Volume 5, Issue 6, June 1964
Index of content:

Segal's Quantization Procedure
View Description Hide DescriptionThe procedure proposed by I. E. Segal for the quantization of nonlinear problems of field theory is here applied to the one‐dimensional oscillator. It is shown that for the linear oscillator Segal's procedure is equivalent to the canonical procedure, but that for the nonlinear oscillator the two procedures lead to quite different results. The differences are reflected in the equations of motion, the energy spectra, and the scattering cross sections.

Convergent Perturbation Expansions for Certain Wave Operators
View Description Hide DescriptionThis paper establishes rigorously the validity of Dyson's perturbation expansion for the Mo/ller wave operators under suitable restrictive assumptions on the interaction potential.

The Necessary and Sufficient Condition in Terms of Wightman Functions for a Field to be a Generalized Free Field
View Description Hide DescriptionThe description of the generalized free field in terms of the Wightman functions is given. The necessary and sufficient condition for a field to be a generalized free field is established.

On Dirac's Wave Equation in a Gravitational Field
View Description Hide DescriptionThe Dirac equation is investigated in the combined electric and gravitational field of a point charge in General Relativity. The wavefunctions are weakly singular at the origin, but still normalizable for a continuous range of the energy. The Hamiltonian, however, is not self‐adjoint over the manifold of its own ``eigenstates,'' though it can be made self‐adjoint by a suitable choice of its domain of definition. The theory, however, is unable to decide how the Hamiltonian should be defined, and what are the bound states.

Properties of a Class of Nonlocal Solvable Interactions
View Description Hide DescriptionA generalization of factorable interactions is taken into account. The direct and inverse problems for the relative Schrödinger equation are investigated and they turn out to be workable. It is shown that such interactions can produce several bound states. The solution of the direct problem is given. A class of interactions of the considered type is constructed which produces an arbitrarily assigned finite set of bound states.

Asymptotic Convergence and the Coulomb Interaction
View Description Hide DescriptionA definition of asymptotic convergence is given for nonrelativistic time‐dependent scattering problems involving Coulomb potentials. Convergence proofs have been found both for potential and for n‐body multichannel scattering. For pure Coulomb potential scattering, the Mo/ller wave matrix is computed explicitly and found to have its usual meaning.

Eigenfunctions of the Electron Spin Operators
View Description Hide DescriptionThe problem of finding linear combinations of Slater determinants which are eigenfunctions of both the S ^{2} and S _{ z } operators is considered. The method of projection operators is used in forming eigenfunctions of S ^{2}. A recursion relation is given for the coefficients in the linear combination. Rules are given for selecting a complete linearly independent set of eigenfunctions and an explicit formula is given for the members of the set.

Perturbation Theory for Strong Repulsive Potentials
View Description Hide DescriptionA conformal mapping of the coupling‐constant plane is used to rearrange the Born series. The new series is guaranteed to converge for any decent repulsive potential. The first few terms do well in actual calculations of the scattering length.

Quantum Kinetic Equations for Electrons in a Periodic System
View Description Hide DescriptionThe approach to equilibrium of a system of electrons in a periodic system is studied by deriving a kinetic equation in the self‐consistent field approximation. The derivation based on the introduction of an hierarchy of equations for the s‐body density matrices and employing a truncation scheme, valid in the self‐consistent field approximation. The irreversibility is introduced by the adiabatic hypothesis of Bogoliubov. The kinetic equation takes proper account of the collective aspects of the electron system and includes the dynamic shielding.

Systematic Characterization of mth‐Order Energy‐Level Spacing Distributions
View Description Hide DescriptionThe mth‐order energy‐level spacing distributions p ^{(m)}(x) for complex spectra are defined in terms of a general joint probability distributionP_{N} (λ_{1}, … λ_{ N }) for N consecutive eigenvalues. The precise limiting processes involved are explained, and are subsequently used to obtain two formal representations of p ^{(m)}(x). Both representations yield p ^{(m)}(x) = x ^{ m }/m! exp (−x) for statistically independent eigenvalues. One of the representations, which is an extension of Dyson's method for m = 0, 1, is applied to the superposition of n independent sequences of levels. General asymptotic results are found for the mth‐order distributions for (a) small x, arbitrary n, and (b) arbitrary x with n → ∞.

Class of Ensembles in the Statistical Theory of Energy‐Level Spectra
View Description Hide DescriptionThe investigation of a large class of ensembles in the statistical theory of energy‐level spectra is initiated. Each member of this class is characterized by a joint probability density for N consecutive eigenvalues of the form ,where a ≤ λ_{ i } ≤ b, and β may be 1, 2, or 4. Formal calculations of the nearest‐neighbor spacing distribution and the level density are made for β = 2. Results are in terms of asymptotic properties of orthogonal polynomials. It is conjectured that spacing distributions are relatively insensitive to the function f(λ) and the interval [a, b]. When f(λ) = 1 and b = −a = 1, the resulting (Legendre) ensemble has the same spacing distribution as the Gaussian and Dyson ensembles. The level density is concave upward and rapidly increasing for λ ≥ 0, qualitatively resembling actual nuclear and atomic densities. This feature is not present in previously investigated ensembles. Certain invariant matrix ensembles introduced by Dyson, which are of the above type, have the same level density and nearest‐neighbor spacing distribution as the Legendre ensemble.

Some Dynamical Properties of an Impurity in the Hard‐Sphere Bose Gas
View Description Hide DescriptionThe dynamical properties of an impurity in the hard‐sphere Bose gas are investigated by extending the pseudopotential method and Hamiltonian of the author's previous work. The interaction potential between two impurities is calculated up to one quasiparticle exchange and is shown to have a Yukawa form of attraction with a range of (16πρa)^{−1/2} in addition to the hard‐core interaction at the origin. (ρ and a are the density of the boson gas and hard‐core diameter of the boson, respectively.) Necessary conditions for impurities to have a bound state are also obtained. It is found that the mass of the impurity must be greater than approximately nine times the boson mass, and the density of the boson gas must lie in some range with a maximum and minimum. Then the differential cross section for Compton‐type scattering and one quasiparticle production on an impurity are calculated. The Čerenkov radiation of sound quanta is studied when the impurity is traveling very fast in the hard‐sphere Bose gas. It is found that there is no radiation unless the velocity v of the impurity is faster than [k ^{2} + (16πρa)]^{1/2} and the radiation has a maximum for the angle θ_{0} such that cosθ_{0} = (k ^{2} + 16πρa)^{½}/v. The cross section for Čerenkov radiation is also calculated. (k is the momentum of the quasiparticle emitted in the Čerenkov radiation.)

Kramers‐Wannier Duality for the 2‐Dimensional Ising Model as an Instance of Poisson's Summation Formula
View Description Hide DescriptionThe well‐known Kramers‐Wannier high‐low‐temperature duality for the 2‐dimensional Ising model is derived by means of the Poisson summation formula for a commutative group.

A Remarkable Connection Between Kemmer Algebras and Unitary Groups
View Description Hide DescriptionStarting from a Kemmer algebra generated by n elements, a new algebra generated by ½(n ^{2} + n) elements is derived. This algebra has only one irreducible representation which is of dimension (n + 1). This representation is equivalent to the first elementary representation of the infinitesimal operators of the unitary group in a space of (n + 1) dimensions. Omitting the identity element, one of course obtains the first elementary representation of the unitary unimodular group in a space of (n + 1) dimensions. The procedure is first illustrated for the case of n = 3 and explicit representations of the infinitesimal operators of U _{4} (SU _{4}) are obtained. Next, the proof for arbitrary n is outlined. Finally, the degenerate case of U _{3} (SU _{3}) is discussed.

Theory of Scattering in Solids
View Description Hide DescriptionThe general theory of the scattering of excitations in solids by localized imperfections is discussed. The solid‐state analog of the usual partial‐wave expansion of the scattering amplitude is derived. In an appendix, the applicability of the general theory to phonons and spin waves as well as electrons is demonstrated.

Singular Potentials and Peratization. II
View Description Hide DescriptionIn a field theory with more than one unrenormalizable interaction (like the Wtheory with weak and electromagnetic coupling), the important problem arises of resumming multiple series of individually divergent terms. In order to get a first insight in the new questions which arise for multiple as compared to single series, we study the analogous question for a superposition of two singular potentials, using a family of exactly soluble cases. We ask whether one can expand resummed series for the zero‐energy scattering amplitude in powers of one coupling constant with coefficients depending on the other. The answer depends both on the relative magnitude of the coupling constants and on the relative degree of singularity of the interactions. Depending on these two conditions one finds three regimes, one where a convergent power expansion holds, another where an asymptotic expansions obtains, and a third where it is impossible to expand in powers of either single constant separately. It is conjectured that a similar situation will be true in a field theory with leading power singularities only (and no logarithmic ones), if such a theory has meaning.

Scattering of Surface Waves on an Infinitely Deep Fluid
View Description Hide DescriptionThe reflection and transmission of small amplitude waves incident on a plane barrier submerged in an infinitely deep fluid are investigated; the barrier has a finite width and is parallel to the undisturbed free surface of the fluid. Green's function techniques are used to represent the velocity potential in terms of its discontinuity (pressure difference) across the barrier. The application of the boundary condition at the barrier leads to a one‐dimensional integral equation for the pressure difference. This equation is extended by introducing the fluid velocity normal to the plane of the barrier, and then it is analyzed by the complex Fourier transform methods of Wiener and Hopf. The velocity transforms are found to be characterized by a pair of dual inhomogeneous integral equations which allow systematic approximation by iteration. A convolution provides representations for the velocity potential in the different regions of the fluid, and these are the basis for investigating the reflection and transmission of waves. Results for some related problems (e.g., scattering by a semi‐infinite barrier, by a finite dock, etc.) are obtained as limiting cases.

Diffraction by a Smooth Transparent Object
View Description Hide DescriptionIn previous investigations the so‐called geometrical theory of diffraction was established. Its application has been limited to problems of diffraction of waves by smooth opaque objects.
In the present paper, the geometrical theory of diffraction is extended and applied to problems of diffraction by a smooth transparent object of any shape. For simplicity only scalar fields and two‐dimensional problems are considered. However, this method can be modified easily to apply to vector fields and three‐dimensional problems.
In Part I, the fields associated with the geometric rays and the diffracted rays are constructed in detail for the case of a circular cylinder. Afterwards a general method for treating diffraction by a smooth transparent cylinder of arbitrary shape is given. In order to determine the coefficients, the exact solution for the case of a circular cylinder is found and evaluated asymptotically for large ka in Part II. Then it is compared with the solution obtained from geometrical theory in Part I.

Generalizations of the Jost Functions
View Description Hide DescriptionThe Jost functions have proved valuable in the study of the analytic properties of the scattering phase for the radial Schrödinger equation. In the present paper we shall present an alternative definition of the Jost functions, prove the equivalence of the new definition to the usual one, and generalize the new definition to the one‐dimensional Schrödinger equation (− ∞ <x< ∞), the three‐dimensional nonseparated Schrödinger equation, and the three‐dimensional nonseparated Dirac equation. It is hoped that these generalizations lead to a better understanding of the analytic properties of the scattering operator for these and related dynamical systems. The generalized Jost functions are shown to be operators in the variables which label the degeneracy of the continuous spectrum of the Hamiltonians which are considered.