Volume 5, Issue 2, February 1964
Index of content:

Group‐Theoretical Models of Local‐Field Theories
View Description Hide DescriptionLocal fields on a finite, discrete, space‐time model are introduced as a guide for axiomatic discussion of quantum field theory.

On the Derivation of Statistical Thermodynamics from Purely Phenomenological Principles
View Description Hide DescriptionSzilard pointed out in 1925 that it is possible to base the foundations of statistical thermodynamics upon ``phenomenological'' principles, analogous to those of the non‐statistical ``classical'' theory. This approach is discussed and developed.

Asymptotic Properties of the Electromagnetic Field
View Description Hide DescriptionFrom the integral form of the general solution for the retarded electromagnetic field of a localized charge‐current distribution, the asymptotic field is shown to have the behavior F_{μν} = N_{μν}/R + III_{μν}/R ^{2} + _{2} J_{μν}/R ^{3}, where the coefficients satisfy N_{μν}k^{ν} = 0, III_{μν}k^{ν} = Ak_{μ} , and k_{μ}k^{ν} = 0. The remainder _{2} J_{μν} is shown to be bounded by using the second‐mean‐value theorem. Thus the algebraically special character of the asymptotic electromagnetic field is exhibited.

Continuous‐Representation Theory. III. On Functional Quantization of Classical Systems
View Description Hide DescriptionThe form of Schrödinger's equation in a continuous representation is indicated for general systems and analyzed in detail for elementary Bose and Fermi systems for which illustrative solutions are given. For any system, a natural continuous representation exists in which state vectors are expressed as continuous, bounded functions of the corresponding classical variables. The natural continuous representation is generated by a suitable set S of unit vectors labeled by classical variables for which, for the system in question, the quantum action functional restricted to the domain S is equivalent to the classical action. When a classical action is viewed in this manner it contains considerable information about the quantum system. Augmenting the classical action with some physical significance of its variables, we prove that the classical theory virtually determines the quantum theory for the Bose system, while it uniquely determines the quantum theory for the Fermi system.

Separation of Angular Variables in the Bethe‐Salpeter Equation for Two Spin‐½ Particles
View Description Hide DescriptionThe Bethe‐Salpeter equation for two fermions, in its usual differential form, is reduced here to one which involves a single spatial and a single temporal variable. Concomitantly, the original underlying 16‐dimensional spinor space is reducible to one of 8 dimensions for angular momentuml ≠ 0 and of 4 dimensions for l = 0. The possibility of this reduction derives from the existence of one (or two) normal divisors of the matrices operating on the original spinor space, in these respective cases. The procedure is illustrated with the positronium problem, but may easily be seen to be quite general.

Regge Behavior of Forward Elastic Scattering Amplitudes
View Description Hide DescriptionA theorem concerning the asymptotic behavior of forward elasticscattering amplitudes in relativistic theories is stated and proved. The assumptions made are (1) identical spinless particles interact via Gφ^{3} and λφ^{4} couplings; (2) a cutoff of the propagators is introduced; (3) the forward scattering amplitude satisfies a Bethe‐Salpeter equation in the crossed channel; (4) the kernel of the equation is an arbitrary finite subset of the Feynman graphs which compose the exact kernel. The theorem states that under these assumptions, the forward scattering amplitude exhibits Regge behavior, i. e., A(s, 0) → s ^{α} + O (1) as s → ∞.

Diffraction by Polygonal Cylinders
View Description Hide DescriptionAsymptotic solutions for the two‐dimensional reduced wave equation in the exterior of a convex polygonal cylinder with both Dirichlet and Neumann boundary conditions are obtained. The method used is the geometrical theory of diffraction. Both the cases of oblique and grazing incidence are treated, and the far fields in all directions including those along shadow and specular lines are found. A feature of the method is that the solutions may be carried to any desired order. The calculation is illustrated for the case of a rectangular cylinder and, for this geometry, cross sections, surfacecurrent densities and radiation patterns are obtained. Graphs of some of these results have been included.

Accuracy of the Semicircle Approximation for the Density of Eigenvalues of Random Matrices
View Description Hide DescriptionCertain statistical properties of the energy levels of complex physical systems have been found to coincide with those for distributions of eigenvalues derived from ensembles of random matrices. However, if ensembles of random matrices give a fair representation for the Hamiltonian of a complex physical system, the density of the characteristic values at the lower end of the spectrum should show some similarity with the exponential dependence found in nuclear spectra. The limiting distribution of the density for very high‐dimensional random matrices is a semicircle, i.e., concave from below if plotted against the characteristic value which represents, in this case, the energy. Hence the deviations from the limiting distribution are investigated and it is shown that there is a region, at the very lowest part of the spectrum, where the density is convex from below, similar to an exponential function. The region of convexity is called the tail of the distribution. It is shown, however, that the avergae number of roots in the tail is very small, of the order of 1. It is concluded that those ensembles of random matrices which have been studied up to now, do not give a fair representation of Hamiltonians of complex systems.

Theory of Vibrational Structure in Optical Spectra of Impurities in Solids. I. Singlets
View Description Hide DescriptionGeneral expressions are derived describing the modifications of the optical spectra of isolated impurities induced by the subclass of interactions with host‐lattice phonons which shift the energy eigen‐values of the internal impurity state but which do not mix those internal states. The results apply to localized phonon modes as well as to extended modes (whose individual coupling decreases with increasing lattice volume), to anharmonic‐ as well as harmonic‐phonon systems, and to phonon‐impurity couplings which need not be linear in the dynamic local strain field. They take the form of ``linked‐cluster'' expansions which, in the linear‐coupling harmonic‐phonon case, terminate after the second term. The relation of these expansions to moment methods is indicated. As a simple application of the linked‐cluster expansions, we compute the dependence of the sharp ``no‐phonon'' spectral line on impurity mass. We conclude that the no‐phonon line should not exhibit isotopic‐mass displacements in those impurity sites for which the lattice is inversion symmetric.

Stroboscopic‐Perturbation Procedure for Treating a Class of Nonlinear Wave Equations
View Description Hide DescriptionA new perturbation procedure is presented for treating initial‐value problems of nonlinear hyperbolic partial differential equations. The characteristic variables of the partial differential equation and the functions of these variables are expanded in powers of ε, and the formal solution is uniformly valid over time intervals O(1/ε). The uniform first‐order solution is evaluated for the equation,subject to the standing‐wave initial conditions: y(x, 0) = a sin πx, y_{t} (x, 0) = 0. This equation is the lowest continuum limit of an equation for which numerical computations are available. The uniform zero‐order solution breaks down after a time t_{B} = 4/εaπ. A detailed study of the solution is made in the vicinity of the breakdown region of the (x, t) plane, and it demonstrates that the formal solution for y_{x} and y_{t} goes from a single‐valued to a triple‐valued function while y_{xx} and y_{tt} become singular. To compare the solutions with the available numerical computations, the y_{x} and y_{t} waveforms are decomposed into spatial Fourier modes. The effect of breakdown is manifest in the modal amplitudes ∝ J_{n} (nT)/nT. The modal amplitudes change their asymptotic behavior, from exponentially decreasing as n → ∞, to algebraically decreasing when t goes from smaller to larger than t_{B} . In the time interval up to breakdown, t < t_{B} , the modal energies are in excellent agreement with the modal energies of the numerical computations, whereas for t > t_{B} they diverge. For t < t_{B} , the total energy calculated from the uniform zero‐order solution is conserved and equal to the initial value,.Thus, the lowest‐continuum‐limit equations describe the dynamics of a discrete model for a finite time. A heuristic discussion is given which suggests that the time of description can be extended beyond t_{B} by including higher spatial derivatives in the continuum model.

Generalization of Laplace's Expansion to Arbitrary Powers and Functions of the Distance between Two Points
View Description Hide DescriptionIn analogy to Laplace's expansion, an arbitrary power r^{n} of the distance r between two points (r _{1}, ϑ_{1}, φ_{1}) and (r _{2}, ϑ_{2}, φ_{2}) is expanded in terms of Legendre polynomials of cos ϑ_{12}. The coefficients are homogeneous functions of r _{1} and r _{2} of degree n satisfying simple differential equations; they are solved in terms of Gauss' hypergeometric functions of the variable (r _{<}/r _{>})^{2}. The transformation theory of hypergeometric functions is applied to describe the nature of the singularities as r _{1} tends to r _{2} and of the analytic continuation of the functions past these singularities. Expressions symmetric in r _{1} and r _{2} are obtained by quadratic transformations; for n = −1 and n = −2; one of these has previously been given by Fontana. Some three‐term recurrence relations between the radial functions are established, and the expressions for the logarithm and the inverse square of the distance are discussed in detail. For arbitrary analytic functionsf(r), three analogous expansions are derived; the radial dependence involves spherical Bessel functions of (r _{<}∂/∂r _{>}) of of related operators acting on f(r _{>}), f(r _{1} + r _{2}) or f[(r _{1} ^{2} + r _{2} ^{2})^{½}].

Three‐Dimensional Addition Theorem for Arbitrary Functions Involving Expansions in Spherical Harmonics
View Description Hide DescriptionFor any vector r = r_{1} + r_{2} an expansion is derived for the product of a power r^{N} of its magnitude and a surface spherical harmonic Y_{L} ^{M} (ϑ, φ) of its polar angles in terms of spherical harmonics of the angles (ϑ_{1}, φ_{1}) and (ϑ_{2}, φ_{2}). The radial factors satisfy simple differential equations; their solutions can be expressed in terms of hypergeometric functions of the variable (r _{<}/r _{>})^{2}, and the leading coefficients by means of Gaunt's coefficients or 3j symbols. A number of linear transformations and three‐term recurrence relations between the radial function are derived; but in contrast to the case L = 0, no generally valid expressions symmetric in r _{1} and r _{2} could be found. By interpreting the terms operationally, an expansion is derived for the product of Y_{L} ^{M} (ϑ, φ) and an arbitrary functionf(r). The radial factors are expansions in derivatives of f(r _{>}); for spherical waves, they factorize into Bessel functions of r _{1} and r _{2} in agreement with the expansion by Friedman and Russek. The 3j symbols are briefly discussed in an unnormalized form; the new coefficients are integers, satisfying a simple recurrence relation through which they can be arranged on a five‐dimensional generalization of Pascal's triangle.

Two‐Center Expansion for the Powers of the Distance Between Two Points
View Description Hide DescriptionThe powers r^{n} of the distance between two points specified by spherical polar coordinates relating to two different origins, or of the modulus of the sum of three vectors, are expanded in spherical harmonics of the angles. The radial factors satisfy simple partial differential equations, and can be expressed in terms of Appell functions F _{4}, and Wigner or Gaunt's coefficients. In the overlap region, first discussed by Buehler and Hirschfelder, the expressions are valid for integer values of n ≥ −1, but in the other regions, for arbitrary n. For high orders of the harmonics, individually large terms in the overlap region may have small resulting sums; as a consequence the two‐center expansion is of limited usefulness for the evaluation of molecular integrals. Expansions are also derived for the three‐dimensional delta function within the overlap region, and for arbitrary functions f(r), valid outside that region.

Explicit Formal Construction of Nonlinear Quantum Fields
View Description Hide DescriptionIt is shown that there can be associated with any given nonlinear relativistic partial differential equation an operator field satisfying the canonical commutation relations, transforming appropriately under the action of the Lorentz group, and propagated in accordance with the given differential equation. This quantization procedure is unique, apart from the scale of the commutators. The treatment is intuitive, but is capable of rigorization in terms of the mathematical theory of analysis in function space (functional integration). The results are in harmony with crucial conventional ones, so far as has been determined, and provide in principle a possible means of computing the collision matrix for particular systems without recourse to perturbation theory.

Irreducible Tensor Expansion of Solid Spherical Harmonic‐Type Operators in Quantum Mechanics
View Description Hide DescriptionA method is proposed to derive the general one‐ and two‐center expansions of quantum mechanical operators that are in the form of regular or irregular solid spherical harmonics of any integral degree. The expansions are expressed as couplings of irreducible spherical tensors. The physical nature of such coupling and the parallelism to vector addition are illustrated. Possible uses of such expansions in the evaluation of molecular integral and their reduction to simple, known expansions obtained from other methods are briefly discussed.

Some Properties of Triangular Representations of SU(3)
View Description Hide DescriptionThe occurrence of an equal‐spacing mass rule in the unitary decuplet can be explained either by observing that the isotopic spin and hypercharge of each particle are related by T = 1 + ½Y, or by making use of a theorem due to Diu and Ginibre. This theorem states that, in all triangular representations of SU(3), the matrix elements of an arbitrary tensor operator depend upon one reduced matrix element instead of two. Here we present a new proof of the Diu‐Ginibre theorem and use it show that relations of the form T = λ ± ½Y exist for all triangular representations. We also show that the L and K spins are related to their corresponding hypercharges by L = λ ± ½Y_{L} and K = λ ± ½Y_{K} . One consequence is that the masses and magnetic moments of particles in a triangular multiplet are equally spaced. Other consequences are also discussed.

The Representations of the Inhomogeneous Lorentz Group in Terms of an Angular Momentum Basis
View Description Hide DescriptionThe irreducible ray representations of the proper, orthochronous, inhomogeneous Lorentz group were originally given by Wigner in terms of a basis in which the energy and linear momenta are diagonal. In the present paper we show how the infinitesimal generators of the irreducible representations act on a basis in which the energy, the square of the angular momentum, the component of the angular momentum along the z axis, and the helicity (or circular polarization) are diagonal.
We consider representations corresponding to particles of nonzero mass, and any spin and of zero mass and finite spin. The continuous‐spin case is to be treated in a later paper.