Volume 9, Issue 4, April 1968
Index of content:

Dyadic Analysis of Spatially Homogeneous World Models
View Description Hide DescriptionThe dyadic formalism is applied to cosmological models, and leads to a convenient set of first‐order ordinary differential equations. The Bianchi‐Behr type of any model is shown to be constant in time, regardless of the state of the matter content. The case of perfect fluid matter content is formulated. Type V models and Type VIII and IX models with incoherent matter are discussed, and some consistent subtypes delineated. The Gödel Hamiltonian for symmetric Type IX models is derived and generalized.

Alpha‐Function Technique for Two‐Center Integrals
View Description Hide DescriptionA closed form has been derived for the function α_{ l }(NML  a,r) introduced by Löwdin for two‐center integrals in molecules and solid states. This expression is general and applied to all values of l, L, and M.

Renormalization and Composite Effects in the Lee Model
View Description Hide DescriptionWe consider the version of the Lee model with relativistic kinematics. The mass renormalization of the V particle, described in purely field‐theoretic terms, is a nonlocal effect. We discuss the composite limit of the model. The natural choice of composite field is nonlocal in the elementary constituents. In the composite limit, Z _{1} does not vanish. The Hilbert‐space formalism of the composite theory is not equivalent to that of an N‐θ theory with a four‐particle interaction. All these results are cutoff independent.

Some Properties of the Contact between Theory and Experiment
View Description Hide DescriptionThis paper uses the sample probability‐space description to review, under general conditions, the measurement process whereby an empirical expectation value is obtained for comparison either with other values or with values calculated from a theory. The emphasis here is on some of the conditions which must be satisfied by general sequences of single measurements, about which an observer may have relatively little knowledge, in order that such sequences yield suitable expectation values. In particular, sequences are considered for which the requirements that the single measurements of the sequence be independent and made of the same physical quantity on ensembles of identically prepared systems are not necessarily satisfied. The differences between sequences which satisfy these requirements and those which do not are discussed in terms of the implications or meaning of the resultant expectation value as a point of contact between theory and experiment.

Structure of the Phonon Propagator for an Anharmonic Crystal
View Description Hide DescriptionThe phonon propagator for an arbitrary crystal is the analytic continuation to the complex z plane of the Fourier coefficient of the imaginary time correlation function, where A _{ k j } is the field operator for phonons with wavevector k and polarization or branch index j. Considering D(k; u) as a 3r × 3r matrix whose elements are labeled by j and j′ (j,j′ = 1, 2, …, 3r), where r is the number of atoms in a primitive unit cell of the crystal, the restrictions imposed on the form of this matrix by the symmetry and structure of the crystal are determined here. In particular, it is proved that the element D_{jj′} (k; u) vanishes unless j and j′ label normal modes of vibration which transform according to the same row of the same irreducible multiplier representation of the point group of the wavevector k, G _{0}(k). As a corollary to this result it follows that if no two modes labeled by the wavevector k exist whose frequencies are different, but whose associated eigenvectors transform according to the same irreducible multiplier representation of G _{0}(k), the matrix D_{jj′} (k; u) is diagonal in j and j′.

Zero‐Mass Representations of the Poincaré Group in an O(3, 1) Basis
View Description Hide DescriptionUnitary irreducible representations of the Poincaré group, corresponding to zero mass and finite helicity, are reduced with respect to the subgroup of homogeneous Lorentz transformations. The action of the energy‐momentum operators in the basis suited to this reduction is examined.

Current Formalism. I. Ordering Theorems for Currents
View Description Hide DescriptionThe basic assumptions of the theory are strong unitarity, Bogoliubov causality, completeness of the in (out) fields on a unique vacuum, and Poincaré invariance. The algebra of II functions, which are integral operators in the form of tempered distributions, is the main technical tool developed here. These II functions can be regarded as generalized step functions where the combination law is such that products with δ functions and their derivatives are well defined. An application of this algebra of II functions in the context of the above assumptions leads to the current‐formalism representations of nth‐order functional derivatives of the current and S _{op}. These are alternatively called the R_{P} and P product representations, respectively, the counterparts to the R and Φ product representations of the nth‐order derivative of the field and S _{op} in the field formalism. In a subsequent paper these relations are used to derive the integro‐differential equations of Pugh [R. E. Pugh, Ann. Phys. (N.Y.) 23, 335 (1961)] in a form amenable to a diagrammatic analysis. The perturbation series is then shown to be unique and finite with no cutoffs and a number of parameters that is independent of (increases with) the order of expansion for renormalizable (nonrenormalizable) interactions.

Current Formalism. II. The S Matrix in Perturbation Theory
View Description Hide DescriptionThis work has accomplished in the context of asymptotic quantum field theory the following objectives. (1) The S‐matrix equations of Pugh [R. E. Pugh, Ann. Phys. (N.Y) 23, 335 (1961)] and their generalization in the manner of Chen [T. W. Chen, Ann. Phys. (N.Y.) 42, 476 (1967)] are derived without the aid of an interacting field. (2) A diagrammatic representation of these integrodifferential equations is demonstrated. (3) The problem of boundary conditions for a self‐interacting system is solved in perturbation theory. This leads to a finite, divergence free, no cutoff expansion in the physical coupling constant. For renormalizable interactions, the only additional parameter is the physical mass, whereas for nonrenormalizable interactions, uniqueness of the expansion requires additional parameters with increasing order of expansion. (4) The success of the perturbation expansion serves as a posteriori justification of the formulation in CF.I., [J. G. Wray, J. Math. Phys. 9, 537 (1968)] upon which the present work is built. The Π or ``generalized step'' function and its associated algebra plays the principal technical role in facilitating this work. The crucial Π‐ordering relations and theorems developed in CF.I. are reviewed here.

Point Transformation of Classical Hard‐Core Potential
View Description Hide DescriptionThe method of an extended canonical point transformation is used to reformulate the singular repulsions in a classical hard‐sphere gas as equivalent velocity‐dependent interactions. The approach provides a Hamiltonian in which the repulsions appear as nonlocal potential interactions between the particles and may therefore be treated within any of the conventional perturbation methods of many‐body analysis. Application of the technique to obtain a kinetic equation for a hard‐sphere gas is outlined.

Modified Born Approximation and Elastic Scattering by Weak Central Potentials
View Description Hide DescriptionWhen only a few partial waves are substantially phase shifted and yet many partial waves are slightly phase shifted, it is possible to use the direct Born approximation, provided that one projects out the inaccurate lower partial waves and replaces them by accurate theoretical or phenomenological phase shifts. We test this technique for central potentials with two different well strengths, i.e., one which will fail to bind the 1Swave, and one which can bind the 1S state. We compare numerically generated angular distributions and total cross sections with those obtained from a modified form of the direct Born approximation. The technique would be useful for weak forces, e.g., the nucleon‐nucleon and electron‐atom interactions, but inapplicable for strong forces, e.g., atom‐atom interaction.

Green's Function of a Particle in a Uniform External Field
View Description Hide DescriptionThe exact propagator of a spinless charged particle in a uniform electromagnetic field with E = H and E·H = 0 is shown to lack the pole corresponding to the mass of the particle.

Analytic Properties of a Class of Nonlocal Interactions. I
View Description Hide DescriptionThe analytic properties of the functions S_{l} (k) for a class of nonlocal interactions are studied in the complex k (wavenumber) plane, for physical angular momenta. The results are compared with those of the local interactions. Two specific examples are discussed.

Asymptotic Gravitational Field of the ``Electron''
View Description Hide DescriptionThe Einstein‐Maxwell equations appropriate to the exterior metric and fields of a source characterized by mass, electric charge, and magnetic dipole moment are formulated. Because of the presence of a nonvanishing Poynting vector, the metric tensor must include off‐diagonal elements. A rigorous reduction of the metric tensor to three unknown functions is accomplished and the field equations are solved to second order in the gravitational constant. Using the parameters of the electron and proton, we find that the magnetic dipole terms dominate the metric at small distances and that general relativistic effects become important at distances of the order of 10^{−22} cm. Possible applications of the asymptotic metric are discussed.

Gravitational Radiation in an Expanding Universe
View Description Hide DescriptionAsymptotic expansions are used to study outgoing gravitational radiation in an expanding, dust‐filled Friedmann universe of negative curvature. It is found that the interaction with the matter modifies the ``peeling‐off'' behavior. A quantity is defined which is interpreted as the total mass of the source and the disturbance, and which monotonically decreases as gravitational radiation is emitted. The group of coordinate transformations that preserve the asymptotic form of the metric and the boundary conditions is the same as the isometry group of the undisturbed Friedmann model. This may indicate that no physical significance attaches to the extra transformations of asymptotically flat space which are not contained in the inhomogeneous Lorentz group.

Perturbation Expansion for Real‐Time Green's Functions
View Description Hide DescriptionThe development of the time‐translation operators in a matrix element of an arbitrary operator is examined. It is noted that we may interpret time as evolving from some remotely early time (t _{0}) to a time in the far future (t _{∝}) and then back to (t _{0}). Using this interpretation, a perturbation expansion is developed for Green's functions defined along this path and a separation of the two‐particle interaction terms into self‐energy parts and single‐particle Green's function terms is justified for quantities on this path. A connection is established between the real‐time Green's functions and the Green's function defined along the path, thereby yielding a perturbation expansion for the real‐time functions and a justification of the separation of the interaction terms in the equations of motion for the real‐time quantities. The transport equations of Kadanoff and Baym are derived without resorting to an analytic continuation from imaginary times and without the correction terms of Fujita.

Extension Theorem and Representations
View Description Hide DescriptionAn extension theorem prescribing a method for constructing the (n + 1)‐dimensional Lorentz groupO(n, 1) from the n‐dimensional inhomogeneous rotation group, proved earlier, has been recapitulated and some comments on the theorem are made. The prescription is used to construct the unitary irreducible representations of the Lorentz groupsO(2, 1), O(3, 1), and O(4, 1), and it is found that only a limited number of representations are allowed.

Screening in the Schrödinger Theory of Scattering
View Description Hide DescriptionThe time‐dependent Schrödinger theory of scattering is studied rigorously with the potential V(x) replaced by e ^{−εxV }(x). Sufficient conditions are given that the Mo/ller wave matrices Ω± be obtainable from this theory as ε → 0. The conjecture that this theory can be used to define a reasonable S matrix when the Ω± do not exist is false for the Coulomb potential.

Note on the Geometric Theory of Neutrinos
View Description Hide DescriptionThe geometric theory of neutrinos proposed by Penney is investigated. It is shown, from the conditions imposed, that the space‐time is conformally flat. All solutions of the field equations are found and attention is paid to the limiting process described by Penney.

Comments on Separability Operators, Invariance Ladder Operators, and Quantization of the Kepler Problem in Prolate‐Spheroidal Coordinates
View Description Hide DescriptionThe Schrödinger equation for the hydrogen atom separates in three coordinate systems: spherical, parabolic, and prolate spheroidal. The separability operators associated with the separation constants for these three systems are exhibited and discussed. Also, for these systems, the invariance ladder operators which transform a simultaneous eigenfunction of the separability operators into a different simultaneous eigenfunction of the same energy are discussed with reference to the elements of the O _{4}Lie algebra. Quantization of the Kepler problem in terms of prolate spheroidal coordinates is accomplished and discussed.

Existence, Uniqueness, and Convergence of the Solutions of Models in Kinetic Theory
View Description Hide DescriptionA recently proved theorem of existence and uniqueness for the linearized Boltzmann equation is extended to two‐ and three‐dimensional domains and general boundary conditions. The proof is valid for collision operators having a finite collision frequency, which can arise either from an angular or a radial cutoff or by assuming a modelequation. Finally, convergence of the solutions of kinetic models to solutions of the actual Boltzmann equation is shown to hold for the boundary‐value problems considered in this paper.