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Volume 11, Issue 4, April 1970

On the Two Majorana Representations
View Description Hide DescriptionAn interesting property of the Majorana representations, symbolized in the Clebsch‐Gordan equations and is reported. refers to the basic spinorial nonunitary representation of the SL(2, c) group in Gel'fand's notation, while (0, ½) and (½, 0) are two Majorana representations, again in the same notation. The Majorana representations represent the only solution of the equation in the sense that, of all the unitary irreducible representations of the SL(2, c) group, only (0, ½), or (½, 0) multiplied tensorially with will yield a unitary result—at least one of the Ys unitary.

Three‐Dimensional Linear Transport Theory
View Description Hide DescriptionA recent technique for extending the singular eigenfunction method in linear transport theory to problems which are not strictly 1‐dimensional is compared to a more naive approach based on the Fourier transform. The latter appears to have advantages with regard to simplicity and directness.

Singular Three‐Body Amplitudes in the Theory of the Third Virial Coefficient
View Description Hide DescriptionWe have shown earlier how the virial coefficients can be expressed in terms of traces taken over products of multiparticle (on‐energy‐shell) scattering matrices. The traces are to be taken in the angular‐momentum representation instead of the momentum representation to avoid the infinite forward N‐particle scattering amplitude for N > 2 caused by certain singular diagrams. Here, we analyze and sum the contribution of the singular diagrams to the third virial coefficient. It is also shown that one can get the same result if off‐shell amplitudes in the momentum representation are utilized.

Generalized Algebraic Mass Formulas. I
View Description Hide DescriptionIt is shown that the SL(2, C)‐Poincaré associative algebraic model of Böhm can be extended without essential difficulty to an SL(2, C)‐de Sitter model which gives rise to a meson mass formula with both spin and isospin dependence.

Equivalence of Some Generalizations on the Spherical Model
View Description Hide DescriptionWe define n‐spherical models as follows: (1) Divide a lattice of N points into n mutually exclusive subsets with N _{1}, ⋯, N_{n} points, respectively; ΣN _{α} = N. With each lattice point j, associate a variable ε_{ j }, and impose the restrictions ; the sums are over the sites of the αth subset. (2) Adopt the energy expression for the Ising model with the ε's playing the role of the ``scalar spins.'' We then prove that, in the thermodynamic limit (N → ∞, N _{α} → ∞ for all α), the thermodynamic functions of the n‐spherical model are equal to those of the ordinary spherical model.

Integral‐Transform Gaussian Functions for Heliumlike Systems
View Description Hide DescriptionThe one‐dimensional Laplace transform of the Gaussian exp (−r ^{2} x),,was used to generate functions which could be useful as basis sets for atomic and molecular calculations. A particular choice of the weighting function G(x) led to functions of the form g(r) = (qr)^{ν} K _{ν}(qr), where K _{ν}(qr) are modified Bessel functions of the second kind. These functions were used as basis functions for the helium isoelectronic series and accounted for 98.98% (H^{−}), 99.89% (He), 99.96% (Li^{+}), 99.98% (Be^{++}), 99.996% (O^{6+}) of the Hartree‐Fock energy.

Model‐Independent Analysis of Nonrelativistic Multiparticle Reactions
View Description Hide DescriptionThe Clebsch‐Gordan coefficients for the reduction of an n‐fold tensor product of nonrelativistic free‐particle states into one over‐all state are calculated. The degeneracy parameters resulting from this decomposition are then cast into Galilean‐invariant functions of the particle momenta. Finally, the reduction is applied to the scattering matrix for nonrelativistic reactions involving an arbitrary number of final particles.

Evaluation of Certain Radial Coulomb Integrals Using Symmetry Properties of the Coulomb Field
View Description Hide DescriptionThe symmetry properties of the nonrelativistic Coulomb field problem allow one to construct an operator calculus for evaluating matrix elements of the multipole operator r ^{−q }. By means of this operator calculus an explanation is given for the vanishing of certain radial integrals treated earlier by Pasternack and Sternheimer, as well as for the value of similar integrals occurring in Coulomb excitation.

Effective Permittivity of a Polycrystalline Dielectric
View Description Hide DescriptionWe use statistical variational principles to determine upper and lower bounds for the effective permittivity of a polycrystallinedielectric. We indicate how to derive bounds containing permittivitycorrelation functions of arbitrary order, and we obtain explicit expressions for bounds depending on one‐ and two‐point correlation functions and for bounds containing one‐, two‐, and three‐point correlation functions. We prove that for two classes of polycrystal, the effective permittivity may be exactly determined, and we use these exact expressions to show that we have obtained the best possible upper and lower bounds.

Spinors in a Weyl Geometry
View Description Hide DescriptionIt was noted by London in 1927 that a mathematical connection exists between the unified field theory of Weyl and the old quantum theory of Bohr. We wish to demonstrate that a Dirac electron in an electromagnetic field described by the Weyl formalism appears to couple to the field via the usual minimal coupling recipe. It is generally acknowledged that the Weyl theory is unsatisfactory for describing macroscopic electromagnetic phenomena. The objections, however, may not be entirely applicable on a quantum‐mechanical scale.

Matrix Elements of Relativistic Electrons in a Coulomb Field
View Description Hide DescriptionMatrix elements for a radiative interaction between states of a Dirac electron in the presence of a Coulomb field are reduced to a closed analytic form in the limit of zero electron mass; corrections for finite electron mass are indicated. The application of these to inelastic electron scattering and radiation problems is discussed.

Reduced Matrix Elements of Tensor Operators
View Description Hide DescriptionReduced matrix elements of a tensorial product of ntensor operators in the basis of N‐particle angular‐momentum eigenstates are expanded into a sum of products of n 1‐particle reduced matrix elements and a single (N + n)‐particle recoupling coefficient. Application of the formula is illustrated by specific examples. The method leaves the coupling schemes of N‐particle states undisturbed, which allows summation over intermediate states in a product of matrix elements to be made for any number of factors. A formula is given for the sum of products of two matrix elements and the extension to a greater number of matrix elements is illustrated by an example which is reduced to a form suitable for numerical evaluation. Such summations over products of matrix elements occur in the perturbation theory of configuration interaction and have hitherto been discussed in terms of effective operators. The connection of the method used here with the effective‐operator approach is demonstrated.

O(5) Harmonics and Abnormal Solutions in the Bethe‐Salpeter Equation
View Description Hide DescriptionExact solutions of the covariant Bethe‐Salpeter equation in the ladder approximation for two scalar particles bound by a massless particle have been obtained for all energies. By using Fock's stereographic projection, the Bethe‐Salpeter equation is transformed on to the surface of a 5‐dimensional Euclidean sphere and the solutions are then expressed as a series in O(5) harmonics. The normalization condition has been imposed by requiring that the expectation value of appropriate components of the energy‐momentum tensor with respect to the bound states is the total energy of the system; it is found that states corresponding to certain values of the quantum numbers do not satisfy the normalization requirement. These are the so‐called abnormal asolutions.

Invariant Representation of All Analytic Petrov Type III Solutions to the Einstein Equations
View Description Hide DescriptionThe results of the application of previously developed techniques to the analysis of the Petrov type III solutions to the vacuum Einstein equations are presented. The procedure involves the computer aided analysis of the Einstein‐Petrov equations to the extent that the functions uniquely and invariantly generating all local analytic solutions are determined. For the case of type III it is shown that, relative to a given fixed point in the manifold, all local analytic solutions are uniquely and invariantly determined by six arbitrary analytic functions of one variable and six others of two variables. These functions, called generating functions, thus provide a representation of all such solutions and may be used for the study of the structure of the family of Einstein empty space metrics.

Reduction of the Direct Product of Representations of the Poincaré Group
View Description Hide DescriptionWe expand the direct product of two representations of the Poincaré group into representations of the Poincaré group in the general case that the factors of the direct product may have any mass, whether real, zero, or imaginary, and the total energy may be indefinite. The representations of the Poincaré group, which appear in the expansion of the direct product, have masses which run through a continuous spectrum of real and imaginary values and are irreducible in terms of the mass and sign of energy (for real mass), but are reducible in terms of the infinitesimal generators of the little groups. To obtain the expansion in terms of irreducible representations, one need only reduce the infinitesimal generators of the little groups. This reduction is carried out for the real mass components and, in principal at least, can be carried out for the infinitesimal generators of the little groups for the imaginary mass components. The factors of the direct product and the representations which appear in the expansion are expressed in terms of a particular momentum representation called ``the standard helicity representation'' which enables us to use a uniform notation for all masses, whether real, zero, or imaginary. The earlier portion of the present paper summarizes the properties of these representations.

Closed‐Form Solution of the Differential Equation by Normal‐Ordering Exponential Operators
View Description Hide DescriptionA closed‐form solution to Lambropoulos' partial differential equation,subject to the initial condition P(x, y, 0) = Φ(x, y), is presented. The applicability of the normal‐ordering method to a class of partial differential equations is briefly discussed.

Analysis of the Newman‐Unti Integration Procedure for Asymptotically Flat Space‐Times
View Description Hide DescriptionAn analysis is given of the procedure of Newman and Unti for solving the vacuum gravitational field equations for all space‐times in which Ψ_{0} = O(r ^{−5}). It was found empirically by Newman and Unti that when the nonradial equations and three of the u‐derivative equations have been satisfied to their lowest nontrivial order in r ^{−1}, they are then found to be identically satisfied to all orders. A general proof of this result is given which avoids the need for direct verification.

Representations of Noncompact Semisimple Lie Groups
View Description Hide DescriptionA method is presented for the construction of unitary representations of semisimple Lie groups (or, more precisely, of the corresponding algebras), proceeding directly from the commutation relations among the canonical generators e _{±α} and h _{α}. In the case of the orthogonal groups, the correspondence between the canonical generators and the more usual tensor generators is written down explicitly. It is shown how this method can be used to construct a certain class of representations (bounded above or below) of the Lie algebras of the noncompact groups SU(p, q) and SO(p, q), which arise in the dynamical group treatment of certain physical systems.

Noncompact Algebras with Additively Labeled Multiplets
View Description Hide DescriptionA large class of noncompact simple Lie algebras is examined to determine those that admit unitary irreducible representations in which the multiplets associated with the compact subalgebra are labeled by the eigenvalues of a generator, with only a finite number of multiplets corresponding to each eigenvalue. It is shown that such representations are precisely those which are bounded above or below, so that the eigenvalues of the labeling generator are bounded above or below. It is found that many of the algebras examined do not have such representations; for example, among the pseudo‐orthogonal groups SO(p, q) with p and/or q even, only those of the form SO(p, 2) do admit representations of this kind.

Statistical Average of a Product of Phase Sums Arising in the Study of Disordered Lattice. II
View Description Hide DescriptionThis recurrence formula, derived in a previous paper [J. Math. Phys. 10, 2263 (1969)] to evaluate 〈S(k _{1})S(k _{2}) ⋯ S(k_{n} )〉,is extended here, for the case when k _{1} + k _{2} + ⋯ + k_{n} = 0 but no other partial sum of the set k _{1}, k _{2}, ⋯, k_{n} is zero. The average is of O(N) as compared to the O(1) when k _{1} + k _{2} + ⋯ + k_{n} ≠ 0. Formulas are then proposed for the situations when more than one partial sum of the set vanish. 〈S(k)S(−k)〉 is considered for a parabolic probability distributions(r) with a cutoff which shows striking similarity with x‐ray diffraction pattern of liquids.