Volume 19, Issue 10, October 1978
Index of content:

Multiplicity free and finite multiplicity indecomposable representations of the algebra su(1,1)
View Description Hide DescriptionA classification is given for the multiplicity free indecomposable representations of the simple Lie algebra su(1,1), which are unbounded on both sides. Formulas have been obtained for the m a t r i x e l e m e n t s of the generators of su(1,1) for all these representations. Representations of su(1,1) are analyzed which have the property that all their weight subspaces are infinite dimensional. Subrepresentations and representations on quotient spaces of this infinite multiplicity representations are considered and their relationship to the multiplicity free indecomposable representations is determined (both, unbounded on both sides, and bounded on one side). F i n i t e m u l t i p l i c i t y i n d e c o m p o s a b l e representations are obtained from the infinite multiplicity representation for special values of the Casimir operator. A decomposition of the infinite multiplicity representation into a direct sum of multiplicity free representations and finite multiplicity indecomposable respresentations is given in two d i f f e r e n t ways. Finally, formulas for the matrix elements of su(1,1) are given for the finite multiplicity indecomposable representations.

On the interaction of the type λx ^{2}/(1+g x ^{2})
View Description Hide DescriptionThe ground state and the first two excited state energy levels for the interaction of the type λx ^{2}/(1+g x ^{2}) have been calculated nonperturbatively as the eigenvalues of the one‐dimensional Schrödinger operator defined by A u=−u′′+x ^{2} u+λx ^{2} u/(1+g x ^{2}). The Ritz variational method in combination with the Givens–Householder algorithm has been used for numerical computations.

Average force and force correlation formulas for momentum transfer cross section
View Description Hide DescriptionGerjuoy has related the momentum transfer cross section to the average force. The force correlation function formula of Rousseau, Stoddart, and March is here shown to be transformable into Gerjouy’s result for a spherically symmetrical scatterer.

Complex p p waves and the nonlinear graviton construction
View Description Hide DescriptionWe show how to construct all of the complex p p waves using the nonlinear graviton construction of Penrose.

On the computation formulas of the SO(n−1,1) representation matrix elements
View Description Hide DescriptionThe formulas for computing the boost matrix elements are found for all classes of the unitary irreducible representations of SO(n−1,1) by defining the invariant scalar product in the space consisting of the D functions of SO(n−1) and assuming that functions with some property exist for the complementary series. The normalization constants of the bases are completely determined by requiring that the boost matrix elements in the finite transformations agree with those obtained by the method of the infinitesimal operators in the infinitesimal transformations.

Symmetries of the stationary Einstein–Maxwell equations. IV. Transformations which preserve asymptotic flatness
View Description Hide DescriptionWe give a series of transformations β^{ k }, k=0,1,... which may be used to generate new stationary axially‐symmetric vacuum solutions from ones already known. These transformations have the important property of p r e s e r v i n g a s y m p t o t i c f l a t n e s s. As one example of their use, we show how to generate the Kerr metric from Schwarzschild. As a second example, we generate a new five‐parameter vacuum solution which contains the δ=2 Tomimatsu–Sato solution as a special case.

On invariant integration over SU(N)
View Description Hide DescriptionWe give a graphical algorithm for evaluation of invariant integrals of polynomials in SU(N) group elements. Such integrals occur in strongly coupled lattice gauge theory. The results are expressed in terms of totally antisymmetric tensors and Kronecker delta symbols.

Almost‐structures and structures in Lorentzian manifolds. I. Almost‐Hermite‐ and almost‐product‐ (2×2) ‐structures
View Description Hide DescriptionWe present a list of the most important almost‐structures which have been found of interest in general relativity, in the null‐bivectors formalism. We discusss some of the relevant properties of such almost‐structures and various new or more or less known results. We also present theorems on the relations between almost‐product‐structures and almost‐Hermitian‐structures.

An investigation of Dubourdieu’s list of space–times which admit holonomy groups
View Description Hide DescriptionAn investigation is made of fourteen space–times given by Dubourdieu which admit holonomy groups. It is shown that, although nine of these space–times admit only trivial vacuum gravitational fields, the remaining ones are of Bel–Petrov types III and N. Many of the latter metrics can be identified with known exact solutions of the source‐free Einstein equations including the Kerr–Goldberg metric and certain type III metrics recently studied by the authors.

Continuum calculus. III. Skorohod’s weak distributions in the evaluation of a class of Feynman path integrals
View Description Hide DescriptionPath integrals for functionals are studied from the point of view of the continuum calculus proposed earlier [J. Math. Phys. 17, 1988 (1976)]. The weak distributions of Skorohod in an infinite‐dimensional Hilbert space and the p‐integral method of continuum calculus are employed to derive a formula for the functional integral, which is in turn evaluated through a natural extension of the weak distribution expression. Generalizations are made to measures with density functions in the function space. As a demonstration, the formula is tested against the polynomial functionals studied by Friedrichs, and valid results are obtained for the general case.

Symmetries of the 3j coefficient
View Description Hide DescriptionAn explicit form of the five Regge symmetries of the 3j coefficient is given. It is shown that a set of six _{3} F _{2}(1) hypergeometric functions is necessary and sufficient to account for the 72 symmetries of the 3j coefficient, each accounting for 12 symmetries. The eight‐element group, recently discussed by Lockwood, accounts for only eight symmetries of the 3j coefficient.

A new form of the Mayer expansion in classical statistical mechanics
View Description Hide DescriptionNew expressions are given for the expansion coefficients in the Mayer expansion (and thus the virial expansion). These promise to be useful in applications, as well as provide a simple rigorous proof of the convergence of the Mayer series and some of its properties.

O(3) shift operators: The general analysis
View Description Hide DescriptionO(3) shift operators are constructed in terms of tensor operators T (j,μ) and the O(3) generators. These are of type B ^{ k } _{ l }, k=−j,...,j, where B ^{ k } _{ l } raises l by k, l (l+1) being the eigenvalue of the O(3) Casimir operator. Various convenient normalizations of these operators are constructed, and their properties and uses considered.

On the polynomial expansion of the Hill–Wheeler integral for the rotational energies
View Description Hide DescriptionA transformation is used which enables one to carry out exactly the integration of the Hill–Wheeler integral for the rotational energies. The expression for the energies E _{ J } is shown to be the ratio of two polynomials in J (J+1), where J is the total angular momentum. A nice feature of the formulation is that the coefficients in the expansion involve only the matrix elements of the lowest rank independent H J ^{2r } and J ^{2r } operators, 0⩽r⩽N−1, N being the number of J states contained in the given rotational band, and H being the Hamiltonian of the system.

Vector constants and their algebras for classical Hamiltonians
View Description Hide DescriptionThe explicit form of a vector constant of the motion for an arbitrary relativistic spherically symmetric time independent classical Hamiltonian is obtained by showing that its construction is achieved on solving a linear second order ordinary differential equation. The solution of this equation is presented and the vector, in conjunction with the angular momentum to which it is normal, is used to generate the algebras of the Euclidean group E(3), the orthogonal rotation group O(4) and the special unitary group SU(3). The mass is assumed to be structureless and to move in an externally prescribed scalar potential field.

Relativity and deformed Lie groups
View Description Hide DescriptionThe concept of a deformed Lie group in which the structure coefficients are functions of the group coordinates is defined. Every Lie group can be deformed outside any of its subgroups. The affine group in four dimensions deformed outside Gl(4,C) has a variable group metric that is closely connected with the Ricci tensor of the four‐dimensional manifold of translations. An analog of Einstein’s vacuum equations expresses the invariance of the metric with respect to the deformation. An enlargement of the affine group leads naturally to the appearance of the energy–momentum terms in the equations, while the gravitational interaction constant plays a role of a fundamental group constant (like c in the Lorentz group) which makes all generators of the gauge group dimensionless.

Para‐Bose coherent states
View Description Hide DescriptionIn place of the usual commutation relation [â,â^{†}]=1 we consider the generalized commutation relation characteristic of the para‐Bose oscillators, viz, [â, Ĥ]=Ĥ where Ĥ is the Hamiltonian (1/2)(ââ^{†}+â^{†}â). The number states and the representation of various operators in the basis formed by these states are obtained. We then introduce the para‐Bose coherent states defined as the eigenstates of â for this generalized case. We consider some of the properties of these coherent states and also show that the uncertainty product 〈 (Δq̂)^{2}〉 〈 (Δp̂)^{2}〉 in this case could be made arbitrarily small.

Coulomb‐like quantization of the electromagnetic field on spacelike hyperboloids
View Description Hide DescriptionMaxwell’sequations relative to a pseudospherical coordinate system in Minkowski space are cast in a form remarkably similar to the usual Cartesian form. The vector spherical harmonics of the unit hyperboloid derived in an earlier work are then applied to the problem of quantizing the electromagnetic field from the standpoint of the method of quantization on spacelike hyperboloids. A Coulomb‐like gauge is introduced for which quantization on spacelike hyperboloids parallels that of ordinary Coulomb gauge quantization. In a kind of ’’turning around’’ of the situation which obtains for ordinary Coulomb gauge quantization, our new Coulomb‐like gauge condition has manifest covariance with respect to transformations of the homogeneous Lorentz group but not with respect to translations.

An inverse scattering transform for potentials of compact support
View Description Hide DescriptionWe consider the inverse scattering problem for the one‐dimensional Schrödinger equation on the whole line, (d ^{2}/d x ^{2}) φ (x,k)+[k ^{2}−V (x)]φ (x,k) =0. In some applications, as for example in the synthesis of electromagnetic media, it is important to have s u f f i c i e n t conditions on the scattering data such that the corresponding potential has compact support in some prescribed interval. The scattering data traditionally used in connection with the above ISP have been either one of the reflection coefficients (we are assuming that the potential V does not support bound states)r and ?—from the left and right, respectively. Although it is easy to obtain simple conditions on r (?) to ensure cutoff of the potential on the left (right), conditions on r (?) that guarantee cutoff on the right (left) are too complicated to be of any practical value. In this paper, we propose to use new scattering data, namely the ratio r/t (where r is either one of the reflection coefficients and t is the transmission coefficient), and give necessary and sufficient conditions for the corresponding potential to have support contained in [−a, a].

Super Clifford algebra
View Description Hide DescriptionAn extension of su(2,2;2n) is presented which is based upon a theory of Z _{4} gradings of Clifford algebras. In the example given su(2,2;4) is extended and the Clifford algebra of an internal six‐dimensional space is manifested. It is suggested that super Clifford algebras of this type can give valuable insight into the study of su(2,2;2n) supersymmetrytheories.