Volume 24, Issue 8, August 1983
Index of content:

Symmetry and generating relations for Clebsch–Gordan coefficients arising from automorphisms
View Description Hide DescriptionA new class of symmetry and generating relations for Clebsch–Gordan coefficients is derived. It is associated with certain elements of the full automorphism group. Our approach turns out to be useful for an efficient computation of Clebsch–Gordan coefficients.

On the quaternion representation of the proper Lorentz group SO(3,1)
View Description Hide DescriptionComplex quaternions are investigated in detail, bringing out some new aspects of the relationship of the multiplicative group of unit complex quaternions (UCQ) with the proper Lorentz group SO(3,1). Constructing the proper Lorentz transformation (PLT) corresponding to a given UCQ, the quaternion parameters of a PLT are determined explicitly in terms of its element L _{ i j }, and this quaternion parametrization is then utilized to obtain an interesting geometrical interpretation of SO(3,1) as the intersection of a hyperboloid with a cone in a real eight‐dimensional Euclidean spaceE _{8}. The UCQ components are then related to the Lie–Cartan parameters of SO(3,1), leading to an identification of complex quantities which may be interpreted as the complex axis and angle of rotation. It is shown that any PLT admits a special type of Euler resolution which is at the same time a resolution into three Lorentz–Synge screws the two angles of which combine to form a complex Euler angle (or Euler–Brauer angle). It is also shown that on taking the rotation parameters in the formula for the D ^{ j } representation of SO(3) to be complex, one obtains the D ^{ j0} representation of SO(3,1), leading at once to its D ^{ j j′} representation. Similarly, a formula for the character χ^{ j0} of the D ^{ j0} representation, having a complete analogy to the character formula for SO(3), but in terms of a complex angle ω is obtained and this in turn yields a formula for the character χ^{ j j′} in the D ^{ j j′} representation of SO(3,1).

Clebsch–Gordan coefficients for SU(5)⊇SU(3)×SU(2)×U(1) theories
View Description Hide DescriptionThe Clebsch–Gordan coefficients are calculated for the following tensor products of SU(5) representations: 5⊗5, 5⊗10, 5⊗1̄0̄, 15⊗5, 15⊗5̄, 10⊗10, 5⊗5̄, 10⊗1̄0̄, 5⊗24, and 10⊗24. Each case is calculated twice: once in a weight vector basis independent of any semisimple subgroup, the second time in a basis which refers to SU(3)×SU(2)×U(1)⊆SU(5).

Tensorial analysis of the hyperfine interaction operator by extensive use of Racah algebra and of translational invariance
View Description Hide DescriptionS t r i c t application of Racah algebra gives an efficient way of deriving the tensorial form of the hyperfine interaction one‐electron operator involved in NMR spin coupling effects. In particular, this procedure avoids the use of classic explicit arguments of integration around the origin to find the Fermi‐contact term. The linear momentum is retained in its explicit translationally invariant form as long as possible during the calculations. This allows the tensorial expansion of the operator to be obtained rapidly at any origin of coordinates. The reduced matrix elements of interest are given by new general closed expressions in the j−j coupling scheme.

Maximal abelian subalgebras of real and complex symplectic Lie algebras
View Description Hide DescriptionWe provide guidelines for classifying maximal abelian subalgebras (MASA’s) of the symplectic Lie algebrass p(2n, R) and s p(2n, C) into conjugacy classes under the Lie groups Sp(2n, R) and Sp(2n, C), respectively. The task of classifying all MASA’s is reduced to the classification of orthogonally indecomposable (OID) MASA’s. Two types of orthogonally indecomposable MASA’s of s p(2n, C) exist: 1. Indecomposable maximal abelian nilpotent subalgebras (MANS’s). 2. Decomposable MASA’s [their classification reduces to a classification of MANS’s of s l(n, C)]. Four types of orthogonally indecomposable MASA’s of s p(2n, R) exist: 1. Absolutely indecomposable MASA’s (MANS’s). 2. Relatively indecomposable MASA’s [their classification reduces to a classification of MANS’s of s u( p, q) for p+q=n]. 3. Decomposable absolutely OID MASA’s [involving MANS’s of s l(n,R)]. 4. Decomposable relatively OID MASA’s [involving MANS’s of s l(n/2, C), for n even]. Low‐dimensional cases of s p(2n, F) (n=1, 2, 3, F=R or C) are treated exhaustively. The algebrass p(2, R), s p(4, R), and s p(6, R) have 3, 10, and 30 classes of MASA’s, respectively; s p(2, C), sp(4, C), and s p(6, C) have 2, 5, and 14 classes of MASA’s, respectively. For n≥4 infinitely many classes of MASA’s exist.

Octonionic description of exceptional Lie superalgebras
View Description Hide DescriptionThe structures of the exceptional Lie superalgebrasG(3) and F(4) are expressed in terms of octonions.

On a new class of gradient formulas in the angular momentum theory
View Description Hide DescriptionRecently B. F. Bayman has derived the formula for Y_{ l m }(∇)φ(r)Y_{ L M }(r), where Y_{ l m }(r) is a solid spherical harmonic, in terms of the derivatives of the function φ(r) by the scalar parameter r. This result is clarified and essentially generalized for the case of the tensor product {Y^{ n } _{ l }(∇)⊗φ(r)Y_{ L }(r)}_{λμ}, where Y^{ n } _{ l m }(∇)=Δ^{ n }Y_{ l m }(∇), in this paper. Applications to the Taylor series in the three‐dimensional Euclidean space, as well as some other expansions, are discussed briefly.

On the higher orders of hyperspherical harmonics
View Description Hide DescriptionWe suggest a procedure to evaluate matrix elements between hyperspherical harmonics of any order. The method is based on the hyperspherical expansion of a Slater determinant constructed with oscillatorwavefunctions. Explicit formulas are given for all matrix elements up to order L _{ m }+2.

A new approach to perturbation theory: star diagrams
View Description Hide DescriptionPerturbation theory for first‐order nonlinear differential equations with source is developed in a new way, and associated with diagrams that we call star diagrams. In some cases the method allows one to express the n‐point functions in explicit form.

Symmetries of differential equations. IV
View Description Hide DescriptionBy an application of the geometrical techniques of Lie, Cohen, and Dickson it is shown that a system of differential equations of the form x ^{(r i } _{ i }=F _{ i } (where r _{ i }>1 for every i=1,...,n) cannot admit an infinite number of pointlike symmetry vectors. When r _{ i }=r for every i=1,...,n, upper bounds have been computed for the maximum number of independent symmetry vectors that these systems can possess: The upper bounds are given by 2n ^{2}+n r+2 (when r>2), and by 2n ^{2}+4n+2 (when r=2). The group of symmetries of x̄^{(r }=0̄ (r>1) has also been computed, and the result obtained shows that when n>1 and r>2 the number of independent symmetries of these equations does not attain the upper bound 2n ^{2}+n r+2, which is a common bound for all systems of differential equations of the form x̄^{(r }=F̄(t,x̄,...,x̄^{(r−1}) when r>2. On the other hand, when r=2 the first upper bound obtained has been reduced to the value n ^{2}+4n+3; this number is equal to the number of independent symmetry vectors of the system ẍ̅=0̄, and is also a common bound for all systems of the form ẍ̅=F̄(t,x̄,x̣).

Completely integrable relativistic Hamiltonian systems and separation of variables in Hermitian hyperbolic spaces
View Description Hide DescriptionThe Hamilton–Jacobi and Laplace–Beltrami equations on the Hermitian hyperbolic space HH(2) are shown to allow the separation of variables in precisely 12 classes of coordinate systems. The isometry group of this two‐complex‐dimensional Riemannian space, SU(2,1), has four mutually nonconjugate maximal abelian subgroups. These subgroups are used to construct the separable coordinates explicitly. All of these subgroups are two‐dimensional, and this leads to the fact that in each separable coordinate system two of the four variables are ignorable ones. The symmetry reduction of the free HH(2) Hamiltonian by a maximal abelian subgroup of SU(2,1) reduces this Hamiltonian to one defined on an O(2,1) hyperboloid and involving a nontrivial singular potential. Separation of variables on HH(2) and more generally on HH(n) thus provides a new method of generating nontrivial completely integrable relativistic Hamiltonian systems.

On a new hierarchy of Hamiltonian soliton equations
View Description Hide DescriptionA method is suggested for studying the Hamiltonian structure of the nonlinear partial differential equations that can be solved by the use of the spectral transform (soliton equations). The method is applied to a new hierarchy of N+1 coupled partial differential equations related to a Schrödinger‐like spectral problem. It is shown that these soliton equations are integrable Hamiltonian equations with commuting flows. For N=1 and N=2 a Miura‐like transformation is computed and the corresponding modified equations are explicitly given.

Connection between the existence of bisolitons for quadratic nonlinearities and a factorization of the associate linear operator
View Description Hide DescriptionIn a first part we investigate the exponential type bisolitons of the class of equations (ε≠0, λ≠0 or ε=λ=0), which are rational solutions and we assume that their denominators have no coupling between the solitons. We have found that their existence requires a factorization property of either the operator ε+L _{ q } for ε≠0, λ≠0, q≤3 or L _{ q } for ε=λ=0, q≤3 with the exception of the Burgers equation. We find three kinds of nontrivial bisolitons: either those associated with the mixed nonlinearity (λ+μ∂_{ x })K ^{2} alone, or those common to the mixed nonlinearity and K K _{ x } or those belonging to K K _{ x } alone. In the second part we look at the two last types of bisolitons for q>3 and give a constructive method leading both to factorized linear operators and to the explicit determination of the bisolitons. We consider mainly l _{ q−2}(t) (a∂_{ t } +b∂_{ x } +c∂_{ x x })G =(∂_{ x } G)^{2}, K=G _{ x }, with the linear operator factorizing the linear part of the Burgers equation,l _{ q−2} (t) is a differential operator in variable t only. The general G solution is a linear combination of logΔ and Δ^{−i }, i=1,...,q−2 with Δ=1+∑^{2} _{1} exp(t+γ_{ i } x). We determine different classes of solutions as well as the associated l _{ q−2}(t) operators.

Application of nonlinear operator theory to the Edwards–Freed equations in the theory of polymer solutions
View Description Hide DescriptionMathematical structure of nonlinear integral equations for ‘‘screened hydrodynamic interaction’’ and ‘‘self‐energy’’ in the static version of Edwards and Freed’s theory of polymer solutions with finite concentrations are analyzed and it is shown that algorithms developed by Lika and Altman for nonlinear operator equations without Fréchet differentiability are applicable. Recipes for successive approximations are presented and questions to be investigated by those techniques are proposed.

Tetrads and arbitrary observers
View Description Hide DescriptionRecent results concerning globally isometric mappings for arbitrary observers in flat space‐time are generalized to space‐times admitting a time orientation. Critical to the method is the use of an orthonormal tetrad which, when it is defined globally, allows the construction of a global isometry which generalizes the pointwise boost on flat space‐time. Connection coefficients are obtained, thereby defining acceleration covariant differentiation for both particle and tensor field equations. An application to orbiting observers in exterior Schwarzschild geometries is presented.

Dynamical symmetries: An approach to Jacobi fields and to constants of geodesic motion
View Description Hide DescriptionIt is shown that every dynamical symmetry (DS) of the Euler–Lagrange equations derived from the LagrangianL =(1/2)g _{ a b } q̇^{ a } q̇^{ b } identifies a Jacobi field on each geodesic of the configuration manifold. Using the connections between Jacobi fields and DS’s, it is proved that DS’s always possess associated conserved quantities, whose expression is explicitly written down. An additional constant of motion concomitant with ‘‘pairs’’ of DS’s, independently of the choice of L, is also determined. Applications to general relativity are emphasized in the course of the discussion.

The asymptotic evaluation of a class of path integrals. II
View Description Hide DescriptionThe asymptotic behavior of a class of Wiener‐like path integrals (functions of the ‘‘local time’’) is determined. These integrals are of interest in themselves and also arise very naturally in the theory of disordered systems. We show that by making use of a Grassman algebra (i.e., a set of anticommuting variables), the earlier treatment of this problem can be greatly simplified. In particular, the previous use of ‘‘replica trick’’ (which involves a difficult to justify analytic continuation in the number of field components) is thus avoided.

The rotating harmonic oscillator eigenvalue problem. I. Continued fractions and analytic continuation
View Description Hide DescriptionThe continued fraction approach to the solution of the rotating harmonic oscillatoreigenvalue problem is examined in detail. It is shown how one may obtain eigenvalue information only from an analytic continuation of the continued fraction accomplished with the aid of modified approximants.

The rotating harmonic oscillator eigenvalue problem. II. Analytic perturbation theory
View Description Hide DescriptionThe theory of self‐adjoint analytic families is applied to the rotating harmonic oscillator Hamiltonian in L ^{2}(0,∞) to obtain weak and strong coupling expansions of the eigenvalues. Various estimates on the radius of convergence of the weak coupling expansion are obtained. The strong coupling expansion is shown to be an asymptotic series which, with the neglect of exponentially small terms, is expressible in terms of a simple formal perturbation of the ordinary harmonic oscillator Hamiltonian in L ^{2}(−∞,∞).

Symmetries, conservation laws, and time reversibility for Hamiltonian systems with external forces
View Description Hide DescriptionA system theoretic framework is given for the description of Hamiltonian systems with external forces and partial observations of the state. It is shown how symmetries and conservation laws can be defined within this framework. A generalization of Noether’s theorem is obtained. Finally a precise definition of time reversibility is given and its consequences are explored.