Volume 11, Issue 1, January 1970
Index of content:

Generalized Addition Theorem for Spherical Harmonics
View Description Hide DescriptionIn analogy with the familiar addition formula for Legendre polynomials, a generalized addition theorem is proved. A general spherical harmonic, depending on the two angles θ_{1} and φ_{1}, is expressed as an expansion involving spherical‐harmonic functions of (θ, φ) and of (θ′, φ′). The six angles are related to each other through the equationsThe expansion is then used in the proof of an integral theorem for spherical harmonics.

Six Integral Theorems for Vector Spherical Harmonics
View Description Hide DescriptionWith the aid of a generalized addition theorem for spherical harmonics [J. Math. Phys. 11, 1 (1970), preceding paper], six integral theorems for vector spherical harmonics are proved. A source‐point direction (θ_{1}, φ_{1}) is first expressed in terms of a field‐point direction (θ, φ) and a polar‐coordinate angle‐pair (θ′, φ′), which has as its polar axis the field‐point direction. For a particular choice of n and m, all the components of the vector spherical harmonics for the source, expressed in terms of (θ_{1}, φ_{1}), are integrated over the relative azimuth angle φ′ while the field‐point direction (θ, φ) and the relative polar angle θ′ are held fixed. The result in each case is a spherical harmonic or vector spherical harmonic of the field‐point direction, with the same n and m but now depending on (θ, φ) instead of (θ_{1}, φ_{1}), multiplied by an explicit function of the relative polar angle θ′.

Vector Spherical Harmonic Expansion in the Time Domain of the Retarded Hertz Vector for a Distributed, Transient Source‐Current Configuration
View Description Hide DescriptionAn equation giving the retarded Hertz vector II(r, t) in terms of a source‐current distribution J(r _{1}, t _{1}) is derived. Both vector functions are written as expansions in vector spherical harmonics, with the expansion coefficients containing the dependence upon radial distance and time, while the angular dependence is kept within the harmonic functions. After integration over two angles, expressions are obtained giving the expansion coefficients for the Hertz vector, which depend upon (r, t), in terms of the corresponding coefficients for the current, which depend upon (r _{1}, t _{1}). The original four‐dimensional problem is thus reduced to two dimensions, but with the four‐dimensional causality requirements satisfied at each step of the analysis.

Magnetic Field Generated by a Transient Current Distribution
View Description Hide DescriptionThe preceding paper [J. Math. Phys. 11, 9 (1970)] gave the retarded Hertz vector in terms of a general transient source‐current distribution. Here, this Hertz vector is differentiated to give the vector potential and the magnetic field vector. All of these vector quantities have been expanded in vector spherical harmonics, and in each case it is the expansion coefficients, which are functions of the radial distance r and the time t, which are expressed in the (r, t) plane as two‐dimensional integrals over the source‐current expansion coefficients having the corresponding values of n and m, where these are the mode parameters characterizing the vector spherical harmonics , and .

Electric Field Generated by a Transient Current Distribution
View Description Hide DescriptionThe retarded Hertz vector, obtained earlier [J. Math. Phys. 11, 9 (1970)], is differentiated to give the scalar potential and the electric‐field vector associated with a general transient source‐current distribution. The scalar and vector quantities are expanded in terms of scalar and vector spherical harmonics. For each mode, characterized by specific values for n and m in the expansions, the scalar potential and electric‐field vector are expressed in the (r, t) plane as two‐dimensional integrals over the causally accessible portion of the source‐current distribution.

Kernel Integral Formulas for the Canonical Commutation Relations of Quantum Fields. II. Irreducible Representations
View Description Hide DescriptionContinuing our investigation of the kernel or group integral for the canonical commutation relations introduced by Klauder and McKenna, we prove that a representation fulfilling any sort of kernel integral formula is irreducible. This has been conjectured by Klauder and McKenna. After collecting some auxiliary results, a complete classification of all representations is given for which a kernel integral formula in the form of a limit superior holds. It is shown that these are just the partial tensor‐product representations and that the limit superior can be replaced by an ordinary limit over a fixed subsequence, thus allowing the transition from norms to scalar products. Then the basis‐independent kernel integral in the form of a sup is investigated. Here the supremum is taken over all bases of the test‐function space. Under a not‐very‐strong irreducibility assumption, we show that this can be reduced to the vacuum functional and that there exists a fixed sequence of subspaces of the test‐function space such that the sup can be replaced by an ordinary limit which again allows a transition to scalar products. The results are strikingly similar to the case of cyclic field. This tempts us to conjecture that a representation fulfilling a kernel integral formula is both irreducible and cyclic with respect to the field just as in the case of finitely many degrees of freedom.

Semisimple Subgroups of Semisimple Groups
View Description Hide DescriptionThe projection theorem for weights of a representation of a semisimple group G on restriction to a semisimple subgroup is derived, and the existence of a subgroup corresponding to a given projection is discussed. Dynkin's definition of the index of a simple subgroup is extended to the case of G being only semisimple, and the geometrical meaning of the index is given. A method is developed for finding branching rules for both regular and nonregular subgroups. Explicit general formulas for the branching multiplicities are obtained for all cases when G is of rank 2 and for B _{3}(R _{7}) → G _{2}. Applications to the construction of weight diagrams and the ``state‐labeling'' problems for B _{2} and G _{2} are mentioned.

Charge and Pole: Canonical Coordinates without Potentials
View Description Hide DescriptionFor particles having both magnetic and electric charge it is shown that (a) in the nonrelativistic many‐particle problem where only Coulomb and Biot‐Savart fields need be considered and (b) in the one‐particle relativistic problem (orbital pole‐charge moving around a fixed pole‐charge), the well‐set classical dynamics can be reduced directly from the equations of motion to Hamiltonian form without the introduction of potentials and Dirac strings. The Lie‐Koenigs theorem, which can give Hamiltonian format to any dynamics, is invoked for this. The essential feature is that canonical coordinates cannot be physical particle coordinates. For (a) and (b), suitable canonical variables are explicitly constructed. Using only Bohr‐Sommerfeld quantization, the Schwinger charge‐pole quantum condition is obtained for pure‐charge‐pure‐pole interactions; but when Coulomb forces are additionally considered, no quantum restriction on charge and pole strength is required.

Field‐Theoretic Description of Massless Particles with Higher Spin and Definite Parity. I. Integer Spin
View Description Hide DescriptionIt is known that there exist S + 1 equivalent covariant formulations of the theory of free massive particles with integer spin S and definite parityP. From these S + 1 equivalent theories one gets by putting m = 0 the description of S + 1 different massless particles, with helicities (k, −k), k = 0, 1, …, S. This property is already known in the case S = 1. In this paper we demonstrate explicitly how to obtain three kinds of massless particles, with different helicities, from three equivalent formulations of free massive theory with S = 2. Further, we outline the general argument for arbitrary integer value of S.

Null Electromagnetic Fields in General Relativity Admitting Timelike or Null Killing Vectors
View Description Hide DescriptionThe paper presents a study of stationary null electromagnetic fields coexisting with a dust distribution. Two exact solutions corresponding to pure radiation fields with the propagation vector as the Killing vector are presented.

Fourier Expansions of Functions of the Distance between Two Points
View Description Hide DescriptionThe expansion of any function f(r) of the distance r between two points is given as a Fourier series. This is a generalization of results given earlier by Ashour for r^{n} and log r. The Fourier‐series expansion for the product r^{N}e^{iM} θ is also given, where now r = (r, θ) denotes the sum of r _{1} and r _{2}, the coordinate vectors of the two points. This is further generalized for the product of a function f(r) of r and a circular harmonic. Expansions of similar products involving spherical harmonics have been given earlier by Sack. Special cases when f(r) is a Bessel function or a modified Bessel function are considered.

Connection between Marchenko Formalism and N/D Equations: Regular Interactions. II
View Description Hide DescriptionIn a previous paper we have shown, for S‐waves, that the resulting integral equations of the f/fequations (equivalent to the N/D approach) can be obtained from the Marchenko formalism. The potential V(μ, r) reconstructed from the discontinuity μΔ(x) is , where is the Fredholm denominator of the Jost solution and that of the resulting integral equation. For ``regular discontinuities'' we find different classes of V(μ, r). First, if , then V(μ, r) is not ``regular at the origin'' [in general, we find that V(μ, r) becomes marginally singular: repulsive and singular like r ^{−2}]. Secondly, if , then V(μ, r) is ``regular'' at the origin and we obtain the following: (i) If μ is less than the smallest modulus root of , then V(μ, r) has no poles for r ≥ 0. This range of μ‐values where the iteration series of the resulting integral equations converge is limited by the smallest μ value where a real or complex ghost can appear or where a bound state can appear at zero energy. (ii) For μ larger than this smallest modulus root but μ inside the interval given by the first positive and negative roots, V(μ, r) has no second‐order poles for r ≥ 0. These results (i) and (ii) are obtained with the restriction that in the considered interval there do not exist (μ, r)‐values such that , and from our study we cannot conclude that this is always true. (iii) For μ outside the above interval, V(μ, r) has poles of the second order for r > 0, the ``bound states'' being, in general, real or complex ghosts, or ``bound states'' corresponding to badly behaved potentials. We find also that the Jost solutions for energy equal to zero are . This gives the connection between ghosts and, in general, possible bound states appearing at zero energy; this gives also the relations between poles of V(μ, r) corresponding to opposite μ values. These results for the Jost function correspond to a normalization at infinity, so we have considered the problem of subtractions with normalization at an arbitrary point. Then the new Fredholm determinant is the product of the old one by the value of the Jost function at the subtracted point. It follows that, if the first μ‐greater‐than‐zero and the first μ‐lessthan‐zero roots of are not opposite (r ≥ 0), the μ interval of convergence of the iteration series is enlarged for the subtracted equation.

Existence and Uniqueness of Crossing Symmetric N/D‐Type Equations Corresponding to the Klein‐Gordon Equation
View Description Hide DescriptionN/D‐type equations satisfying crossing symmetry are established inside the relativistic wave‐mechanics formalism (Klein‐Gordon equations). We show that, for a superposition of exponential potentials with finite zeroth moment, the N/D‐type equations have unique solutions. Furthermore, for sufficiently weak couplings the solutions are physically available.

Diffraction of Waves by a Conducting Cylinder Coated with a Moving Plasma Sheath
View Description Hide DescriptionThe scattering of plane electromagnetic waves by a perfectly conducting cylinder coated with a moving dielectric or plasma sheath is investigated theoretically. The homogeneous sheath is assumed to be moving in the axial direction with a uniform velocity v_{z} with respect to the conducting cylinder. Solutions of this problem are obtained by making use of the special theory of relativity, the covariance of Maxwell's equations, and the Lorentz transformations. Results are given in terms of the radiation patterns of the scattered fields. A rather unique feature concerning mode coupling between the incident wave and the scattered wave is found. Even at normal incidence for v_{z} ≠ 0, an incident Ewave or Hwave will produce a scattered wave which contains both E and Hwaves. Detailed discussions are presented.

Symmetrized Tensor Geometry
View Description Hide DescriptionFor Weyl symmetrized tensors, the inner product structure has previously been proven to be identical within equivalent multiplets. Here the inner product structure between a set of equivalent multiplets is proven to be essentially the same as that within the equivalent multiplets. This property is important because it separates the problem of orthogonalization within equivalent multiplets from the problem of orthogonalization between equivalent multiplets. Thus, full orthogonalization can be achieved by, first, identically orthogonalizing each of a set of equivalent multiplets and, then, recoupling the equivalent multiplets with coefficients which do not depend on individual vectors within the multiplets. These statements apply to symmetrized tensor multiplets of both the permutation group and an underlying group (such as U_{n} ).

Relativistic Continuum Theory for the Interaction of Electromagnetic Fields with Deformable Bodies
View Description Hide DescriptionA variational principle is formulated which yields the balance laws and constitutive equations of a nonconducting, charge‐free elastic solid interacting with electromagnetic fields. It is found that the form of the total energy‐momentum tensor and the constitutive equations that follow from a Lagrangian action which depends arbitrarily on the inverse deformation gradients and the electromagnetic field tensor are identical to those obtained by formulating a constitutive theory of a nondissipative material based on the basic mechanical, thermodynamical, and electromagnetic balance laws of a continuum.

Legendre Polynomial Expansions of Hypergeometric Functions with Applications
View Description Hide DescriptionThe expansion of a class of hypergeometric functions in a series of Legendre polynomials is derived. The range of validity and the meaning to be attached to the sums is investigated. Several applications to the problem of the scattering of charged particles are presented.

New Method for Finding Eigenvalues
View Description Hide DescriptionThe eigenvalue equation of a function of a single complex variable is shown to be equivalent to the Cauchy‐Riemann equations so that the eigenvalue problem is reducible to the problem of finding the analytic regions of the function. The eigenvalues of a function of several complex variables are also found from the analytic regions on each plane. A self‐consistent treatment of two relativistic fields is developed and applied to the interaction of spinor and scalar fields.

Degeneracy of the Dirac Equation with Electric and Magnetic Coulomb Potentials
View Description Hide DescriptionInvestigation is made of the symmetry and degeneracy of the Dirac equation for a Coulomb potential with a fixed center bearing both electric and magnetic charge. Seen from the viewpoint of classical mechanics, relativistic precession removes the accidental degeneracy of the nonrelativistic potential, and may be so severe as to lead to spiral rather than precessing elliptic orbits. The degeneracy may be restored by the introduction of a vector potential which combats the precession and leads to closed relativistic orbits. An angular momentum and a Runge vector are found for the ``symmetric'' potential for arbitrary values of electric and magnetic nuclear charges. A related symmetric Hamiltonian and constants of the motion may be constructed for the Dirac equation, which reduce to those of Biedenharn and Swamy in the absence of magnetic charge.Magnetic charge must be quantized—a requirement seen from the angular part of the wavefunction exactly as in the nonrelativistic problem. The Dirac Hamiltonian is singular for the lowest admissible angular momentum state, corresponding to the spiral orbits, when the magnetic charge is nonzero. The remaining states show an accidental doubling of degeneracy, whose presence may be deduced from an operator which reduces to that of Johnson and Lippman, or the algebra of Malkin and Manko, without the magnetic charge.

High‐Temperature Properties of Random Spin Systems
View Description Hide DescriptionThe free energy and the correlation functions are proved to be analytic functions of the inverse temperature β, the external magnetic fieldH, and the impurity probability p, provided the temperature is high enough and the spin‐spin interaction has finite range. In particular, this excludes spontaneous magnetization at high temperatures.