Volume 11, Issue 5, May 1970
Index of content:

Representations of the Lorentz Group: New Integral Relations between Legendre Functions
View Description Hide DescriptionNew integral relations between Legendre functions of the first and second kind are derived. These functions figure as the basis functions of the irreducible representations of the homogeneous Lorentz group, so that the derived formulas have direct applications to invariant expansions of relativistic amplitudes.

Nonexistence of Global Solutions to the Nonhomogeneous Wave Equation, Regardless of Boundary Conditions
View Description Hide DescriptionThe fact that the ordinary derivative y′ is known to be a Darboux function implies that the right‐hand side of the differential equationy′ = f(x, y) must also be such a function. Since it is now known that the mixed hyperbolic derivative u_{xy} is also a Darboux function, this implies that under certain mild conditions the equation u_{xx} − u_{tt} = f may not have a classical global solution unless f has the proper Darboux structure. As with the ordinary differential equation, this nonexistence result does not depend on boundary conditions.

Lattice Dynamics of Cubic Lattices with Long‐Range Interactions
View Description Hide DescriptionAn analytic study is made of the dispersion relations and frequency spectra of unbounded cubic lattices in which there exist long‐range pair potentials of the form r^{−p} , where r is the distance between particles. This paper is an extension of work by Davies and Yedinak in which simple cubic lattices are studied. Cubic lattices having two particles per unit cell are now considered. The analytic behavior of the optical branches near the origin of the first Brillouin zone is determined for the case of 1 ≤ p ≤ 3 and the resulting contribution of the longitudinal optical branch to the frequency spectrum is obtained. A special case of such lattices is Kellermann's model for NaCl. This model is studied in detail.

Derivation of a Lower Bound on the Free Energy
View Description Hide DescriptionFor a Hamiltonian with a ground‐state energy E _{0} = 0, we show that for any normalized state , where From this inequality, a general formula for a lower bound on the free energy is derived.

Diffraction of Electromagnetic Waves by Planar Dielectric Structures. I. Transverse Electric Excitation
View Description Hide DescriptionA formalism is developed for solving a class of electromagnetic diffraction problems involving heterogeneous planar dielectric structures. Using analytic properties of finite Fourier transforms, we show that the electromagntic field distributions can be obtained by solving a single‐variable Fredholm integral equation of the second kind. The special case of TE‐wave propagation through a finite metallic guide with a dielectric insert is discussed in detail. Our numerical approach spans the Rayleigh to geometric optics range. Field distributions are presented for typical configurations involving positively and negatively absorbing media. Excitation of bound‐mode resonances is discussed.

Kronecker Products and Symmetrized Squares of Irreducible Representations of Space Groups
View Description Hide DescriptionApplication is made to space groups of a theorem by Mackey on the symmetrized and antisymmetrized squares of induced representations. A new and constructive proof is given of Mackey's theorem which enables the representations appearing in the decomposition of the symmetrized squares to be easily identified. The method is used to extend Birman's tables for the zinc‐blende structure, to include products involving its double‐valued representations

Neutron Transport in a Nonuniform Slab with Generalized Boundary Conditions
View Description Hide DescriptionWe prove the existence and the uniqueness of the solution of the initial‐value problem for neutron transport in a nonuniform slab with generalized boundary conditions, which include the vacuum and the perfect reflection boundary conditions as particular cases. Moreover, we show that the position‐dependent transport operator has at least one real eigenvalue and we indicate the asymptotic behavior of the neutron density as t → + ∞.

Localized States on a Hyperplane
View Description Hide DescriptionThe localized states of Newton and Wigner are reconstructed as a superposition of the canonical states of Foldy, and these states are then generalized to an arbitrary hyperplane via a procedure similar to that used in a previous work for the hyperplane generalization of helicity states. The corresponding hyperplane position operator, with mutually commuting components, is constructed and is seen to be equivalent to that local position operator given by Fleming.

Branching of Solutions to Some Nonlinear Eigenvalue Problems
View Description Hide DescriptionThe multiplicity of solutions to nonlinear eigenvalue problems is analyzed by a method originally proposed by Hammerstein, and the results are given in tables which display the relation between the number of solutions for eigenvalues in the neighborhood of a critical eigenvalue and the properties of the nonlinear function and its derivatives. Several general types of nonlinear functions are considered, and a simple method of estimating the critical eigenvalue for each type is presented. Since these functions describe physical phenomena, a stability analysis is given for the cases of multiple solutions in order to determine which solution represents the observed physical state. The results are applied to the following nonlinear eigenvalue problems: (i) nonlinear heat generation and the temperature distribution in conducting solids; (ii) temperature distribution in a heat‐conducting gas undergoing chemical reactions, leading to a thermal explosion; (iii) nonlinear effects of temperature‐dependent viscosity on the temperature distribution of a fluid flowing in a pipe; (iv) neutron flux distribution in a reactor for temperature‐dependent cross sections.

Exact Solution of a Family of Integral Equations of Anisotropic Scattering
View Description Hide DescriptionIt is shown that the solution of a certain Cauchy system provides the solution of a family of integral equations occurring in the theory of anisotropicscattering in a finite slab. Numerical experiments show that the Cauchy system is readily solved numerically, even in the case of very strong forward scattering.

Groups of Curvature Collineations in Riemannian Space‐Times Which Admit Fields of Parallel Vectors
View Description Hide DescriptionBy definition, a Riemannian space V_{n} admits a symmetry called a curvature collineation (CC) if the Lie derivative with respect to some vector ξ^{ i } of the Riemann curvature tensor vanishes. It is shown that if a V_{n} admits a parallel vector field, then it will admit groups of CC's. It follows that every space‐time with an expansion‐free, shear‐free, rotation‐free, geodesic congruence admits groups of CC's, and hence gravitational pp waves admit such groups of symmetries.

of SL(2, C)
View Description Hide DescriptionWe consider the of unimodular 2 × 2 matrices A ∈ SL(2, C), which is of the form. is a covering of the group of Euclidean motions in the plane. We compute the correspondingly factorized matrix elements of the unitary representations of SL(2, C) in an the result is given in Eq. (6). As a fringe benefit we obtain an integral transform which amounts to expansion in terms of Meijer G‐functions and which generalizes the familiar Hankel transform. The results of this paper are useful, e.g., for computing vertex functions in the theory of massless particles with continuous spin.

Problem of the Disordered Chain
View Description Hide DescriptionThe equivalence of the methods of Dyson and Kac is proven in the 1‐dimensional problem of the disordered chain, i.e., the determination of the spectrum of the frequencies of normal modes of a system of coupled oscillators, where the masses and/or spring constants are random variables.

Irreducible Cartesian Tensors. III. Clebsch‐Gordan Reduction
View Description Hide DescriptionThe reduction of products of irreducible Cartesian tensors is formulated generally by means of 3‐jtensors. These are special cases of the invariant mappings discussed in Part II [J. A. R. Coope and R. F. Snider, J. Math. Phys. 11, 993 (1970)]. The 3‐j formalism is first developed for a general group. Then, the 3‐jtensors and spinors for the rotation group are discussed in detail, general formulas in terms of elementary invariant tensors being given. The 6‐j and higher n‐j symbols coincide with the familiar ones. Interrelations between Cartesian and spherical tensor methods are emphasized throughout.

Inequalities for Appell Functions
View Description Hide DescriptionThe asymptotic expansion of one of Appell's generalizations of the Jacobi function is given for one parameter becoming large while the other is kept fixed. Inequalities are given which may be useful when both parameters become large.

Diffraction by a Circular Cavity
View Description Hide DescriptionThe reduced wave equation Δu + k ^{2} n ^{2} u = 0 is considered, where n = 1 in the exterior of a circular cylinder of radius a and n = N < 1 in the interior. The solution and its first derivatives are required to be continuous for r = a. The behavior of the solution for large ka is described, especially in the vicinity of a ray incident upon the cylinder at the critical angle. The principal novelty of the present results is a description of the solution in certain regions by a combination of geometrical optics (ray contributions) and whispering gallery modes (a kind of diffracted modes). These results are in qualitative agreement with those of Chen, but certain details in Chen's treatment require elaboration. The present results do not agree with those of Nussenzveig for N < 1, especially in the geometrical interpretation. This revision of Nussenzveig's work brings it into agreement with Keller's geometrical theory of diffraction. In fact, part of the present work can be combined with Chen's results to give a fairly complete geometrical theory for diffraction by a convex, transparent object. However, the present treatment does not give a uniform asymptotic expansion of the solution, since the expansion is not given in a transition region in the immediate vicinity of the critically reflected ray.

Noninvariance Groups in the Second‐Quantization Picture and Their Applications
View Description Hide DescriptionWe investigate the existence of noninvariance groups in the second‐quantization picture for fermions distributed in a finite number of states. The case of identical fermions in a single shell of angular momentumj is treated in detail. We show that the largest noninvariance group is a unitary group U(2^{2j+1}). The explicit form of its generators is given both in the m scheme and in the seniority—angular‐momentum basis. The full set of 0‐, 1‐, 2‐, ⋯, (2j + 1)‐particle states in the j shell is shown to generate a basis for the single irreducible representation [1] of U(2^{2j+1}). The notion of complementary subgroups within a given irreducible representation of a larger group is defined, and the complementary groups of all the groups commonly used in classifying the states in the j shell are derived within the irreducible representation [1] of U(2^{2j+1}). These concepts are applied to the treatment of many‐body forces, the state‐labeling problem, and the quasiparticle picture. Finally, the generalization to more complex configurations is briefly discussed.

Geometrical Derivation of the Conservation Laws
View Description Hide DescriptionConservation laws which are customarily obtained by invoking the invariance of a Lagrangian density under the transformations of some intrinsic symmetry group may be given a completely geometrical treatment within the context of the theory by Yang and Mills. The formalism is extended from unitary transformations to general linear transformations and the concepts of parallel transfer, covariant differentiation, and intrinsic curvature tensor are discussed. Conservation laws follow from the assumed invariance of the Lagrangian under parallel transfer defined with a general affine connection which is a direct sum of intrinsic and space‐time affinities. Conservation of generalized charge is a consequence of the arbitrariness of that part of the affinities which operates in the intrinsic space, and conservation of energy and momentum is related to the arbitrariness of the part of the affinities which operates on the space‐time indices. Some comments on Palatini's derivation of Einstein's equation of general relativity are made.

Lorentz‐Invariant Gravitational Perturbations and the Evaluation of Generalized Green's Functions
View Description Hide DescriptionThe Lorentz‐invariant perturbation theory of classical nonlinear field theories, notably the fast‐motion approximation for a system of gravitationally interacting particles, is used to illustrate the appearance of generalized Green's functions (GGF's) in physics. The (retarded) GGF's are defined as the convolution of the usual (retarded) Green's function with certain other linear functionals. A manifestly Lorentz‐invariant technique for the evaluation of the convolution integral is described and applied to two important classes of GGF's. Time‐symmetric GGF's are also discussed briefly. The technique exhibited provides the tools for the calculation of the higher‐order approximations in a Lorentz‐invariant perturbational approach to the classical n‐body problem.

Breaking of Euclidean Symmetry with an Application to the Theory of Crystallization
View Description Hide DescriptionWe present a systematic study of the processes by which the original Euclidean invariance of a quantum statistical theory can be broken to produce pure phases with a lower symmetry. Our results provide a rigorous basis for Landau's argument on the nonexistence of critical point in the liquid‐solid phase transition. A classification of the possible residual symmetries is obtained, and its connection with spectral and cluster properties is established. Our tools are those of the algebraic approach to statistical mechanics; in particular, we make an extensive use of the KMS condition. None of our proofs involves the separability of the algebra of quasilocal observables.