Volume 9, Issue 3, March 1968
Index of content:

Minimum Uncertainty Product, Number‐Phase Uncertainty Product, and Coherent States
View Description Hide DescriptionThe number‐phase uncertainty products proposed by Carruthers and Nieto are studied to determine whether they are minimized by coherent states. It is found that coherent states do not minimize these products. States that do minimize some of the uncertainty products are constructed. Variational techniques for the study of arbitrary uncertainty products are developed.

Onset of ODLRO and the Phase Transition in the Ideal Boson Gas
View Description Hide DescriptionThe divergence of the constant‐pressure specific heatC_{P} and the isothermal compressibility K_{T} as one lowers the temperature of the ideal boson gas to the transition temperature T_{c} is discussed in terms of the onset of off‐diagonal long‐range order (ODLRO) of the one‐particle density matrix ρ_{1}.

Electron Correlations, Magnetic Ordering, and Mott Insulation in Solids
View Description Hide DescriptionA model is constructed for the purpose of investigating electron correlations pertinent to magnetic ordering and Mott insulation in solids. The model consists of an assembly of interacting itinerant electrons in a periodic atomic lattice, such that the intra‐atomic coupling between electrons is extremely strong. The correlations due to this latter coupling serve to prevent electrons of opposite spin from occupying the same atomic state, except in virtual transitions. Thus their net effect is to renormalize certain interactions and, also, to confine the state vectors of the entire system to a subspace, of the Hilbert space, that is kinematically available to them. The observables are thus represented by operators on whose algebraic properties are different from thsoe of the corresponding operators on Thus, the correlations due to intra‐atomic forces are imbedded in the theory in the form of the new algebra. In cases of one electron per atom, these correlationslead simply to both magnetic ordering and Mott insulation. In cases of nonintegral number of electrons per atom, they can lead to magnetic ordering, subject to specified conditions.

Foundations for Quantum Statistics
View Description Hide DescriptionA derivation of Fermi and Bose statistics is given, based on the general structure of quantum mechanics, together with a simple axiom of direct physical significance. The axiom concerns an operation, denoted by ∘, forming the union of two states; denotes the state of a compound system whose parts are in the states Let denote 1‐particle states and assume: (a) exists whenever (b) (c) the transition probability between and is zero if and and (d) the product of the transition probabilities from and from if and It is then shown that, at least in so far as 2‐particle states are concerned, the particles obey either Fermi or Bose statistics.

Transmission of Electromagnetic Waves through a Conducting Slab. I. The Two‐Sided Wiener‐Hopf Solution
View Description Hide DescriptionThis is the first of a series of papers dealing with propagation of electromagnetic waves through a metallic slab of finite thickness. In this first paper, we present a method for solving the integrodifferential equation governing the electric field in the interior of the metal when the electrons in the interior of the metal suffer diffuse reflection at each surface. The method is potentially of use in a wide class of problems, namely, the finite‐slab generalization of all those semi‐infinite‐medium problems which are conventionally studied by the Wiener‐Hopf technique. The solution given here is an iterative one with successive terms converging as e^{−L/l} , where L is the thickness of the slab and l is the range of the kernel of the integral term in the equation.

State Labeling of the Irreducible Representations of SU_{n}
View Description Hide DescriptionState labeling of the irreducible representations of SU _{3} is done by using Littlewood's rules for the analysis of products of representations of unitary groups. The method is generalized to any SU_{n} .

Unitarity of Dynamical Propagators of Perturbed Klein‐Gordon Equations
View Description Hide DescriptionAfter discussing the basic notions of quantizations as representations of the Weyl relations, a criterion is established for a symplectic transformation on a classical linear system to be unitarily implementable in the free (zero‐interaction) representation. The result is applied to the temporal propagators of to obtain a condition which is sufficient to ensure that they are unitarily implementable in the free representation of the quantized Klein‐Gordon field of mass m. Necessary conditions are also obtained when K commutes with m ^{2} I − Δ. Several examples are discussed, the most interesting of which is that of a mass jump (i.e., K = m′ ^{2} I), where the results given are fairly complete.

Relationships among the Wigner 9j Symbols
View Description Hide DescriptionSeveral identities satisfied by the 9j symbols or by the product of 6j and 9j symbols are derived by means of the symmetry properties of the Möbius strip type 15j symbol; in particular, the identity.The resulting recursion relations for the 9j symbols are also considered.

Embeddings of the Plane‐Fronted Waves and Other Space‐Times
View Description Hide DescriptionThe plane‐fronted waves of general relativity are embedded in a six‐dimensional pseudo‐Euclidean space of signature −2. Two distinct families of embeddings are found. Embeddings of several well‐known space‐times are obtained. Certain results of J. Rosen are improved.

Structure of the Dirac Bracket in Classical Mechanics
View Description Hide DescriptionWe discuss the structure of the Dirac bracket in classical mechanics. We consider a generalization of the usual Poisson bracket and show the close connection of this generalization to the Lagrange brackets of classical mechanics. We show how the Dirac bracket appears as a particular case of the generalized Poisson bracket, thus giving a simple reason why the Jacobi identity holds for the Dirac bracket. We also discuss the nature of the transformations generated via the Dirac bracket and the relation of these to canonical transformations.

Unitary Representations of the Lorentz Groups: Reduction of the Supplementary Series under a Noncompact Subgroup
View Description Hide DescriptionUnitary representations of O(2, 1) belonging to the exceptional class are reduced with respect to the noncompact subgroup O(1, 1). We recover the result that the spectrum of the generator of this subgroup covers the real line twice. Unitary representations of O(3, 1) belonging to the supplementary series are reduced with respect to the noncompact subgroup O(2, 1). These representations of O(3, 1) may be labeled by a parameter ρ in the range 0 < ρ < 1. Representations corresponding to 0 < ρ ≤ ½ yield upon reduction only those representations of O(2, 1) that belong to the continuous nonexceptional class; each of these appears twice. A representation corresponding to ½ < ρ < 1, however, yields upon reduction a single representation of O(2, 1) of the exceptational class (with parameter σ = ρ − ½) and, in addition, all the representations of O(2, 1) of the nonexceptional continuous class. The exceptional representation appears only once, while the nonexceptional ones appear twice each.

Asymptotic Properties of Perturbation Theory
View Description Hide DescriptionThe perturbation expansions are derived by a technique which does not assume that convergent expansions exist. The theory is shown to be asymptotic, and criteria are developed to determine if a finite number of terms underestimates or overestimates the exact result for sufficiently small values of the coupling constant.

Numerical Solution of a Singular Integral Equation Encountered in Polymer Physics
View Description Hide DescriptionA numerical method for the solution of an integral equation of the type encountered in the Kirkwood‐Riseman theory of intrinsic viscosities of flexible macromolecules is investigated. The absolute accuracy and rate of convergence of the method are evaluated for a special case and the results of this method are compared with another recently proposed method of solution.

Scattering of Scalar Waves by a Convex Transparent Object with Statistical Surface Irregularities
View Description Hide DescriptionThe scattering of time‐harmonic spherical scalar waves by a large, convex, transparent, dense, and three‐dimensional object with statistically corrugated surface is considered. The maximum deviation of the corrugated surface from the smooth one is assumed to be small, and hence the boundary‐perturbation technique is utilized in this study. First, the scattering of scalar waves by a large, transparent, and dense sphere with statistical surface irregularities is treated as a canonical problem in the general discussion. After the perturbation solution is expanded asymptotically for large ka, it is found that the higher‐order solutions can be obtained from the zeroth‐order solution in a simple and straightforward manner. Then this relationship is generalized to scattering by a large, convex, transparent, and dense object with statistical surface irregularities; a general recipe is given. Finally, the asymptotic expressions of mean values of the scatteredwavefunction and the scattered intensity are given for the general problem.

Local Characterization of Singularities in General Relativity
View Description Hide DescriptionWe formulate a new approach to singularities: their local description. Given any incomplete space‐time M, we define a topological space, the ``g boundary,'' whose points consist of equivalence classes of incomplete geodesics of M. The points of the g boundary may be thought of as the ``singular points'' of M. Local properties of the singularity may now be described in a well‐defined way in terms of local properties of the g boundary. For example, the notions: ``dimensionality of a singularity,'' ``past and future of a singular point,'' ``neighborhood of a singular point,'' ``spacelike or timelike character of a singularity,'' and ``metric structure of a singularity'' may all be expressed as properties of the g boundary. Two applications of the g boundary outside of the realm of singularities are discussed: (1) In the case in which the space‐time M is extendable (for example, Taub space), the g boundary is shown to be that regular 3‐surface across which M may be extended [in this case, the Misner boundary between Taub and Newman‐Unti‐Tamburino (NUT) space]. (2) With a slight modification of the definitions, the g boundary of an asymptotically simple space‐time is shown to be Penrose's surface at ``conformal infinity.'' The application of the g boundary technique to singularities is illustrated with a number of examples. The g‐boundary structure of one particular example leads to our consideration of non‐Hausdorff space‐times.

General Griffiths' Inequalities on Correlations in Ising Ferromagnets
View Description Hide DescriptionLet N = (1, 2, ⋯, n). For each subset A of N, let J_{A} ≥ 0. For each, let σ_{ i } ± 1. For each subset A of N, define . Let the Hamiltonian be − Σ_{ ACN } J_{A} σ^{ A }. Then for each A, , and . This weakens the hypothesis and widens the conclusion of a result due to Griffiths.

Tail of a Gravitational Wave
View Description Hide DescriptionA first‐order quadrupole sandwich wave of gravitational radiation exploding from a first‐order Schwarzschild mass is examined to second order. If the second‐order field preceding the sandwich wave vanishes, it is shown that the region of space‐time following the sandwich wave contains a second‐order, imploding quadrupolewave. The rest of the second‐order field in the space‐time region following the sandwich wave is also given, and it is seen to consist of monopole, quadrupole, and 16‐pole nonradiative motions.