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Volume 11, Issue 9, September 1970

Characters of the Poincaré Group
View Description Hide DescriptionWe calculate the characters of the Poincaré group P̄ as solutions of differential equations in a way which is valid for a large class of Lie groups. We discuss the solutions of linear differential equations on an analytic manifold in a space of distributions. Then, we calculate the characters of SL(2, R) and the central distributions on P̄. Finally, we give the characters of all unitary irreducible representations of P̄, including mass 0, and some expressions which may be ``characters'' of nonunitary representations.

Mathematics of the N/D Method
View Description Hide DescriptionA complete mathematical account is given of the N/D method, subject to certain restrictions on the given functions L and ρ (left‐hand cut contribution and phase‐space factor). The restrictions are the weakest possible if the phase shift is to be Hölder continuous. Any asymptotic behavior ρ ∼ x ^{β}, 0 < β < 2, is allowed. No analyticity is assumed for L. Exhaustive existence and uniqueness theorems are given: Any allowed function F (= e ^{ iδ} sin δ/ρ) possesses an N/D decomposition, and the integral equation for N acts in a space of L _{2} functions; this equation satisfies the Fredholm alternative theorems [though its kernel has a continuous spectrum if δ(∞)/π is not integral]; any L _{2}solution of the equation yields an allowed function F; all allowed solutions may be obtained by varying the CDD parameters (which enter linearly); each solution has a uniqueness index κ; there will usually be a κ parameter infinity of solutions with each κ ≥ 0, and none with κ < 0, but precise conditions on L and ρ are given in order that there will be a negative κ solution.

Quantum Theory of the Electromagnetic Field in a Variable‐Length One‐Dimensional Cavity
View Description Hide DescriptionThe quantum theory of linearly polarized light propagating in a 1‐dimensional cavity bounded by moving mirrors is formulated by utilizing the symplectic structure of the space of solutions of the wave equation satisfied by the Coulomb‐gauge vector potential. The theory possesses no Hamiltonian and no Schrödinger picture. Photons can be created by the exciting effect of the moving mirrors on the zero‐point field energy. A calculation indicates that the number of photons created is immeasurably small for nonrelativistic mirror trajectories and continuous mirror velocities. Automorphic transformations of the wave equation are used to calculate mode functions for the cavity, and adiabatic expansions for these transformations are derived. The electromagnetic field may be coupled to matter by means of a transformation from the interaction picture to the Heisenberg picture; this transformation is generated by an interaction Hamiltonian.

Intermediate Statistics
View Description Hide DescriptionThe distinctions between intermediate statistics, parastatistics, and Okayama statistics are discussed and it is pointed out that the distribution function of the intermediate statistics does not follow from the para‐Fermi statistics. The partition function, the pressure, and the specific heat of free particles which obey intermediate statistics are calculated in one, two, and three dimensions.

Finite and Infinitesimal Canonical Transformations
View Description Hide DescriptionThe general relation between the infinitesimal generator of a 1‐parameter subgroup of canonical transformations and the usual finite generating functions is obtained. This relation is found to be simply a generalization of the Hamilton‐Jacobi equation. When the latter relation is solved for the finite generating function, the connections for a finite transformation can be determined without the need for integration of the associated infinitesimal transformation.

Zeros of the Grand Partition Function for a Lattice Gas
View Description Hide DescriptionThe following three statements about the zeros of the grand partition function of a lattice gas with negative (attractive) interactions are proved: (1) Not all the zeros will be on the unit circle in the high‐temperature limit if forces of higher order than 2‐body are included; (2) in the low‐temperature limit they will, in general, lie on the unit circle; (3) it is possible to have the zeros dense in the complex plane. It is also shown that not all polynomials with positive coefficients and roots on the unit circle are a grand partition function of a lattice gas.

Exactly Solvable Electrodynamic Two‐Body Problem
View Description Hide DescriptionThe equations of motion of the physical 2‐body problems of relativistic electrodynamics are differential‐difference equations. However, in the unphysical 2‐body problem in which one particle responds only to retarded fields and the other only to advanced fields, the equations of motion are differential equations. These differential equations are solved in the center‐of‐momentum frame by the elementary method of finding integrating factors. Solutions in an arbitrary Lorentz frame are found by a method which parametrizes the world lines on the particle velocities. All of the constants of the motion contain interaction contributions; this appears to be a characteristic feature of relativistic particle dynamics.

Comparison of SCF and k _{ν} Functions for the Helium Series
View Description Hide DescriptionWavefunctions for He, Li^{+}, Be^{2+}, and O^{6+} are presented. They were determined by using reduced modified Bessel functions of the second kind, k _{ν}(qr). The z dependence of energies calculated using such functions for high values of z is found to be the same as for the Hartree‐Fock functions.

Global Singularities and the Taub‐NUT Metric
View Description Hide DescriptionSeveral examples of singular behavior which are not apparent in the line element are discussed. It is shown that the presence of such singularities depends on the manner in which coordinate patches are assembled to form the entire manifold. It is shown that the usual connection of a coordinate patch with the Taub metric to a patch with the NUT metric is singular in the sense of this particle.

Stark Effect in Hydrogen Atoms for Nonuniform Fields
View Description Hide DescriptionThe correction for the energyeigenvalues of the Schrödinger equation for a hydrogenic atom in a non‐uniform field resulting from the inhomogeneity of the field is expressed in terms of expectation values involving the eigenfunctions of the system for a uniform field. Only the first‐order terms in the inhomogeneity of the field are retained. An examination of the symmetry of the eigenfunctions for the uniform field, followed by an application of Gauss' law, shows that the correction depends only on one component of the field gradient tensor, regardless of the symmetry of the field, except for states with magnetic quantum number m = ±1. For the latter states we find the degeneracy is removed provided that the field is not cylindrically symmetric. We evaluate the correction by applying Feynman's theorem to a pair of 1‐dimensional eigenvalue equations similar to those obtained in the separation of the uniform field problem in parabolic coordinates. All the necessary eigenvalues are calculated by the WKB method that has been previously employed in obtaining the eigenvalues for the uniform field problem. As the final result we present an expression for the zz component of the quadrupoletensor of the electron labeled according to parabolic quantum numbers. Finally, we discuss the use of this expression in the study of line broadening caused by interatomic interactions (pressure broadening).

SL(2, C) Symmetry of the Gravitational Field Dynamical Variables
View Description Hide DescriptionWe represent the spin coefficients and the Riemann tensor in the form of linear combinations of the infinitesimal generators of the group SL(2, C). This representation is similar to the way Yang and Mills write their dynamical variables in terms of the Pauli spin matrices. The spin coefficients take the role of the Yang‐Mills‐like potentials, whereas the Riemann tensor takes the role of the fields.

Radiative Transfer in a Rayleigh‐Scattering Atmosphere with True Absorption
View Description Hide DescriptionThe singular‐eigenfunction‐expansion technique is used to solve the equation of transfer for partially polarized light in a Rayleigh‐scattering atmosphere with true absorption. The normal modes for the considered nonconservative vector equation of transfer are established; two discrete eigenvectors and two linearly independent continuum solutions are thus derived. Further, the necessary full‐range completeness and orthogonality theorems are proved, so that all expansion coefficients can be determined explicitly, and, in order to illustrate the technique, an exact analytical solution for the infinite‐medium Green's function is developed. Finally, a numerical tabulation of the required discrete eigenvalue, as a function of the single‐scatter albedo, is given.

Proper Orientation of Space‐Time
View Description Hide DescriptionAn observer determining space‐time around his world line by various measurements may encounter the problem of not being able to define a unique time order and a spatial orientation along world lines of other objects. The problem is discussed and resolved with the aid of geometrical considerations.

Physical‐Region Discontinuity Equation
View Description Hide DescriptionA Cutkosky‐type formula for the discontinuity around an arbitrary physical‐region singularity is derived from precisely formulated S‐matrix principles.

Relationship between Geometrical‐Optical and Full‐Wave Solutions to the Problem of Propagation over a Nonparallel Stratified Medium
View Description Hide DescriptionThe problem of propagation over a wedge‐shaped overburden has been analyzed using a full‐wave‐solution approach. The relationship between this solution and an earlier one employing the compensation theorem is discussed in detail. The compensation theorem method makes use of the simplifying concept of of the surface impedance which is determined by a geometrical‐optical approach. The applicability and limitations of the latter approach are carefully studied for both dissipative and nondissipative over‐burdens.

Kruskal's Perturbation Method
View Description Hide DescriptionA general method for eliminating the angle variable from the equations of a perturbed periodic motion and for deriving an ``adiabatic invariant'' J has been given by Kruskal and, for a special class of Hamiltonian systems, McNamara and Whiteman have shown (to order ε^{2}) that J is related to a set of invariants I obtained from the expansion of Poisson‐bracket relations. In this work, an order‐by‐order algorithm for Kruskal's method is introduced, and a new set of invariants Z _{1} is obtained. It is shown that these invariants bear a close relation to those obtained from the Poisson‐bracket expansion and, in the special case investigated by McNamara and Whiteman, the relation between I and Z _{1} may be brought to the same form as the relation between I and J derived by those authors. Finally, the relationship between Z _{1} and J is examined, and arguments are presented that in certain cases the two are equal to all orders.

Direct Canonical Transformations
View Description Hide DescriptionSome of the perturbation methods in classical Hamiltonian mechanics lead to near‐identity transformations of the variables, with the new variables explicitly given as functions of the old ones. Two methods are used for identifying and characterizing the subclass of all such transformations which are also canonical: one approach is related to the conventional method of generating canonical transformations, while the other one uses the properties of Poisson brackets and is related to an operator method of Lie. Either of the methods may be used to derive certain steps in a perturbation method devised by Lacina, inadvertently omitted by that author.

Second Quantized Atomic Wavefunctions with Definite Unitary and Rotational Symmetry
View Description Hide DescriptionSome atomic wavefunctions for equivalent electrons in the group scheme are constructed in terms of electron fermion creation and annihilation operators. The concept of semiconjugacy is defined and shown to reduce the number of states that must be explicitly calculated. The states for the d shell are calculated and tabulated.

Some Topological Properties Connected to the Parametrized Feynman Amplitudes
View Description Hide DescriptionAn approach first developed by one of the present authors for computation involving Poincaré incidence matrix is now carried over to the so‐called ``loop‐matrix'' formalism. A matrix is introduced to prove some important properties of transformations among different sets of basic loops. Several properties in the form of lemmas and theorems are presented here. In particular, a formula of very concise form is found for the total number of possible tree graphs of a given Feynman graph in terms of its corresponding Poincaré incidence matrix. Also derived here is the formula for the total number of tree graphs in terms of any loop matrix. Their proofs show that the previously developed approach for Poincaré incidence matrix can be nicely generalized to a surprising extent to the loop matrix formalism and thus demonstrate the duality between the two formalisms.

Nonlinear Light Propagation in a Resonant Medium and Causality
View Description Hide DescriptionThe propagation of an optical pulse through a system of two‐level atoms embedded in a host medium of constant refractive indexn is considered. It is shown that the values of the solutions of the nonlinear system of coupled Maxwell and Schrödinger equations at time t at a point z, in the McCall‐Hahn approximation, depend only on data contained in the interval [z ‐ c/nt, z] at t = 0. Thus, information cannot be transmitted through this medium with velocity greater than c/n. Apparent violations of this property in the case of self‐induced transparency are explained.