Volume 5, Issue 5, May 1964
Index of content:

Matrix Elements of the Octet Operator of SU _{3}
View Description Hide DescriptionAll the nonvanishing matrix elements of all the components of the tensor operator which belongs to the regular representation (the octet) of SU _{3} have been evaluated. Of special interest is the component Y, for it is usual in the broken unitary symmetry theory of strong interactions to assume that the interactions which break exact SU _{3} invariance have the same transformation properties as Y. Previously, matrix elements of Y connecting states of the same irreducible representation of SU _{3} have been given by Okubo in the form of the mass formula. Knowledge of all the matrix elements of Y is essential however if one is to do more than evaluate one‐particle matrix elements in the broken unitary symmetry theory. Our method provides such knowledge for all components of the octet tensor operator with little more effort than is needed to treat Y alone.

Generalized Shmushkevich Method: Proof of Basic Results
View Description Hide DescriptionWe here derive certain orthogonality properties of the Clebsch‐Gordan (CG) coefficients of an arbitrary compact group G. Our discussion recognizes the fact that the irreducible representations (IR's) of G need not be equivalent to their complex conjugates and that the same IR can appear more than once in the reduction of the direct product of two IR's of G. The properties obtained allow the development of a generalized Shmushkevich method for directly writing down consequences of the invariance of particle interactions under G. The discussion given is sufficiently general to apply to the currently interesting cases of SU _{3} and G _{2}.

Intensive Observables in Quantum Theory
View Description Hide DescriptionThe notion of strictly intensive observables is introduced in a theory of local observables (such as that of R. Haag). If , are two disjoint regions, then the value of a strictly intensive observable in is the sum of its values in and . It is shown that energy‐momentum can never be strictly intensive. This result is used to prove that the algebras of observables is not of Type I for some regions. By analogy with the energy‐momentum tensor density for the free field, the definition is weakened, so that an intensive observable in a region is only ``approximately'' in . This leads to the introduction of germs of intensive observables. It is proved that the unitary intensive operators form a sheaf F of groups, and the Hermitian intensive operators form a sheaf G of Abelian groups with operators in F, and on which the inner derivative Y → i(XY − YX) is defined.

Coulomb Green's Functions and the Furry Approximation
View Description Hide DescriptionThe Coulomb Green's function for the nonrelativistic Schrödinger equation is obtained in closed form starting from the partial‐wave expansion and using an integral representation for a product of two Whittaker functions with different arguments. The Neumann's series for J _{ v }(kz) is required in evaluating the sum on states. Using the same methods, the Coulomb Green's functions for the Klein‐Gordon and iterated Dirac equations are obtained in closed form in the ``Furry approximation,'' a ^{2}/(J + ½)^{2} ≪ 1, a = Ze ^{2}/4πh/c. The Klein‐Gordon Green's function in this approximation is shown to be at the same time the exact Green's function for the Klein‐Gordon equation without the potential squared term. An alternate and very simple derivation of the approximate Green's function for the iterated Dirac equation is given using perturbation theory. From this Green's function, an approximate Coulomb Green's function in closed form for the Dirac equation itself is constructed. Certain known results for Coulomb wavefunctions with modified plane‐wave behavior at large distances are rederived using the foregoing methods and results.

Development of Singularities of Solutions of Nonlinear Hyperbolic Partial Differential Equations
View Description Hide DescriptionIn a recent paper Zabusky has given an accurate estimate of the time interval in which solutions of the nonlinear string equationy_{tt} = c ^{2}(1 + εy_{x} )y_{xx} exist. A previous numerical study of solutions of this equation disclosed an anomaly in the partition of energy among the various modes; Zabusky's estimate shows that at the time when the anomaly was observed the solution does not exist. The proof of Zabusky uses the hodograph method; in this note we give a much simpler derivation of the same result based on an estimate given some years ago by the author.

Boundary Conditions for a Partial‐Wave Amplitude
View Description Hide DescriptionA partial‐wave amplitude is frequently subject to constraints which specify its values at a given set of points. This happens, for instance, when we insist on its correct threshold behavior. We investigate such constraints in this paper and derive a class of inequalities which are necessary conditions for the existence of such an amplitude.

Study of Exactly Soluble One‐Dimensional N‐Body Problems
View Description Hide DescriptionIn this paper it is shown that several cases of one‐dimensional N‐body problems are exactly soluble. The first case describes the motion of three one‐dimensional particles of arbitrary mass which interact with one another via infinite‐strength, repulsive delta‐function potentials. It is found in this case that the stationary‐state solution of the scattering of the three particles is analogous to an electro‐magnetic diffraction problem which has already been solved. The solution to this analogous electro‐magnetic problem is interpreted in terms of particles. Next it is shown that the problem of three particles of equal mass interacting with each other via finite‐ but equal‐strength delta‐function potentials is exactly soluble. This example exhibits rearrangement and bound‐state effects, but no inelastic processes occur. Finally it is shown that the problem of N particles of equal mass all interacting with one another via finite‐ but equal‐strength delta functions is exactly soluble. Again no inelastic processes occur, but various types of rearrangements and an N‐particle bound state do occur. These rearrangements and the N‐particle bound state are illustrated by means of a series of sample calculations.

Application of Nonlocal Field Operators to a System of Hard‐Sphere Bose Gas
View Description Hide DescriptionSome relationships of the nonlocal field operators developed by Siegert et al. and the usual free‐field operators are obtained. The interaction Hamiltonian derived with the help of these relationships is compared with the pseudopotential recently obtained by Liu and Wong. A study of the fluid dynamicalequations at extreme low temperature has been made.

Analytic Solution of the Percus‐Yevick Equation
View Description Hide DescriptionThe properties of the Percus‐Yevick approximate integral equation for the pair distribution function in classical statistical mechanics are examined for the class of pair potentials consisting of a hard core plus a short‐range tail. For one‐dimensional systems, some elementary theorems of complex variable applied to the Laplace‐transformed equations enable one to express the direct correlation function in a very simple form, one which becomes explicit and trivial in the absence of a short‐range tail. In the presence of the tail, the direct correlation function satisfies a (coupled) integral equation over a finite domain. The impossibility of a phase transition in one dimension is strongly indicated. Analysis of the case of three dimensions proceeds similarly, but is complicated by the appearance of essential parameters other than the density and compressibility. The character of the direct correlation function is qualitatively unchanged. Principal differences in three dimensions are that a phase transition is no longer prohibited, and the pair distribution function cannot be reasonably expressed as a sum of nth‐neighbor contributions.

The Influence of Initial Correlations on the Approach to Equilibrium
View Description Hide DescriptionThe irreversible behavior of a particle under the influence of fixed scattering centers is investigated in the weak coupling limit. An ensemble is introduced which statistically describes the scattering centers as well as the particle. This allows the treatment of correlations between the particle and the scattering centers. A new diagrammatic method is used to investigate high‐order terms in the perturbation theory. This method yields quite explicit information about the way in which the influence of initial correlations will disappear. In particular, it is quite clear that as one goes to higher order in the perturbation theory one must wait longer times for the influence of initial correlations to disappear. For completeness, the Boltzmann equation is derived and solved to lowest order in the interaction strength.

Multiple Scattering in the Diffusion Approximation
View Description Hide DescriptionThe passage of classical particles through a grainy scattering medium can be described by a line‐arized Boltzmann equation. A discussion is given of the physical conditions which justify the use of the Fokker‐Planck diffusion approximation to this equation. Some limiting properties of the solutions of the diffusionequation are first discussed for the initial‐value problem in an infinite medium characterized by a diffusion length D. For a total path length ≪ D convenient formulas are given for the distribution of scattering angles θ and, for given θ, the first few moments of the final position vector are computed. These results are taken as a basis for approximate treatment of steady‐state boundary‐value problems. The case of a particle beam incident on a thin plane parallel slab of thickness d ≪ D is considered. Approximate formulas are given for the angular distribution of the transmitted beam and for the (very small) fraction of the beam which emerges from the entrance face. Errors are assessed, and the behavior for grazing angles of incidence or exit is discussed in a conjectural way.

One‐Speed Neutron Transport in Two Adjacent Half‐Spaces
View Description Hide DescriptionUsing Case's method for solving the one‐speed transport equation with isotropic scattering, the Milne problem solution, the solution for a constant source in one half‐space, and the Green's function solution are obtained for two adjacent half‐spaces. These problems have been solved previously by other methods. Here the derivations are greatly simplified by using Case's method.

Simple Derivation of the Faxén Solution to the Lamm Equation
View Description Hide DescriptionIn this paper a Hankel transform technique is used to derive the Faxén solution to the Lamm equation when the sedimentation coefficient is constant and when it varies linearly with concentration.

Phase‐Space Formulation of the Dynamics of Canonical Variables
View Description Hide DescriptionStatistical reformulation of quantum mechanics in terms of phase‐space distribution functions as given by Moyal using Weyl's correspondence rule between classical functions and operators has been extended to various different correspondence rules. The dynamical bracket in the Weyl correspondence (the ``Moyal'' or the ``sine'' bracket) is shown to be a Lie bracket. It is further shown that if the theory is restricted to Lie brackets of the form ,evaluated for p _{1} = p _{2} = p; q _{1} = q _{2} = q after differentiation, then the only admissible functional form of f is f(x) = β[(sin αx)/α], where α and β are constants. A law of multiplication which is associative and distributive with respect to addition is also introduced in each case. It gives a correct correspondence between operator multiplication and the multiplication of classical functions. The dynamical brackets obtained in each case are also found to be Lie brackets. Conditions on the phase‐space distribution functions to describe pure states are also given.

Norm Invariance of Mass‐Zero Equations under the Conformal Group
View Description Hide DescriptionIt is known that a suitable collection of solutions of the free‐field Maxwell's equations is a Hilbert space with respect to an appropriate norm, and that the inhomogeneous Lorentz group acts on this Hilbert space in a unitary and irreducible manner. It is shown that this representation extends to a unitary representation of the conformal group of Minkowski space. Similar results are obtained for other mass‐zero relativistic equations.

Existence of a New Conservation Law in Electromagnetic Theory
View Description Hide DescriptionTen new extensive quantities that appear to be independent of stress‐energy, but that analogously characterize the physical state of an electromagnetic field, are exhibited and are shown to be conserved in vacuum because of Maxwell's equations. These new quantities are shown to be capable of retrograde flow in a circularly polarized plane‐wave field.