Volume 4, Issue 1, January 1963
Index of content:

Expectation Value Formalism in Quantum Field Theory. I
View Description Hide DescriptionA formalism has been developed to evaluate the expectation values of projection operators directly. The formalism is applied to the production of particles when the corresponding quantized fields are coupled to external sources and external electromagnetic fields.

Expectation Value Formalism in Quantum Field Theory. II
View Description Hide DescriptionThe expectation value formalism has been developed to treat the problems which require the coupling of quantized Fermi and Bose fields.

Relativistic Invariance and the Square‐Root Klein‐Gordon Equation
View Description Hide DescriptionAlthough the usual operator invariance requirements and corresponding commutation conditions encountered in the study of the invariance of relativistic wave equations (and other equations of physics) are sufficient conditions for invariance, they are by no means necessary. More general conditions are given and illustrated with the square‐root Klein‐Gordon equation. A new proof is thereby given of the Lorentz invariance of this equation. The methods developed are extended to cover the presence of external fields, and it is proved that the usual gauge invariant modification of the relativistic Hamiltonian of a spinless particle which takes into account the presence of an external electromagnetic field leads, in the quantum mechanical case, to an equation which does not admit the proper Lorentz group. This theorem and its generalization are discussed in connection with Dirac's statement that the square‐root equation cannot be extended to include interaction without losing Lorentz invariance.

Normalization Condition for the Bethe‐Salpeter Wavefunction and a Formal Solution to the Bethe‐Salpeter Equation
View Description Hide DescriptionBy the use of an inhomogeneous Bethe‐Salpeter equation, a normalization condition for the Bethe‐Salpeter wavefunction is obtained. This condition requires the normalization integral to be positive. A formal solution is obtained in the ladder approximation, and convergence of the normalization integral is proved by the use of this solution. This solution is also used to prove a dispersion relation for the vertex function of the compound particle and to give an approximate solution The positiveness of the normalization integral is proved in the nonrelativistic limit. The bound state of nucleon and antinucleon is studied in the ladder‐chain approximation and it is found that the normalization condition gives a finite wavefunction in spite of divergency of the normalization integral.

A Summation Procedure for Certain Feynman Integrals
View Description Hide DescriptionA mathematically rigorous summation procedure is developed for Feynman integrals involving a repulsive potential. The central idea of the proof is to connect certain Wiener integrals with the Feynman integral in question by making use of holomorphic vector‐valued functions and holomorphic semigroups.

A Unified Variational Formulation of Classical and Quantum Dynamics. I
View Description Hide DescriptionWe have critically re‐examined the role and status of Lagrangian concepts in both classical and quantal contexts. We find a new, unified variational principle which, subject to the superadded postulates of determinism or indeterminism, respectively, leads uniquely to the classical concept of state and Lagrange equations of motion or to the quantal concept of state and Schrödinger equations of motion.

Complex Scalar Field in General Relativity
View Description Hide DescriptionThis paper introduces complex scalar fields in general relativity to describe charged, gravitating particles with zero spin. A class of exact solutions of the combined Maxwell‐Einstein‐Klein‐Gordon field equations is found that may represent complete models of matter. One of the models discussed is spherically symmetric and the remaining class does not assume any specific symmetry. Such exact solutions are possible only if the mass parameter equals the charge parameter in magnitude, so that physically speaking, matter is in equilibrium under the mutual action of electromagnetic and gravitational forces.

Mathematical Deductions from Some Rules Concerning High‐Energy Total Cross Sections
View Description Hide DescriptionMathematical implications of the Pomeranchuk rule and the Pomeranchuk‐Okun rule are discussed.

Low‐Energy Expansion of Scattering Phase Shifts for Long‐Range Potentials
View Description Hide DescriptionPhase shifts can be used to describe the quantum mechanical scattering of a particle by a spherically symmetric potential. They are odd functions of the wave number k, which is proportional to the square root of the energy of the incident particle. For short‐range potentials, they are analytic at k = 0 and can be expanded in odd powers of k. However for long‐range potentials they are not analytic at k = 0 so they cannot be expanded in powers of k. Since electron‐atom, atom‐atom, proton‐neutron, and multipole‐multipole potentials are of long range, it is important to consider such potentials. A method is presented for determining the appropriate expansions around k = 0. The method is applied to potentials which are O(r^{−v} ) as r → ∞, and the phase shifts for any angular momentum are obtained up to and including the first nonanalytic term. For L = 0 and 3 < v < 4 the next term is also obtained. The nonanalytic terms involve ln k and fractional powers of k. The results for v = 4 and the k dependence for v = 2L + 3 agree with those obtained in a different way by O'Malley, Spruch, and Rosenberg.
The method involves a function η(r) whose asymptotic value η(∞) is the desired phase shift. A nonlinear first‐order ordinary differential equation is derived for η(r). This equation is solved by expansion and iteration methods. To expand the solution around k = 0, two simple theorems are proved concerning the asymptotic forms of certain integrals containing a parameter.

The Diffraction of Waves By a Penetrable Ribbon
View Description Hide DescriptionThe exact solution of the diffraction of waves by a dielectric ribbon or by an elliptical dielectric cylinder is obtained. Results are given in terms of Mathieu and modified Mathieu functions. It is found that each expansion coefficient of the scattered or transmitted wave is coupled to all coefficients of the series expansion for the incident wave, except when the elliptical cylinder degenerates to a circular one. Both polarizations of the incident wave are considered: one with the incident electric vector in the axial direction and the other with the incident magnetic vector in the axial direction. It is noted that the technique used in this paper to satisfy the boundary conditions may be applied to similar types of problems; such as the plasma‐coated ribbon radiator and the corresponding acoustical problems.

The Inverse Problem in the Quantum Theory of Scattering
View Description Hide DescriptionThis report is a translation from the Russian of a survey article by L. D. Faddeyev, which appeared in Uspekhi Matem. Nauk., 14, 57 (1959). Our own interest in this article lies in its relevance to the inverse scattering problem—that is, the problem of determining information about a medium from which an electromagnetic wave is reflected, given a knowledge of the reflection coefficient. Similar questions concerning scattering phenomena in other branches of physics, e.g., in quantum mechanics, can be investigated by means of the same theory. We have therefore thought it worth‐while to reproduce and distribute the translation. A good indication of the contents is given in the Introduction.

The Line Shape in a Harmonic Lattice
View Description Hide DescriptionA general formalism is developed for the treatment of lattice vibrations. The problem is reduced to the mathematics of noncommuting operators. Hausdorff's equation is then solved for a lattice interacting with linear forces. The cases treated in detail are the line shapes of a radiating atom as affected by the displacement (Franck‐Condon effect) and by the nuclear recoil (Mössbauer effect). Only formal results are obtained, which are more difficult to compute than those of the method of normal modes. However, they are more general and do not require the data of vibrational frequencies.

Statistical Thermodynamics of Nonuniform Fluids
View Description Hide DescriptionWe have developed a general formalism for obtaining the low‐order distribution functionsn_{q} (r _{1}, …, r _{ q }) and the thermodynamic parameters of nonuniform equilibrium systems where the nonuniformity is caused by a potential U(r). Our method consists of transforming from an initial (uniform) density n _{0} to the final desired density n(r) via a functional Taylor expansion. When n _{0} is chosen to be the density in the neighborhood of the r's we obtain n_{q} as an expansion in the gradients of the density. The expansion parameter is essentially the ratio of the microscopic correlation length to the scale of the inhomogeneities. Our analysis is most conveniently done in the the grand ensemble formalism where the corresponding thermodynamic potential serves as the generating functional [with U(r) as the variable] for the n_{q} . The transition from U(r) to n(r) as the relevant variable is accomplished via the direct correlation function which enters very naturally, relating the change in U at r _{2} due to a change in n at r _{1}. It is thus essentially the matrix inverse of the two‐particle Ursell function. The recent results of Stillinger and Buff on the thermodynamic potentials for non‐uniform systems follow as a special case of our analysis without any recourse to the virial expansion. Thus, they hold also in the liquid region. In a succeeding paper we apply our analysis to obtain the asymptotic behavior of the radial distribution function in a uniform system.

Perpendicular Susceptibility of the Ising Model
View Description Hide DescriptionA high‐temperature expansion of the partition function for a lattice of N spins with Hamiltonianis derived and thence an expansion for the zero‐field perpendicular susceptibility is found. By perturbation theory, χ⊥(T) is also expanded at low temperatures and seen, in general, to increase with T from the value χ⊥(O) = Nm ^{2}/q  J , where q is the coordination number of the lattice. The perpendicular susceptibility is re‐expressed in terms of near neighbor pair and higher‐order spin correlation functions in zero field. This yields exact closed formulas for the linear chain, the Bethe pseudolattices, and for the plane square and honeycomb lattices. The behavior of χ⊥(T) in the critical region is investigated for these lattices and for the plane superexchange lattice.

Quaternionic Representations of Compact Groups
View Description Hide DescriptionThe main purpose of this paper is to show the conditions under which a finite dimensional representation of a group, irreducible over the complex field, is reducible over quaternions. The answer is simply stated in terms of the Frobenius‐Shur classification of group representations.

Uniqueness and Existence of the Solution to the Static London‐Maxwell Equations in Two Dimensions
View Description Hide DescriptionA theorem is given for the existence and uniqueness of the time‐independent solution of the exterior‐interior problem associated with determining the distribution of superconducting current, according to the London model, in simply‐ or doubly‐connected two‐dimensional regions. The proof of the corresponding theorem in three dimensions is outlined. A discussion is also given of the relationship between two ``different'' solutions which already exist for rectangular regions.