Volume 8, Issue 8, August 1967
Index of content:

Dispersion of Electromagnetic Waves in Molecular Crystals
View Description Hide DescriptionThe dispersion of electromagnetic waves in molecular crystals has been studied using the second quantization formalism. The excitation spectrum, the Green's functions for the optical exciton and photon field, as well as the corresponding distribution functions, are calculated and discussed. The ground state energy of the crystal is derived in a closed form with the polarization of the medium taken into account explicitly. In the low‐density limit the expression for the ground state energy corresponds to the summation of an infinite sequence of terms in a perturbation‐theory approach.

Scattering of Electromagnetic Waves in Molecular Crystals
View Description Hide DescriptionThe scattering of electromagnetic waves in molecular crystals has been investigated by considering the polarization operator of the system that results from the Dyson equation in the first approximation. The Green's function for the photon field is then calculated and an expression for the frequency‐ and wavevector‐dependent index of refraction of optical waves is developed including nonlinear effects. The excitation spectrum has been studied by evaluating the spectral intensity for the photon field. Expressions are derived for the electronic contribution to the spectral width and energy shift for physical processes: (i) resonance Raman scattering and (ii) resonances occurring when either two excitons are created or one exciton is created and the other is absorbed by a single incident photon. A theory for the dielectric permeability of the crystal is developed and a relation between the dielectric permeability, the polarization operator, and the photonGreen's function is established. Contributions to the binding energy of the crystal resulting from the dispersion and scattering of the polarization waves at finite temperatures have been calculated and expressed in terms of the excitation energies and the index of refraction of the medium.

Ground‐State Energy of a Finite System of Charged Particles
View Description Hide DescriptionA trial wavefunction of superconducting type is postulated for the ground state of a system of N positive and N negative charges with Coulomb interactions in the absence of any exclusion principle. The ground‐state binding energy is rigorously proved to be greater than Ry, where A is an absolute constant. Results of earlier perturbation‐theoretic calculations for an infinite system are confirmed. The author, with A. Lenard, has previously proved that the exclusion principle, holding for particles with one sign of charge only, is a sufficient condition for the stability of matter; the present paper shows that the exclusion principle is also necessary for stability.

Lower Bound on the Mean Kinetic Energy of a System of Particles
View Description Hide DescriptionIt is shown that K ≥ ⅗μ_{0} for a system of fermions. K is 〈P ^{2}/2m〉, and μ_{0} is the Fermi energy of an ideal gas of fermions of the same mass, and at the same density, as the system under consideration.

Empty Space‐Times Algebraically Special on a Given World Line or Hypersurface
View Description Hide DescriptionThe null tetrad formalism of Newman and Penrose is used to investigate empty space‐times which are algebraically special on a given world line. It is found that, when the world line is timelike, the space‐time admits a congruence which is geodesic and shear‐free on the world line. A similar result is obtained for those empty space‐times which are algebraically special on a given space like hypersurface.

Associative Algebra in the Problem of Mass Formulas
View Description Hide DescriptionAn associative algebra of continuous operators in a rigged Hilbert space, which contains the enveloping algebra of the Poincaré group and gives rise to a discrete mass spectrum, is studied. In Appendix B some general results on the representation of Lie algebras in a rigged Hilbert space are derived.

Klein‐Gordon Equation for Spinors
View Description Hide DescriptionSpinors obeying the Dirac equation also obey the Klein‐Gordon equation, but the converse is not true. In this paper we make a systematic study of four‐component spinors obeying the Klein‐Gordon equation, with special regard for the additional solutions. The starting point is the Lagrangian density , and we first develop from it the theory of a classical spinor field. We then proceed to the canonical quantization of this field and are confronted by some anticommutators of creation and annihilation operators equal to −1, and the subsequent need for an indefinite metric in Hilbert space.Quantum electrodynamics can be reformulated, and in spite of a modified fermion propagator, gradient coupling, and vertices with two photon lines, the amplitude for Compton scattering to order e ^{2} is the usual one. Special problems arising for massless fermions are indicated, and we note that the four‐fermion point interaction is now renormalizable. Some interesting variations of strong interactions also become possible.

Connection between Complex Angular Momenta and the Inverse Scattering Problem at Fixed Energy
View Description Hide DescriptionThe paper contains a number of remarks occasioned by Sabatier's enlargement of the class of solutions to the inverse scattering problem at fixed energy found previously by the author. The implications of the large class of scattering‐equivalent potentials for the angular momentuminterpolation problem are discussed. The additional angular momenta that appear in the expansion of the potential are directly related to the singularities of its Mellin transform. It is shown that the expansion coefficients must not converge too rapidly to zero unless the first moment of the potential vanishes. Finally we analyze the information contained in an ``angular momentumdispersion relation'' obeyed by the Jost function which is found as a by‐product.

Matrix Elements for Irreducible Representations of the U(6, 6) Algebra in Harmonic Function Space
View Description Hide DescriptionIn this paper we realize the algebra of the group U(6, 6) as an algebra of differential operators acting in the Hilbert space of functions defined on a 23‐dimensional pseudosphere. We then calculate the matrix elements of the generators of this algebra between certain harmonic function states.

Superconductivity in One and Two Dimensions. II. Charged Systems
View Description Hide DescriptionIn an earlier paper the Ginzburg‐Landau free energy functional was used to calculate the effect of thermodynamic fluctuations on the off‐diagonal correlation function and we found no off‐diagonal long‐range order in one‐ and two‐dimensional systems. It has been pointed out that, for a charged system, the use of the Ginzburg‐Landau free energy functional is in error for arbitrary nonequilibrium values of the order parameter since the electrostatic energy of the charge fluctuations associated with an arbitrary order parameter is not included in the free energy functional. We have not succeeded in a direct generalization of the free energy functional so we are forced to proceed by inference from the generalized random phase approximation (RPA). We find that, for uncharged systems, the RPA gives a linearization of the results obtained earlier using the Ginzburg‐Landau theory. For charged systems we find in the RPA results similar to those obtained for uncharged systems. From this we conclude that it is very likely that, as in uncharged systems, there will be no ODLRO in charged infinite one‐ and two‐dimensional systems.

Description of Extended Bodies by Multipole Moments in Special Relativity
View Description Hide DescriptionThe description of an extended charged body in a given electromagnetic field in flat space‐time is considered, and it is shown that such a body may be completely specified by a certain set of multipole moments of the energy‐momentum tensorT ^{αβ} and the charge‐current vector J ^{α}. These moments include the momentum vector, spin tensor, and total charge of the body, and they completely determine T ^{αβ} and J ^{α}. It is shown that the only relations between the moments due to the ``generalized conservation equations'' ∂_{β} T ^{αβ} = −F ^{αβ} J _{β} and ∂_{α} J ^{α} = 0 are the constancy of the total charge and equations of motion for the momentum vector and spin tensor, in contrast to previous descriptions by moments, such as that of Mathisson, which have an infinite number of such relations. The equations of motion are given exactly, as infinite series in the moments, without assuming the applied electromagnetic field to be analytic, and an approximation procedure is developed, based on the smallness of the body compared with a typical length scale for the external field.

Finiteness of the Number of Positive‐α Landau Surfaces in Bounded Portions of the Physical Region
View Description Hide DescriptionIt is shown that if the spectrum of physical particle rest masses contains neither accumulation points nor the zero point, then the number of different positive‐α Landau surfaces that enter any bounded portion of the physical region of any multiple‐particle scattering process is finite. This implies that if the physical‐region singularities of scattering functions are confined to the closure of the set of points lying on positive‐α Landau surfaces, then the scattering functions are analytic at almost all points of the physical region. The proof is made by proving an equivalent property of systems of classical point particles scattering via point interactions.

Convergence of the Bremmer Series for the Spatially Inhomogeneous Helmholtz Equation
View Description Hide DescriptionThe convergence of the Bremmer series expansion for the solution of the one‐dimensional Helmholtz equation with varying wave number is investigated. It is proved that the series converges provided the quantity εσ is sufficiently small, where ε measures the relative rate of change 1/k ^{2}(dk/dx) and σ the relative total change of the wave number k. An exactly soluble example is discussed in order to show that convergence fails if the above criterion is significantly relaxed. It is shown that if the Bremmer method is applied to the calculation of the time‐dependent response to an externally imposed signal, the series is convergent at any finite time after the signal is turned on.

Solutions to the Neutron Transport Equation for a Critical Slab by Perturbation Theory
View Description Hide DescriptionA perturbation‐theoretic method is developed by which solutions to the monoenergetic neutron transportequation for an unreflected critical slab can be obtained in the case of anisotropicscattering and/or nonconstant cross sections. The method rests upon the use of a generalized Green's function to solve the perturbationequations. The Green's function is derived by an eigenfunction expansion technique and as the solution to two coupled singular integral equations, and the results are compared numerically.

Drukarev Transformation of Dirac Equation
View Description Hide DescriptionDrukarev has transformed the Fredholm equation that is the Green's function formulation of the solution of the Schrödinger equation into a Volterra equation. The present paper exhibits the corresponding result for the Dirac equation. The advantages of this technique in the numerical evaluation of phase shifts (and wavefunctions) are discussed.

Relativistic and Causal Theories with Four‐Fermion Interaction Hamiltonians
View Description Hide DescriptionPossible ways of constructing field‐theoretic operators satisfying the commutation relations of the inhomogeneous Lorentz group are investigated along lines laid down by Dirac. They can be satisfied in both the instant form, in which operators representing rotations and translations in space remain unchanged, and the point form, in which operators representing the homogeneous Lorentz group remain unchanged, provided one can find a causal Hamiltonian density such that [H(t, x), H(t, y)] is proportional to δ(x − y) and which transforms as a scalar under Less restrictive sufficient conditions in the instant form are found, similar to those of Dirac. The commutator can be proportional to derivatives of δ(x − y) if the coefficients on the derivatives satisfy a certain condition. The only way found to satisfy these conditions for an interaction Hamiltonian constructed from fields for identical spin ½ particle (in the interaction picture) is to have the commutator proportional to δ(x − y), which implies local coupling with no derivations. The possibility of having relativistic theories in the instant form without causality is also investigated for the case of a four‐fermion interaction Hamiltonian constructed from creation and annihilation operators for a spin ½ particle, but no definite conclusion is arrived at.

Some Theorems Concerning the Phase Problem of Coherence Theory
View Description Hide DescriptionThe spectral density of a fluctuating light beam may be determined from the knowledge of both the modulus and the phase of the complex degree of self‐coherence γ(τ) of the beam. The phase itself may be determined from the modulus and from the location of the zeros of the analytic continuation of γ(τ) in the lower half of the complex τ plane. In the present paper results of an investigation are presented which show that the determination of the zeros is equivalent to the solution of a certain inhomogeneous eigenvalue problem of the Sturm‐Liouville type on a semi‐infinite frequency range. This eigenvalue problem is found to be equivalent to a certain stability problem in mechanics. Although no general technique for the solution of this type of an eigenvalue problem appears to be known, the new formulation may be used to determine spectral profiles for which the associated degree of self‐coherence has zeros at prescribed points in the complex τ plane. Some illustrative examples are given.

Green's Functions and Double‐Time Distribution Functions in Classical Statistical Mechanics
View Description Hide DescriptionThis paper rederives the Bogoliubov/Sadovnikov classical‐equilibrium‐correlation Green's function hierarchy by using the double‐time theory of Rostoker. Thus the traditional variational technique is avoided.

Existence and Uniqueness in the Large for Boundary Value Problems in Kinetic Theory
View Description Hide DescriptionThe boundary‐value problem for the linearized Boltzmann equation is shown to have a unique solution for a bounded domain (two walls separated by an arbitrary distance). The proof applies to a general class of models with finite collision frequency and appears to be easily extendible to similar problems in two and three dimensions. It differs essentially from previously known proofs because no limitations are put on the distance between the walls.

States of Classical Statistical Mechanics
View Description Hide DescriptionA state of an infinite system in classical statistical mechanics is usually described by its correlation functions. We discuss here other descriptions in particular: as (1) a state on a B* algebra; (2) a collection of density distributions; (3) a field theory; (4) a measure on a ``space of configurations of infinitely many particles.'' We consider the relations between these various descriptions and prove, under very general conditions, an integral representation of a state as superposition of ``extremal invariant states'' corresponding to pure thermodynamical phases.