Volume 10, Issue 5, May 1969
Index of content:

Considerations on the Classical Spinning Electron
View Description Hide DescriptionThe present work is concerned with the asymptotic properties of the field radiated by a classical spinning (and nonspinning) point charge in arbitrary motion. It is demonstrated that the expression found for the radiated angular momentum is satisfactory in that it is a 4‐tensor which is spacelike surface‐independent. The angular‐momentum radiation rate is then expressed in terms of the retarded kinematic properties of the particle. In addition, a ``center‐of‐energy'' theorem is proved for nonspinning charges in arbitrary motion—a result which follows from the demonstrated result that nonspinning charges also radiate angular momentum. Finally, it is demonstrated that the (linear) energy‐momentum radiation rate is independent of the spin of the involved charge. The discussion is manifestly covariant throughout and many mathematical details are deferred to appendices.

Nonsaturation of Gravitational Forces
View Description Hide DescriptionRigorous inequalities are derived for the ground‐state energy of a nonrelativistic quantum‐mechanical system of N particles in gravitational interaction. It is shown that gravitational forces do not saturate, the binding energy per particle increasing with N, like N ^{2} for a Bose system, like (N ^{4/3}) for a Fermi system. As a by‐product, we obtain a generally valid Heisenberg‐like inequality for N‐fermion systems, expressing very simply the effect of the Pauli exclusion principle. These results are extended to the case of a system of oppositely charged particles which is shown to behave, with respect to gravitational forces, as a Fermi system as soon as particles with one sign of charge only are identical fermions. This explains quantitatively how and when gravitational forces finally predominate over Coulomb forces for large enough bodies (planets). A further extension to the case where relativistic effects enter only at the kinematical level permits us to derive rigorously from first principles the existence and an estimate of the Chandrasekhar mass limit, above which no collection of cold matter is stable (white dwarfstars).

Zero‐Mass Representation of Poincaré Group and Conformal Invariance
View Description Hide DescriptionBy explicit construction it is shown how to extend zero‐mass, discrete spin representations of the Poincaré group to corresponding representations of the conformal group of Minkowski space.

Best Error Bounds for Padé Approximants to Convergent Series of Stieltjes
View Description Hide DescriptionWe consider series of Stieltjes with a nonzero radius of convergence R. We establish by way of Padé approximants the allowed range of values for such functions at any point in the cut (−∞ < z ≤ − R) complex plane when a finite number of Taylor‐series coefficients are known. The previous results for z real and positive are sharpened. We investigate the fitting problem and again give error bounds throughout the cut complex plane and we give necessary and sufficient conditions that the set of values fitted be values of a series of Stieltjes with radius of convergence at least R.

Criteria for Slater Determinants and Quasiparticle Vacuum States
View Description Hide DescriptionNew criteria characterizing Slater determinants and quasiparticle vacuum states are obtained. These criteria are expressed as quadratic homogeneous equations in the coefficients of the development of the trial wavefunction in a basis of Slater determinants. A redefinition of quasiparticle vacuum states permits the introduction of quasiparticle transformations which are more general than the generalized Bogoliubov transformations.

Macroscopic Causality Conditions and Properties of Scattering Amplitudes
View Description Hide DescriptionTwo causality conditions that refer only to mass‐shell quantities are formulated and their consequences explored. The first condition, called weak asymptotic causality, expresses the requirement that some interaction between the initial particles must occur before the last interaction from which final particles emerge. This condition is shown to imply that if a two‐body scattering function is analytic except for singularities in the energy variable at normal thresholds, then (a) the physical scattering functions in two adjacent parts of the physical region separated by any normal threshold are parts of a single analytic function; (b) the path of continuation joining these two parts bypasses the singularity in the upper half‐plane of the energy variable; and (c) the integral over the physical function can be represented as an integral over a contour that is distorted into the upper‐half energy plane (hence not, for example, by a principal‐value integral). Singularities possessing finite derivatives of all orders with respect to real variations of the energy are not encompassed by this result. The second causality condition, called strong asymptotic causality, expresses the requirement that, apart from contributions whose effects fall off faster than any inverse power of Euclidean distance, momentum‐energy is carried over macroscopic distances only by stable physical particles. This condition implies that all n‐particle scattering functions (n ≥ 4) are analytic, apart from infinitely differentiable singularities, at physical points not lying on any positive‐α Landau surface. Moreover, the scattering functions on the two sides of any such Landau surface are analytically connected by a path that passes around the singularity surface in a well defined manner, which is the same as in perturbation theory. Thus, apart from possible infinitely differentiable singularities, the physical region singularitystructure is derived from a mass‐shell causality requirement. Several properties of the set of physical region positive‐α Landau surfaces are derived.

Application of the Kirillov Theory to the Representations of O(2, 1)
View Description Hide DescriptionThe Kirillov construction is applied to the semisimple Lie groupO(2, 1). All the unitary irreducible representations (except the supplementary series) are found, provided that complex subalgebras and complex points on orbits are admitted. The characters of the representations are calculated and the relation of their Fourier transforms to the orbits is examined.

Solution of the Dimer Problem by the S‐Matrix Method
View Description Hide DescriptionThe S‐matrix approach of Hurst is applied to rederive the partition function for the dimer problem. By using every two lattice sites as a unit cell and a set of six creation and annihilation operators at each cell, the expression for the partition is reduced to a form equivalent to the vacuum expectation value of an S matrix with a quadratic interaction Lagrangian.

Lie‐like Approach to the Theory of Representations of Finite Groups
View Description Hide DescriptionUsing characters, one can set up the theory of representations for both finite and continuous groups. For continuous groups, another approach—the Lie theory—is also possible. It is shown that a similar theory, based on commutators, can be developed also in the case of finite groups. Essentially, the group algebra of a finite group is converted into a Lie algebra by replacing the usual associative product by the product . Then the resulting Lie algebra is a direct sum of special unitary Lie algebras.

Analytical Solutions of the Neutron Transport Equation in Arbitrary Convex Geometry
View Description Hide DescriptionThe integral equation describing the transport of monoenergetic, isotropically scatteredneutrons in a one‐, two‐, or three‐dimensional body of arbitrary convex shape, containing distributed sources, is considered. An exact representation of the neutron density ρ(r) is obtained, involving a superposition of functions belonging to the null space of a simple differential operator. In general, when a countable basis is chosen to span the null space, the coefficients in the expansion of ρ(r) satisfy a coupled system of singular integral equations which is reducible to a system of Fredholm equations. If no sources are present, an exact criticality condition is also obtained. Some techniques for evaluating the expansion coefficients are given and several examples are considered.

Wave Propagation in Sinusoidally Stratified Plasma Media
View Description Hide DescriptionThe problem of the propagation of electromagnetic waves in a sinusoidally stratified plasma media is treated analytically. The propagation characteristics of TE and TM waves are determined, respectively, from the characteristic equations of the resultant Mathieu and Hill equations. Detailed dispersion characteristics of TE and TM waves in an infinite stratified plasma medium and in waveguides filled longitudinally with this stratified dispersion material are given. It is found that, although the stop‐band and pass‐band structures exist for the ω‐β diagrams of both TE and TM waves, detailed dispersion properties for TE and TM waves are quite different for most frequency ranges except when , where ω is the frequency of the propagating waves, ω_{ p0} is the average plasma frequency of the inhomogeneous plasma medium, and δ is the amplitude of the sinusoidally varying term for the electron‐density profile (0 ≤ δ ≤ 1).

Expansion of Lie Groups and Representations of SL(3, C)
View Description Hide DescriptionWe show that in any unitary representation of SL(3,C) belonging to the principal series, the generators can be expressed as functions of the generators of a suitably chosen unitary representation of the non‐semisimple group SU(3) × T _{8}. Both the nondegenerate and the degenerate series are considered. Since the representations of SU(3) × T _{8} in an SU(3) basis are known, this provides a solution to the multiplicity problem in setting up the SL(3,C) representations in an SU(3) basis.

Statistical Properties of Polycrystalline Dielectrics
View Description Hide DescriptionIn this paper we consider the determination of the two‐point electric field correlation tensor in a random polycrystalline medium subjected to a constant average electric field. The medium is supposed to be statistically homogeneous and isotropic and to be composed of crystals all of the same kind. For media whose principal permittivities do not differ greatly from one another, we employ perturbation techniques to linearize the governing equations and derive explicit expressions for both the correlation function and the cross correlation of the electric field and permittivitytensor. We also determine the effective permittivity in this limit. For media with arbitrary principal permittivities, we derive bounds on the effective permittivity which depend on certain two point correlation functions and which reduce to our perturbation solution in the limit of small differences in principal permittivities.

Analytic Representations of Two‐Point Functions with Noncanonical Light‐Cone Singularities. I
View Description Hide DescriptionWe introduce the classes of analytic functions describing by means of its boundary values the sets of commutator functions with singularities δ^{(k)}(x ^{2}) and on the light cone. Their relation with the Källen‐Lehmann representation having nonintegrable spectral functions is given. The generalized wave‐renormalization constants, measuring noncanonical singularities, are introduced. The formulas exhibit in a clear way the light‐cone behavior and provide a proper scheme for studying the equal‐time limits.

Determination of Invariant Amplitudes from Experimental Observables
View Description Hide DescriptionNecessary and sufficient conditions are formulated for deciding whether a set of 2n − 1 bilinear products of n complex amplitudes determines these amplitudes with no continuum of ambiguity or not. The conditions can be translated into very simple geometrical prescriptions which in most cases provide quick and easy practical tests for such decisions.

Hamilton‐Dirac Theory of Hamilton's Equations
View Description Hide DescriptionIt is shown that in a very general way two distinct canonical formalisms can be used to describe a classical system. No corresponding nonuniqueness is introduced into the canonical quantization procedure if the Dirac bracket correspondence to the quantum‐mechanical commutators is employed.

Measure‐Theoretical Description of Klein‐Gordon Multiparticle States
View Description Hide DescriptionIt is shown that a multiparticle state constructed from solutions of the Klein‐Gordon equation can be also described by a family of complex measures having simple properties. These measures are the expectation values for that state of products of projectors corresponding to the spectral decompositions of certain self‐adjoint operators. The results obtained are used in discussing the problem of defining quantized free fields at a point.

Orear Behavior in Potential Scattering. II
View Description Hide DescriptionHere we will extend certain results obtained in a previous paper [J. Math. Phys. 9, 712 (1968)] concerning high‐energy large‐angle scattering upon nonsingular potentials V(r) which are even in r and analytic in a finite strip about the real r axis. (In the previous paper, it was required that the potentials have a larger domain of analyticity than is required here.) The scattering amplitude is compared with the first Born approximation for this wider class of potentials.

Stationary ``Noncanonical'' Solutions of the Einstein Vacuum Field Equations
View Description Hide DescriptionThe complete set of nonflat normal‐hyperbolic solutions of the Einstein vacuum field equations is obtained for the metric tensor defined by the quadratic differential form ds ^{2} = α du ^{2} − 2γ du dv − β dv ^{2} − e ^{φ}(dx ^{2} + dz ^{2}) subject to the condition that αβ + γ^{2} is constant. These solutions are characterized by the existence of a null hypersurface‐orthogonal Killing vector, which is also a four‐fold degenerate Debever vector with vanishing covariant derivative, and therefore are a special case of the class of plane‐fronted gravitational waves.