Volume 22, Issue 11, November 1981
Index of content:

A note on stochastic dynamics in the state space of a commutative C* algebra
View Description Hide DescriptionIn this paper a functional characterization of stochastic evolutions within the state spaces of commutative C* algebras with identity is derived. Consequences concerning the structure of those linear evolution equations (master equations) that give occassion to stochastic evolutions are discussed. In part, these results generalize facts which are well known from the finite‐dimensional classical case. Examples are given and some important particularities of the W* case are developed.

On the triangle anomaly number of SU(n) representations
View Description Hide DescriptionA simple geometrical interpretation of the triangle anomaly number of SU(n) representations is demonstrated. The number is equal to the sum of cubes of projections of weights on the U(1) direction; U(1) is the group in the reduction SU(n)⊇U(1)×SU(n−1). Properties of the anomaly number are then consequences of properties of weight systems. Explicit formulas are derived for the anomaly numbers for reducible representations corresponding to general plethysms based on an arbitrary representation of SU(n). Generalizations to ’’anomaly numbers’’ corresponding to degrees other than three are discussed.

Wigner 9j symbols and the product group
View Description Hide DescriptionThe 9 j symbols of a simply reducible Lie groupG are related to the 3 j symbols of the product group G×G. It is shown that this relationship leads to certain identities satisfied by the 9 j symbols. Two known identities involving 6 j and 9 j symbols are shown to be reduced forms of a new identity involving 9 j symbols.

Nonscalar extension of shift operator techniques for SU (3) in an O(3) basis. I. Theory
View Description Hide DescriptionA set of relations is set up which connect quadratic products of the shift operators O ^{ k } _{ l } (k = 0,1,2), which are nonscalar with respect to the O(3) subgroup of SU(3). The usefulness of these relations is illustrated by the calculation of the eigenvalues of the scalar shift operator O ^{0} _{ l } for various irreducible representations ( p,q) of SU(3).

Nonscalar extension of shift operator techniques for SU (3) in an O(3) basis. II. Applications
View Description Hide DescriptionIn a preceding paper relations have been derived which connect nonscalar quadratic shift operator products. Here, the extreme usefulness of these relations is demonstrated by the example of the O ^{0} _{ l } ‐eigenvalue calculation for the cases l = p−i (i = 0,1,2,3, and 4), where ( p,q) is any SU(3) representation. For the first time a case of threefold l‐degeneracy is completely solved in a pure analytical way.

Lorentz and SU(2)⊗U(N) analysis of the Lie algebra sp(4N;r) [gl(4N;r)] for any integer N≥1
View Description Hide DescriptionThe generators of the fundamental representation of the Lie group GL(4N;r), integral N?1, are constructed from Kronecker products of smaller matrices in such a way that their tensor character under the action of the (unique) full null‐plane Lorentz subgroup is apparent. Commutation relations of these tensors are given in terms of symmetric and antisymmetric structure constants for the fundamental representation of U(N) used in their construction. Generators of the Sp(4N;r) subgroup are classified according to transformation character under a U(N) subgroup. Commutation relations of sp(4N;r) are given in terms of SU(2)_{spin} ⊗U(N) multiplets.

Wigner coefficients for a semisimple Lie group and the matrix elements of the O(n) generators
View Description Hide DescriptionA purely algebraic infinitesimal method for obtaining multiplicity‐free Wigner coefficients is presented. The method is applied to obtain analytic expressions for the complete matrix elements of all O(n) generators. Moreover the structure of these matrix elements in terms of reduced matrix elements, Wigner coefficients, and reduced Wigner coefficients is made explicit. By comparison with the Wigner–Eckart theorem explicit analytic expressions are obtained for the fundamental Wigner coefficients of O(n). Finally the results are presented in a form which is directly analogous with the corresponding results for U(n).

Classical and quantal systems of imprimitivity
View Description Hide DescriptionWe discuss from a group‐theoretical point of view, a simple framework in which the classical and the quantal state spaces appear in a unified way. The framework is characterized by the use of (possibly continuous) direct unions of Hilbert spaces. Symmetries are correspondingly represented by families of (anti‐) unitary operators. We consider here the associated notions of projective representations as well as the corresponding observables. These observables, classical or quantal, are defined in terms of a slight generalization of the notion of a system of imprimitivity of Mackey.

State spaces for classical and quantal, relativistic and nonrelativistic elementary particles
View Description Hide DescriptionWe apply a recently developed mathematical formalism to the kinematical classification of elementary physical systems, more precisely to the derivation of the possible state spaces for single (massive) particles and to their corresponding group theoretical interpretation. We show in particular that in all considered cases, relativistic or not, we find in a unified way two solutions, a quantal one and a classical one.

Noncomplex representations and their relation to antiunitary symmetry
View Description Hide DescriptionEquivalent eigenvalue problems in real, complex, and quaternionic Hilbert spaces are discussed. In all cases a compact symmetry group is assumed to exist, in the complex case also an antiunitary operator ϑ commuting with both the self‐adjoint operator and the elements of the symmetry group and satisfying ϑ^{2} = ±1. It is shown that the extra degeneracies caused by the existence of ϑ can also be obtained by consideration of an equivalent eigenvalue problem in a noncomplex Hilbert space.

Representation theory of compact groups over fields of characteristic zero
View Description Hide DescriptionThe group algebras over the reals, the complex numbers, and the quaternions, are discussed and compared for a fixed compact group. Special emphasis is put on the properties of ring bases of the minimal two‐sided ideals and their relation to the matrix representation of the group irreducible over the fields cited above.

Analytic evaluation of certain zeroth order coulombic hyperangular interaction integrals
View Description Hide DescriptionAs has been presented in a previous paper, the use of hyperspherical coordinates in research on a system of electrically charged particles carries mathematical complications into the evaluation of certain kinds of hyperangular interaction integrals. These integrals contain the hyperangular interaction potential and various powers of the inverse of the total angular momentum operator on the space of hyperharmonics, from which the zeroth order hyperharmonic is excluded. In this work the simplest one of these integrals has been taken into consideration. After some intermediate steps, it has been shown that it can be expressed in terms of elementary functions of the cosine of the angle (γ) between the hyperaxes of the potential term for an odd number of particles. In the case of an even number of particles, these integrals can be given in terms of a generalized hypergeometric function of the same argument (γ) and its derivatives.

Solution of Poisson’s equation: Beyond Ewald‐type methods
View Description Hide DescriptionA general method for solving Poisson’s equation without shape approximation for an arbitrary periodic charge distribution is presented. The method is based on the concept of multipole potentials and the boundary value problem for a sphere. In contrast to the usual Ewald‐type methods, this method has only absolutely and uniformly convergent reciprocal space sums, and treats all components of the charge density equivalently. Applications to band structure calculations and lattice summations are also discussed.

Linear equations invariant under arbitrary coordinate changes
View Description Hide DescriptionLinear homogeneous equations of the type i α^{λ}∂ψ/∂x ^{λ} = βψ are considered, where α^{λ} and β are constant Hermitian N×N matrices and ψ is an N component column vector. The conditions that this equation is invariant under arbitrary changes of coordinates are shown to be α^{λ} C ^{κ} _{ι}+C ^{†κ} _{ι}α^{λ} = α^{κ} δ^{λ} _{ι}−α^{λ}δ^{κ} _{ι}, βC ^{κ} _{ι}+C ^{†κ} _{ι}β = −βδ^{κ} _{ι}, α^{λ} C ^{κ} _{ι}+α^{κ} C ^{λ} _{ι} = 0, where C ^{κ} _{ι} is a set of 16 N×N matrices which specify the transformation law for the field components ψ. Some theorems are proved about solutions of these matrix relations and some explicit representations are given.

Similarity solutions for the Ernst equations with electromagnetic fields
View Description Hide DescriptionA class of exact similarity solutions is shown to exist for the Ernst equations with electromagnetic fields. We reduce the underlying coupled nonlinear partial differential equations for the potentials into a system of coupled ordinary nonlinear differential equations for the similarity variables. The reduced system is exactly solvable in terms of elementary functions. These solutions are bounded everywhere.

Bounds on Green’s functions of second‐order differential equations
View Description Hide DescriptionWe estimate the diagonal part of the Green’s function for the equation (−Δ/2+V(x)+∂/∂t)ψ(x, t) = 0, t≳0, x∈ B, where B is a finite region of the Euclidean spaceR ^{ d } with a regular boundary. In the special case V(x) = 0, x∈B, we also obtain bounds for the nondiagonal part of the Green’s function which are uniform in t.

Decay mode solution of the two‐dimensional KdV equation and the generalized Bäcklund transformation
View Description Hide DescriptionWe consider the generalization of the customary Bäcklund transform (≡BT) for the two‐dimensional (2D) KdV equation, (u _{ t }+6u u _{ x }+u _{ x x x })_{ x } +3α^{2} u _{ y y } = 0, with α being constant. A nonlinear superposition formula has been obtained and it is shown that the present generalized BT can produce multiple soliton–multiple decay mode solutions.

Evolution equations associated with the discrete analog of the matrix Schrödinger spectral problem solvable by the inverse spectral transform
View Description Hide DescriptionThrough the generalized Wronskian technique we derive the whole class of nonlinear differential difference equations associated with the discrete analog of the matrix Schrödinger spectral problem. For such equations we briefly discuss soliton solutions, continuum limit, and Bäcklund transformations.

Some properties of Borel summable functions
View Description Hide DescriptionClasses of Borel summable functions are defined and studied. Some properties of these functions, which are useful for high‐order calculations pertaining to certain physical theories, are proved and discussed. They include reciprocal Watson‐like theorems, sufficient conditions for membership in the classes, and asymptotic behaviors of the expansion coefficients after composition of functions.

The Coulomb unitarity relation and some series of products of three Legendre functions
View Description Hide DescriptionWe obtain from the off‐shell Coulomb unitarity relation a closed expression for J^{ ∞ } _{ l = 0}(2l+1)P _{ l }(x) ×Q _{ l } ^{ iγ} ( y) Q _{ l } ^{−iγ} (z), and we consider some related series of products of Legendre functions.