Volume 5, Issue 11, November 1964
Index of content:

Crossing Matrices for Helicity Amplitudes, Application to Crossed Channel PartialWave Analysis, and Reggeization
View Description Hide DescriptionThe crossing relations of twobody scattering amplitudes for reactions involving particles withspin are derived for the helicity amplitudes of Jacob and Wick. From the crossing relations, thedifferential cross section and polarization of the direct channel are related to the analytic continuationof the crossed channel helicity amplitude. The differential cross section is then expressed interms of the crossed channel partial waves, and rules are given for treating the exchange of fixedangular momentum poles and Regge poles for twobody processes involving particles with higher spin.

Asymptotic Behavior of Feynman Integrals with Spin
View Description Hide DescriptionSome general features are investigated of the dependence of the asymptotic behavior of Feynmanintegrals upon factors in the numerator of the integrand resulting from particles with spin. Theseresults are used to analyze the highenergy behavior of ladder diagrams for nucleonsinteractingwith neutral vector mesons. The leading contribution is shown to consist of terms corresponding to areggeised nucleon together with certain other terms. The expected cancellation of these other termsby terms associated with a welldefined class of crossed diagrams is verified in detail for the sixthordercase. Finally, other significant diagrams, different from the ladders and their associated crosseddiagrams, are investigated and it is shown that they only provide higherorder corrections to thetrajectory of the reggeised nucleon.

Partial Differential Equations with Periodic Coefficients and Bloch Waves in Crystals
View Description Hide DescriptionBloch waves are special solutions of Schrödinger's equation with a periodic real potential. Theyare plane waves multiplied by periodic functions. In this paper we prove the existence and completenessof Bloch waves and of the related KohnLuttinger waves in unbounded domains for a class of partialdifferential equations which includes the Schrödinger equation. In addition, we discuss the dependenceof these waves and the corresponding eigenvalues on the wave vector of the associated plane wave.The results may be interpreted as the analogs for certain partial differential equations of Floquet'stheory for ordinary differential equations or as the determination of the spectral representation ofcertain periodic Hamiltonian operators.

A Modified WKB Approximation for Phase Shifts
View Description Hide DescriptionExtending an idea of Good, a modified WKB approximation using radial wavefunctions having the form of freeparticle solutions to the radial wave equation rather than an exponential form is developed. The lowestorder phase shifts are the same as those of the usual WKB approximation, but are improved by the contribution of the next order. The method is applied to two examples: the radial Dirac equation in the highenergy limit and the radial Schrödinger equation.

Particle Moderation: Random Functional Approach
View Description Hide DescriptionThe slowing down of a particle by a homogeneous isotropic moderator is considered. It is shown that finding the collision density of the particle as a function of space and energy is equivalent to finding the probability distribution of a certain random functional. By means of this random functional, expressions for the spatial moments are obtained without imposing any restrictions on the variations of the scattering kernel or cross section with energy. These moments are then used to obtain the age equation, the derivation given here differing from others in that no a priori assumptions are made on the collision density itself. Finally, as a special case of the above, the timeenergy moments are found.

Solution of the InitialValue Transport Problem for Monoenergetic Neutrons in Slab Geometry
View Description Hide DescriptionThe initialvalue transport problem of monoenergetic neutrons migrating in a thin slab is solved by applying the normalmode expansion method of Case to the results of Lehner and Wing. Fredholm integral equations are derived for the expansion coefficients. In addition, exact expressions for the eigenvalues of the problem are derived and the results of calculations are presented. The solution is shown to have properties expected from elementary diffusion theory.

On Scattering of Waves by Objects Imbedded in Random Media: Stochastic Linear Partial Differential Equations and Scattering of Waves by Conducting Sphere Imbedded in Random Media
View Description Hide DescriptionA new inhomogeneous linear partial differential equation satisfied by the mean value of the solution of the corresponding inhomogeneous stochastic linear partial differential equation is derived. This new equation has the interesting phenomenon that the differential operators couple with the inhomogeneous terms to form new inhomogeneous terms of the equation. Physically, this means that the randomness of medium and source are coupled together to form new sources. The above approach is then used to derive the equation characterizing wave motions in random media due to random sources. Finally, the problem of scattering of a plane wave by a perfectly conducting sphere of radius a imbedded in a random medium is considered. By utilizing a “pseudopotential” to incorporate the effect of the boundary condition into the reduced wave equation and by the above result, for both ka large and small it is found that up to and including terms of order the mean value of the scattered field can be calculated from the same deterministic scattering problem with k replaced by an effective propagation constant kñ. The specialization of the new formulation to the problem of scattering of a plane wave by a perfectly conducting semiinfinite space checks with a previous result of Chen.

Some Remarks Concerning a Pathological Matrix of Interest in the InverseScattering Problem
View Description Hide DescriptionA Hermitian matrix which occurs in the theory of the quantummechanical inversescatteringproblem has apparently contradictory properties. It has a wellbehaved inverse in spite of havingzero as one of its eigenvalues. The properties of the matrix are investigated and the relevance of theresults to the theory are discussed.

The Effective Resistance of Passive Networks
View Description Hide DescriptionA recently found solution to the problem of a random walk over a lattice with reflecting boundariesis used to evaluate the resistances of several passive electrical networks. The solutions are particularlysimple for rectangular planar networks and various types are considered. The extension of the solutionto the corresponding threedimensional array, the simple cubic lattice, is given.

Characters of Irreducible Representations of the Simple Groups. II. Application to Classical Groups
View Description Hide DescriptionThe general formulas found in a preceding paper for the characters of irreducible representationsof simple Lie groups are developed in the case of the classical groups: the four series , , , ,plus the exceptional case . Groups of rank 1 and 2, plug are studied in detail.

Kinetic Approach to Thermal Transport Phenomena. I. Classical Theory
View Description Hide DescriptionAssuming noninteracting particles, which are scattered by randomly arranged static potentials,such as free electrons in a lowtemperature semiconductor containing impurity atoms, the expressionsof thermal transport coefficients are derived by means of the kinetic approach established by Prigogineet al. In this part, classical law of mechanics is assumed, and the case of zero magnetic field is discussed.It is shown that the thermal transport coefficients thus obtained can be expressed in the terms of thecorrelation functions of suitable currents in the same way as the Kubo formula for the electric conductivity.

Perturbation Variation Methods for a Quantum Boltzmann Equation
View Description Hide DescriptionFor molecules with degenerate internal states, the singleparticle distribution function must bereplaced by a density matrix, or better, if the translational motion is treated classically, by a Wignerdistributionfunction density matrix. The modified BoltzmannIntegrodifferential equation for thisquantity has been previously derived but so far only limited solutions of the resulting equation havebeen obtained. Methods are herein discussed which enable the standard methods for the solution ofthe classical Boltzmann equation to be applied to the solution of this equation. Complications involvingcommutation properties are resolved.

Observables in Relativistic Quantum Mechanics
View Description Hide DescriptionThe conventional statement of statistical determinism is that “the expectation values of all (Heisenberg)observables are determined by the expectation values of the observables at one time.” This requiresthat a full algebra of selfadjoint operators be in onetoone correspondence with measurementprocedures performed at one time. For instance, it requires that if two noncommuting observables pand q are defined at , there should exist a measurement procedure at corresponding to .No such procedure is known. The contrast between the positive assertion of the existence of certainlaboratory procedures and the inability to describe them constitutes perhaps the weakest point ofquantum mechanics. However, the conventional statement of statistical causality is shown to be untenablein a relativistic theory. This paper proposes a weaker form of causality which (1) uses measurementsmade within a truncated light cone rather than at one time for predictive purposes, and (2)which involves only strictly localized states, i.e., states which are vacuumlike outside a finite volume.Failure of the conventional causality statement implies that the set of quasilocal observables is notnecessarily linear, i.e., if A and B are in a set, is not necessarily in it. This remark may open theway to a systematic inquiry into the problems of associating laboratory procedures to selfadjointoperators.

Relativistic Coulomb Scattering of Electrons
View Description Hide DescriptionA simple and useful relation between the Coulomb amplitudes F and G (in Mott's notation) is derived and F and G are evaluated analytically up to terms for arbitrary . These results are valid for all angles, but are particularly useful at small angles. The general analytic behavior of F and G in the variable is discussed. The method is applicable to higherorder terms ( and up). A double integral representation of F is also derived by using the SommerfeldWatson transformation. This integral representation exhibits the dependence on α, q, and θ separately.

Possible Relationship between Electric Charge and Dual Charge
View Description Hide DescriptionElectric charge has no direct meaning for strong interactions, yet is involved in the octet model insuch a way that the size of the elementary unit of electric charge becomes tied to the topology of thegroup of strong interaction symmetries. This tie may indicate a relationship of electric charge toanother kind of charge got directly from topology of the group.

Coulomb Green's Function
View Description Hide DescriptionA oneparameter integral representation is given for the momentum space Green's function of the nonrelativistic Coulomb problem.

On the Construction of a Unitary Matrix with Elements of Given Moduli
View Description Hide DescriptionGiven nonnegative real numbers which form a matrix with row and column vectors of unit magnitude, it is shown under what conditions there exists a unitary matrix , such that . The results may be shown to contain a theorem on unitary matrices.

On the Generation of Anisotropic Tensors
View Description Hide DescriptionThe problem of generating a complete set of linearly independent nthorder tensors which are invariant under a crystallographic group is considered. A number of methods for the solution of this problem such as the use of tensor bases, the addition of tensors of lower symmetry, and the method of polynomial invariants are discussed. The limitations of these methods are outlined.

Lagrangian Formulation of the Phonon Field Equations
View Description Hide DescriptionThe equations of motion of lattice vibrations are formulated with the action principle as a starting point. Aa a result, one obtains, in addition to the equations of motion, the conservation laws for energy and momentum. The latter are contained in a set of finite difference equations.Boundary conditions on the field variables must be specified over a region equal to one lattice spacing in order for the entire procedure to be meaningful. The quantized version of the theory can be constructed in a conventional way, and the commutators of the field variables exhibit a set of periodically spaced singularities. In this way we construct a field which is nonlocal with respect to its dependence on space variables, but is local with respect to its time dependence.

Multiple Scattering of Electromagnetic Waves by Random Scatterers of Finite Size
View Description Hide DescriptionThe problem of multiple scattering of waves by randomly positioned objects has been treated byseveral authors, for example, Foldy, Lax, Twersky, Waterman, and Truell. The present work extendsthe theory to electromagneticvector fields and to scatterers of arbitrary size and properties. A generalformulation has been made for scattering by any type of discrete and identical scatterers which aresimilarly oriented. The case of spherical scatterers has been treated by using the rigorous Mie theoryboth for sparse and dense concentration. Results indicate that in case of sparse concentration, thestatistical expectation of the total field has a polarization similar to that of the normally incidentwave and the distribution of scatterers is equivalent to a homogeneous medium with a modifiedrefractive index. In case of dense concentration the medium can sustain a number of planewavemodes. A dispersion relation for the modified medium has been obtained. When the special cases ofsmall spheres is considered, the wellknown results obtained by other authors are recovered.