maximum number of S‐Killing vector fields. In particular, the important case where S is umbilical is treated in detail.

A gravitational lens produces an odd number of images
View Description Hide DescriptionRigorous results are given to the effect that a transparent gravitational lens produces an odd number of images. Suppose that p is an event and T the history of a light source in a globally hyperbolic space‐time (M,g). Uhlenbeck’s Morse theory of null geodesics is used to show under quite general conditions that if there are at most a finite number n of future‐directed null geodesics from T to p, then M is contractible to a point. Moreover, n is odd and 1/2 (n−1) of the images of the source seen by an observer at p have the opposite orientation to the source. An analogous result is noted for Riemannian manifolds with positive definite metric.

Topological closure as the necessary condition for frustration or phase transitions
View Description Hide DescriptionIt is shown that the same conditions are necessary for both frustration in Ising models with negative multispin interactions present and phase transition in analogous ferromagnetically coupled models.