Volume 11, Issue 8, August 1970
Index of content:

New Integral Formulation of the Schrödinger Equation
View Description Hide DescriptionThe 1‐dimensional Schrödinger equation is replaced by a pair of coupled integral equations. The equations are solved by iteration. The zeroth‐order solution is the WKB approximation. The solutions are valid even in the presence of classical turning points and discontinuous potentials. Thus, connection formulas are not required. Low‐order calculations are carried out in several cases of interest. In each case, a first‐order calculation gives results comparable to or better than those obtained with connection formulas.

Canonically Conjugate Pairs, Uncertainty Relations, and Phase Operators
View Description Hide DescriptionApparent difficulties that prevent the definition of canonical conjugates for certain observables, e.g., the number operator, are eliminated by distinguishing between the Heisenberg and Weyl forms of the canonical commutation relations (CCR's). Examples are given for which the uncertainty principle does not follow from the CCR's. An operator F is constructed which is canonically conjugate, in the Heisenberg sense, to the number operator; and F is used to define a quantum time operator.

Solution of Mathieu's Type of Secular Equations
View Description Hide DescriptionA discussion is given for the general solution of a secular equation with all other matrix elements equal to zero except H_{ii} , H_{i i+h} , and H_{i+h i} . Both the eigenvalue and the amplitudes of the corresponding eigenfunction are expressed in terms of continued fractions of the matrix elements. A solution of Mathieu equation is used as an example to compare the present method with other methods.

Diffraction Theory of Holography
View Description Hide DescriptionA diffraction theory of side‐band holography for transmission objects is formulated. Conditions are derived under which good quality images are formed. A simple geometrical construction is found by means of which the location of the images may be determined. It is also shown that, under conditions that are often satisfied in practice, one of the two images will be completely absent. This effect is entirely caused by diffraction (conversion of a homogeneous wave into evanescent waves on diffraction by the hologram) and has nothing to do with the finite resolving power or the finite thickness of the photographic plate. Numerical examples illustrating this phenomenon are given.

Commuting‐Operator Approach to Group Representation Theory
View Description Hide DescriptionA study is given of the class‐sum‐operator approach to the representation theory of finite groups, the group D _{3h } being specifically studied. The class sum approach is shown to simplify the decomposition of Kronecker products. By using the class‐sum‐operator approach, it is shown that the ``indirect'' group‐projection operators of Löwdin can be used in finite group theory, where they lead to useful factorizations of the finite group‐projection operators. It is also shown that tensor operators of certain symmetry types can be constructed within the group itself, and may be used analogously to the usual operator equivalents of crystal field theory.

O(4) Symmetry and the BS Equation for a Spin‐1 + Spin‐0 System
View Description Hide DescriptionA set of vector functions analogous to the 3‐dimensional vector spherical harmonics is defined in four dimensions. These functions are employed to separate the Bethe‐Salpeter (BS) equation for a spin‐1 + spin‐0 system at vanishing total 4‐momentum K, where O(3) symmetry degenerates into O(4) symmetry. The angular parts of the matrix elements of all possible tensor operators, symmetry conserving and otherwise, are reduced. Finally, the BS equation at small K is briefly discussed.

Block Diagrams and the Extension of Timelike Two‐Surfaces
View Description Hide DescriptionThe work of Finkelstein, Kruskal, Graves and Brill, Carter, and Boyer and Lindquist on the extension and schematic representation of 2‐dimensional metrics is systematized and generalized. As a result, a number of extensions may be found by inspection. Some well‐known examples are given, and the technique is applied to the ``soluzioni oblique'' of Levi‐Civita.

Classical and Quantum Mechanical Correlation Functions of Fields in Thermal Equilibrium
View Description Hide DescriptionThe two‐point, two‐time correlation functions of classical and quantum mechanical fields in thermal equilibrium in an arbitrary domain are considered. For classical fields that satisfy linear equations of motion, the correlation functions are expressed in terms of certain Green's functions of the domain. It is also shown, by evaluating the characteristic functional of the field, that a classical field is Gaussian distributed, so that all higher‐order correlations can be expressed in terms of the second‐order correlations. Then the second‐order correlations are evaluated in various cases, and the classical and quantum mechanical results are compared. They are found to agree except within layers, about a thermal wavelength wide, near the boundaries of the domain and near the characteristic surface or light cone emanating from either of the two points. Thus the quantum mechanical correlation function can be approximated by the classical one with quantum‐mechanical boundary‐layer corrections.

Relation of the Inhomogeneous de Sitter Group to the Quantum Mechanics of Elementary Particles
View Description Hide DescriptionA relativistic dynamical group recently introduced by Aghassi, Roman, and Santilli for the quantum mechanics of elementary particles is briefly reviewed. It is shown in detail that the algebra of this group can be obtained by contracting the Lie algebra of the inhomogeneous de Sitter group ISO(3, 2). Some crucial concepts of the proposed new group are shown to appear in a new light when viewed in the context of a de Sitter world. The emergence of proper time as an additional kinematical variable is discussed in some detail.

Summability of the Many‐Fermion Perturbation Series with Attractive Forces
View Description Hide DescriptionThe summability of the many‐fermion perturbation series in the presence of attractive forces is considered. We find that, although it is an asymptotic series, the potential‐strength perturbation series is summable. We find that some rearrangements to handle the hard‐core potentials also lead to the correct sum; however, the ``hole‐line'' summation procedure using Brandow's choice for intermediate state energies is either wrong or deceptive in its convergence.

Eigenvalues and Eigenfunctions of the Spheroidal Wave Equation
View Description Hide DescriptionA technique is presented for the calculation of the oblate and prolate spheroidal wave equationeigenvalues and eigenfunctions. The eigenvalue problem is cast in matrix form and a tridiagonal, symmetric matrix is obtained. This formulation permits the immediate calculation of the eigenvalues to the desired accuracy by means of the bisection method. The eigenfunction expansion coefficients are then obtained by a recursion method. This technique is quite simple to program, and the computation speed is rapid enough to allow its use as a function subroutine where values not previously tabulated or large numbers of values are required.

New Theorems about Spherical Harmonic Expansions and SU(2)
View Description Hide DescriptionDespite the heroic efforts of Laplace, Legendre, Maxwell, Hobson, and company, there are still new chapters to be written in the theory of spherical harmonics. This paper describes some of the special windfalls that result when one expands analytic functions. In technical language, we are concerned with Fourier series of analytic functions on SU(2).

Theory of Particles with Variable Mass. I. Formalism
View Description Hide DescriptionThe equivalence principle (through the mechanism of the gravitational red shift) allows one to set up a classical formalism with the proper time as an extra degree of freedom, independent of the coordinate time, and with an immediate physical interpretation. Then proper time and mass occur as conjugate variables in a canonical formalism, leading to a gravitational theory of particles with variable mass. The nonrelativistic theory and a relativistic vector theory of gravity are described as models. The theory is capable of providing a dynamical framework for cosmological models with the creation of matter. Some simple examples are discussed, including the steady‐state universe with continuous creation, where the correct relation between the density of matter and the Hubble constant appears automatically, with no free parameters.

Theory of Particles with Variable Mass. II. Some Physical Consequences
View Description Hide DescriptionThe formalism of the previous paper, in which mass and proper time are treated as independent dynamical variables in a canonical formalism, is shown to imply certain physical consequences. There will exist a mass vs proper time uncertainty relation; trajectories and proper time will be exactly determinable in an external gravitational field, while mass will be determinable in an external electromagnetic field; and conventional quantum mechanics will imply that equivalence is invalid for low‐lying quantum states. This leads to a second possible way to quantize a system in a gravitational field, which introduces a fundamental length. It is shown that it is possible to test for quantum interferenceeffects of gravitational systems with present technology and conventional techniques, using the earth's gravitational field.

Moments and Correlation Functions of Solutions of a Stochastic Differential Equation
View Description Hide DescriptionThis paper shows how to obtain exact, closed‐form expressions for various moments and correlation functions of the solutions of the stochastic, ordinary differential equation,where T(z) is the so‐called ``random telegraph'' wave and β_{0} ^{2} and η are positive real constants. These moments and correlation functions are calculated by two different methods, one a phase space method and the other a matrix method familiar from optics. It is found that the moments are sums of exponentials. The first‐order moments decay exponentially but the second‐order moments grow exponentially. The correlation functions are also sums of exponentials and show that the solutions do not form a stationary process. An important application of these results is obtained in the problem of a plane electromagnetic wave normally incident on a randomly stratified dielectric plate. It is shown that, if S is the amplitude transmission coefficient of the plate, then can be expressed in terms of the second‐order moments of the solutions and derivatives of solutions of the stochastic differential equation.

Application of the Smoothing Method to a Stochastic Ordinary Differential Equation
View Description Hide DescriptionWe study the use of the ``smoothing method'' to calculate the second‐order moments of solutions of stochastic, ordinary, linear differential equations. We consider in detail the equation,where N(z) is a real, zero mean, wide‐sense stationary stochastic process and β_{0} and η ≪ 1 are positive constants. We show that, for one choice of N(z), the conventional use of the smoothing method yields correct first‐order moments of the solutions, but badly incorrect second‐order moments. We develop what we believe is a better way to use the smoothing method to calculate second‐order moments. For the special choice of N(z), this method yields exact results. The method can be extended to the calculation of moments of all orders for arbitrary stochastic, ordinary, linear differential equations.

Canonical Unit Adjoint Tensor Operators in U(n)
View Description Hide DescriptionA complete, fully explicit, and canonical determination of the matrix elements of all adjoint tensor operators in all U(n) is presented. The class of adjoint tensor operators—those transforming as the IR [1 0̇ −1]—is the first exhibiting a nontrivial multiplicity. It is demonstrated that the canonical resolution of this multiplicity possesses several compatible (or equivalent) properties: classification by null spaces, classification by degree in the Racah invariants, classification by limit properties, and the classification by conjugation parity. (The concepts in these various classification properties are developed in detail.) A systematic treatment is presented for the coupling of projective (tensor) operators. Six appendices treat in detail the explicit evaluation of all Gel'fand‐invariant operators (I_{k} ), the structural properties of Gram determinants formed of the I_{k} , the zeros of the norms of the adjoint operators, and the conjugation properties of the canonical adjoint tensor operators.

Derivation of the Wave and Scattering Operators for an Interaction of Rank One
View Description Hide DescriptionExpressions for the wave and scattering operators in terms of singular integral operators are rigorously derived for a simple scattering system, for which the interaction Hamiltonian V is of rank 1. The wave operators, defined by the strong limits, lim e^{iHt}e^{−iH0t} as , are known to exist for this interaction, and the existence of these limits is assumed throughout the paper. Using complex contour integrals to write down representations for certain operators, we find an identity for the evolution operator in terms of a family of bounded operators, from which singular integral operators are obtained in the limit as t → ∞. The analysis differs from previous applications of time‐dependent scattering theory to this system in that at no stage need ``smoothness'' or Hölder conditions be assumed for the element of L _{2}(− ∞, ∞) from which V is constructed.

Singular Integral Operator Encountered in Scattering Theory
View Description Hide DescriptionThe domain of an unbounded singular integral operator occurring in scattering theory is investigated and is proved to be everywhere dense. An application is made to a simple scattering system, for which the interaction Hamiltonian is of rank one, and the existence of the wave operator is proved by a ``time‐dependent'' method which does not assume ``smoothness'' of the element of L _{2}(− ∞, ∞) from which the interaction is constructed.

Rotation in Closed Perfect‐Fluid Cosmologies
View Description Hide DescriptionA Lagrangian is presented which describes the evolution of anisotropy in rotating closed (Bianchi type IX) cosmologies. It is assumed that the matter is a perfect fluid with an adiabatic equation of statep = P(ρ) such that 0 < dp/dρ < 1 − γ, γ > 0. The matter terms enter the Lagrangian as a potential which for large anisotropy is essentially identical to one found previously for dust. Because the dust potential has been thoroughly investigated by Ryan, we confine our discussion of the dynamics to a few brief remarks. The matter terms are crucial for the rotation since the constraint equations require matter for the presence of rotation. The chief effect of the rotation is the appearance of centrifugal terms in the Lagrangian which exclude certain parameter values to the system. An appendix gives a proof that R = 0 is a true infinite‐density singularity.