Volume 13, Issue 9, September 1972
Index of content:

The Dielectric Ring in a Uniform, Axial, Electrostatic Field
View Description Hide DescriptionThe exact solutions for the fields inside and outside a dielectric ring in a uniform, axial, electrostatic field are derived in the system of toroidal coordinates. Comparison is made with the perturbationsolution generated by inverse aspect ratio expansion, and it is shown how the exact solution may be truncated to any desired accuracy. The cylindrical limits of the exact and truncated solutions are obtained.

Unitary Representations of the Homogeneous Lorentz Group in an Basis and Some Applications to Relativistic Equations
View Description Hide DescriptionUnitary irreducible representations of the homogeneous Lorentz groupO(3, 1) belonging to the principal series are reduced with respect to the subgroup . As an application we determine the mixed basis matrix elements between O(3) and bases and derive recurrence relations for them. This set of functions is then used to obtain invariant expansions of solutions of the Dirac and Proca free field equations. These expansions are shown to have the correct nonrelativistic limit.

Dynamics of Harmonically Bound Semi‐Infinite and Infinite Chains with Friction and Applied Forces
View Description Hide DescriptionThe dynamics of semi‐infinite and infinite linear chains of identical masses and ideal springs is studied. In addition to the harmonic coupling between nearest neighbors, each particle is harmonically bound to ts equilibrium position and is subject to friction and time‐dependent applied forces. The Laplace transform method is used to express the motion of all the particles. The exact solutions are found and discussed for four different cases: (a) an infinite chain, (b) a semi‐infinite chain, (c) a semi‐infinite chain with the position of the end particle specified as a function of time, and (d) an infinite chain with the position of one particle specified as a function of time. By specializing some results of the present work, those of previous calculations on simpler systems by other authors are recovered.

Lattice Wind‐Tree Models. II. Analytic Property
View Description Hide DescriptionThree lattice versions of the wind‐tree model of Ehrenfest are studied. It is shown that various moments, including the recurrence time and the Cesaro limits of the mean‐square displacement Δ(t) and of the one‐particle distribution ρ(t, x) at time t, are analytic functions of the reciprocal of the fugacity of the trees, or equivalently of the deviation 1‐ρ of the density ρ of the trees from their close packing density 1, in certain disks in the complex plane. Two of the models were considered in Paper I, but the third is new.

Exact Nearest Neighbor Statistics for One‐Dimensional Lattice Spaces
View Description Hide DescriptionIt is shown that A(n _{11},q,N), the number of ways of arranging q indistinguishable particles on a one‐dimensional lattice space of N compartments in such a way as to create n _{11} nearest neighbor pairs is . A similar expression is also derived for n _{00}, the number of pairs of vacant nearest neighbors. The normalization, first moment, and most probable value of these statistics are also discussed.

Anharmonic Oscillator with Polynomial Self‐Interaction
View Description Hide DescriptionA quantum anharmonic oscillator with a polynomial self‐interaction is defined in coordinate space by a Hamiltonian of the form H = −d ^{2}/dx ^{2} + ¼x ^{2} + g[(½x ^{2})^{ N } + a(½x ^{2})^{ N−1} + b(½x ^{2})^{ N−2} + ⋯]. Using WKB techniques we derive a secular equation which determines the eigenvalues of H for small g. We find that the qualitative analytic structure of these eigenvalues as functions of complex g remains unchanged for all fixed values of a, b,…, including a = b = ⋯ = 0. The secular equation also implies an elegant theorem which predicts how the a,b,⋯ terms in H affect the large‐order growth of perturbation theory. We use this theorem to compare the perturbative behavior of non‐Wick‐ordered and Wick‐ordered field theories in one‐dimensional space‐time. In particular, we show that the perturbation series and for the energy levels of the (gψ^{2N })_{1} and (:gψ^{2N }:)_{1}field theories differ in large order by an over‐all multiplicative constant lim_{ n→∞} A_{n}/B_{n} = exp[N(2N − 1)/(2N − 2)].

Quantum Electromagnetic Zero‐Point Energy of a Conducting Spherical Shell
View Description Hide DescriptionThe quantum zero‐point energy of a conducting spherical shell was first calculated by Boyer [Phys. Rev. 174, 1764 (1968)]. Because of the importance of this calculation and also of Boyer's uncertainty about the analytical dependence of the energy on the cutoff function, we have checked the calculation independently. We determine an analytic continuation of the energy function using the Mellin transform, and thereby show how an exact value of the self‐energy can be obtained from the divergent series. We also compute an approximate value of the self‐energy by extrapolating a direct numerical evaluation of the cutoff integrals. These calculations confirm Bover's result.

On the Symmetric Tensor Operators of the Unitary Groups
View Description Hide DescriptionThe algebraic expressions for the matrix elements of symmetric tensor operators (the powers of infinitesimal operators) of the unitary groups in the Gel'fand basis have been studied. The expressions for the isoscalar factors of the related Clebsch‐Gordan coefficients, one of the two representations to be coupled being symmetric, as well as the elements of a special recoupling matrix have been found. The supplementary symmetry properties of the isoscalar factors corresponding to the Regge symmetries of the Wigner and 6j coefficients of SU _{2} have been examined.

Deformable Magnetically Saturated Media. II. Constitutive Theory
View Description Hide DescriptionThis article is devoted to the development of constitutive equations of deformable magnetically saturated media in three dimensions. In Sec. 1 we recapitulate the local balance laws and jump conditions derived previously. A thorough study of the consequences of the objectivity requirement is given in Sec. 2. In the following sections, the material symmetry restrictions are examined and exact and approximate constitutive equations are obtained for a variety of material classes.

Procedures in Quantum Mechanics without Von Neumann's Projection Axiom
View Description Hide DescriptionThe results of previous work are generalized to include procedures whose measurement operations correspond to expectations as defined by Davies. For such procedures Von Neumann's projection axiom is not in general applicable. Finite and infinite sequences of measurements and transformations as well as finite and infinite decision procedures are considered. It is shown that with each such procedure there is associated in a unique manner a probability operator measure.

Ward‐Takahashi Relations in Massive Yang‐Mills Theory
View Description Hide DescriptionThe equivalence of massive Yang‐Mills theory in the simplest gauge to a theory of vector electrodynamics is reviewed. The resultant Feynman rules and SU(2) invariance are used to derive Ward‐Takahashi relations among the corrected vertices which are valid to all orders. One of these is employed to find a new Ward identity. A brief discussion of divergence problems is given.

Quantum Dynamics of Higher‐Derivative Fields
View Description Hide DescriptionThe generalized quantum field theory which follows from Lagrangians containing arbitrarily high‐order derivatives is formulated in an indefinite metric space. Particular attention is given to conservation laws and canonical commutation relations. The Heisenberg equations of motion are derived.

Physical Interpretation of Higher‐Derivative Field Theories
View Description Hide DescriptionPolynomial‐type field equations are shown to have a realistic physical interpretation in terms of particle form factors, both for classical fields and for ``dipole‐regularized'' quantized fields.Form factors arising from such field equations are found to give a reasonable description of the electromagnetic structure of the proton.

Scattering by Two Charged Centers
View Description Hide DescriptionThe scattering of a charged particle by two fixed charged centers is discussed. The scattering potential is long‐range and spheroidal. It is pointed out that the general method for handling the short‐range spheroidal potential is not directly applicable here. The integral differential equation for the Coulomb spheroidal phase shift is given in the text. The behavior of the phase shift is discussed in Born's approximation. A method for solving for the radial spheroidal Coulomb wavefunction is also given.

On the Decomposition of Direct Products of Irreducible Representations
View Description Hide DescriptionA lemma concerning irreducible representations contained in the decomposition of a direct product of irreducible representations of simply reducible groups is generalized to arbitrary decomposable unitary and nonunitary groups.

The Symmetric Group and the Gel'fand Basis of U(3). Generalizations of the Dirac Identity
View Description Hide DescriptionIt is shown that the symmetrization of N particle states by means of the orthogonal units of the algebra of the symmetric group S_{N} yields the Gel'fand basis states of the irreducible representations of U(3). The existence of generalizations of the Dirac identity is demonstrated, and a connection between the symmetrized two‐ and three‐body exchange operators and the invariants of U(3) is established.

Lie Theory and the Lauricella Functions F_{D}
View Description Hide DescriptionIt is shown that the Lauricella functions F_{D} in n variables transform as basis vectors corresponding to irreducible representations of the Lie algebra. Group representation theory can then be applied to derive addition theorems, transformation formulas, and generating functions for the F_{D} . It is clear from this analysis that the use of symmetry in atomic and elementary particle physics will lead inevitably to the remarkable functions F_{D} .

Averages of the Components of Random Unit Vectors
View Description Hide DescriptionIt is shown that a parametrization of the orthogonal and unitary groups due to Hurwitz can be used to evaluate averages of components of random unit vectors for those two spaces. Explicit results are given for moments which are general enough to include most cases of interest in applications.

Wave Propagation in a Random Lattice. I
View Description Hide DescriptionThe small amplitude periodic classical motion of a lattice of particles about their equilibrium positions in a lattice is considered. The effect of random masses and random spring constants upon the coherent or mean motion is treated by using an equation for the coherent motion derived previously by Keller and others. From this equation the dispersion equation for coherent wave motion is determined. It is solved for the case in which the spring constants are not random but the masses are random. Explicit results are obtained in the one‐dimensional case for both uncorrelated and exponentially correlated mass defects. They show an alteration of frequency or of wavelength and of phase velocity, as well as an attenuation due to scattering by the defects. In addition new highly attenuated modes are found. These results are utilized in Part II in which various reflection and Green's function problems are treated.

Lattice Green's Functions for the Triangular and Honeycomb Lattices
View Description Hide DescriptionThe lattice Green's function for the triangular lattice at an arbitrary lattice site is expressed in terms of the complete elliptic integrals of the first and second kind. The lattice Green's function for the honeycomb lattice is shown to be expressed in terms of the one for the triangular lattice. The results obtained are shown by graphs.