Volume 15, Issue 3, March 1974
Index of content:

New commutator identities on the Riemann tensor
View Description Hide DescriptionA number of new generally covariant identities which involve second derivatives of the Riemann tensor are presented. Each of these new identities can be expressed by equating to zero either (a) a particular sum of terms each of which contains an operator of the form (▿_{μ}▿_{ν} − ▿_{ν}▿_{μ}) acting on the Riemann tensor; or (b) a particular sum of terms each of which contains an operator of the form ▿_{ a } acting either on the expression or on the expression ; or (c) a particular sum of algebraic terms each of which contains no derivatives of the Riemann tensor, but rather is quadratic in the Riemann tensor. Each of the new identities can be expressed in all three of the above‐described forms. Furthermore, each of these new identities can be thought of as an integrability condition derived from the equations that define the Riemann tensor in terms of the or the g _{μν}. The requirements of Riquier's existence theorem are used to guide the derivation of the identities. The operator ▿_{μ} denotes covariant differentiation. All the new identities assume the existence of a symmetric connection and one of the new identities assumes the existence of a metric. Schouten's identity and Walker's identity are also discussed.

Lowest nonnegative solution of the Yukawa Hamiltonian
View Description Hide DescriptionTwo recent papers [J. Math. Phys. 13, 1825 (1972); 14, 1140 (1973)] obtained rough limits 1.68 ≥ b _{0} ≥ 1.67 for the quantity b _{0} such that an energy eigenvalue of the Yukawa Hamiltonian must be zero or positive if . We point out that (along with other aspects of the Yukawa equation) the quantity b _{0} has long been known to a much higher precision.

On ergodic limits of normal states in quantum statistical mechanics
View Description Hide DescriptionThe asymptotic behavior for and its time average is discussed. Here is S an element of the Banach space , constituted by the trace class of operators on the (separable or nonseparable) Hilbert space, and H is the Hamiltonian, i.e., a bounded or unbounded self‐adjoint operator on . Necessary and sufficient conditions are given for the existence of the limits and S(± ∞) with respect to the weak topology on , for the latter under the assumption that the continuous spectrum of H is absolutely continuous. In addition it is shown that if, for a normal state (density operator) ρ, has a weak limit, then the limit is again a normal state. This provides further insight in the nature of Emch's ``first ergodic paradox'' [G. G. Emch, J. Math. Phys. 7, 1413 (1966)].

Green's function for electromagnetic scattering from a random rough surface
View Description Hide DescriptionThe stochastic Green's matrix is calculated for a random rough surface with Gaussian statistics and a magnetic boundary condition. The techniques we use are similar to those developed for the scalar and elastic cases. The coupled surface integral equations which are derived are the Green's function version of the Franz formulas. These integral equations are represented in k‐space in a certain gauge and a Feynman‐diagram‐like interpretation is given to each term in the equations. The diagram rules have many formal similarities with the scalar and elastic rules. By using partial summation techniques, the mean and second moment of this Green's function are shown to be solutions to Dyson and Bethe‐Salpeter equations respectively. The Green's function is applied to a scattering problem. Some approximations and simple examples are presented. The lowest order approximations agree with the standard literature results. The main advantage of the diagram method, its systematic presentation of higher order approximations, is stressed.

An initial value method for an inverse problem in wave propagation
View Description Hide DescriptionKay and Balanis have reduced various inverse problems in wave propagation to the solution of Fredholm integral equations. These integral equations can be further reduced to Cauchy systems.

Time‐dependent multichannel Coulomb scattering theory
View Description Hide DescriptionThe formulation by Mulherin and Zinnes of two‐particle Coulomb scattering theory is extended to the multichannel case. The wave operators so obtained are proved by a direct method to be identical with those of Dollard.

Some differential‐difference equations containing both advance and retardation
View Description Hide DescriptionAn explicit solution is given to the boundary value problem for certain linear differential‐difference equations. The solution is well behaved even in the presence of advanced interactions. Interest in these equations arises from study of time symmetric electrodynamics.

Solutions of the steady, one‐speed neutron transport equation for small mean free paths
View Description Hide DescriptionThe solution of the steady, one‐speed neutron transport equation with isotropic scattering and small mean free path is obtained. The solution is asymptotic with respect to a small parameter ε, defined as the mean free path in terms of a unit length of the same order of magnitude as a typical dimension of the domain. The solution, the leading two terms of which are given, consists of a boundary layer solution plus an interior solution. The boundary layer solution decays exponentially with distance from the boundary, the decay rate being proportional to ε^{−1}, and it shows the effects of boundary curvature and variations in the incoming flux along the boundary. The interior solution is a multiple of the source for subcritical domains, and depends on a diffusion equation for near critical domains. The boundary condition for the diffusion equation and an asymptotic criticality condition are derived.

Spinor calculus in five‐dimensional relativity
View Description Hide DescriptionA consistent spinor calculus is developed within the framework of five‐dimensional relativity. The formalism is manifestly five‐covariant, and in special coordinate systems and special spin frames it reduces to a familiar spinor formalism in curved space‐time. The five‐dimensional formulation is free of the difficulties involving the coupling of the electromagnetic field, which characterize the four‐dimensional approach to spinor calculus. Some theorems and useful identities in the five‐dimensional spinor calculus are proved.

Five‐dimensional approach to the neutrino and electron theories
View Description Hide DescriptionThe theories of the neutrino and the electron are studies from a five‐dimensional point of view. The concept of periodic 5‐spinors is examined, and the formalism of five‐dimensional spinor calculus is applied to the manipulations of spinor equations. It is shown that the five‐dimensional approach reproduces the results of the familiar four‐dimensional approach in a unified and coherent manner. Furthermore, the present approach predicts in a natural way the correct coupling to the electromagnetic field, without ad hoc assumptions.

Smoothed boundary conditions for randomly rough surfaces
View Description Hide DescriptionThe problem of scalar wave propagation in the half‐space bounded by a rigid, randomly rough surface which is a small perturbation of an infinite plane is considered. It is shown that, if the boundary roughness is statistically homogeneous, the coherent (or average) wave satisfies a generalized impedance boundary condition on the average boundary. This is referred to as a ``smoothed'' boundary condition. Applying it to plane waves yields an expression for the effective plane‐wave reflection coefficientC_{e} of the boundary. For the case of isotropic roughness, approximate expressions for C_{e} are obtained for both long (relative to the correlation length of the boundary roughness) and short waves. These expressions show that generally C_{e}  < 1, and therefore that the amplitude of the coherent wave is diminished upon reflection from the boundary. This is the result of scattering of energy out of the coherent wave by the boundary roughness. It is also shown that this type of boundary can support a surface wave. This wave propagates at near‐grazing incidence with a speed slightly less than the free‐space propagation speed. Its amplitude decreases with propagation distance, also as a result of scattering by the boundary roughness.

A one‐dimensional fermion model exhibiting an anomalous type of phase transition
View Description Hide DescriptionWe construct a one‐dimensional fermion model (rigorously reducible to a mean field theory). We show that this model exhibits a second order phase transition associated with a spontaneous breakdown of continuous space translational symmetry in favor of a periodic symmetry. However, Landau and Lifshitz conjectured that a phase transition in which there is a spontaneous breakdown of Euclidean symmetry in favor of a crystallographic symmetry must be a transition of the first order. Thus we obtain a counterexample to this conjecture in the case of one dimension.

Conformal Killing vector fields on timelike two‐surfaces
View Description Hide DescriptionA topological description of Killing vector fields as well as conformal Killing vector fields on a 2‐space‐time is given, and the block diagram extension technique is generalized.

Renormalized number operators
View Description Hide DescriptionA number operator for a Weyl system is called renormalized essentially if it is obtained from the total number operator by subtraction of a (possibly infinite) constant (in exponentiated form). Necessary and sufficient conditions for the existence of a renormalized number operator are obtained.

The three‐particle S matrix
View Description Hide DescriptionIt is proved that a reduced T matrix, defined by factoring out of the three‐particle S matrix the product of the three two‐particle S matrices, is a compact operator on the energy shell, in spite of the double‐scattering singularity. As a result there exists a discrete, complete set of eigenphase shifts at every energy.

Generalized free fields and the representations of Weyl group
View Description Hide DescriptionThe transformation properties of generalized free fields under the transformations of the Weyl group, and particularly under the subgroup of dilatations, are discussed. It is shown that there exists, for any complex value d of the dimensionality parameter, a generalized free field, defining by means of its one‐particle states the suitable irreducible representation of the Weyl group.

On algebraically irreducible representations of the Lie algebra sl(2)
View Description Hide DescriptionAn algebraic study of the irreducible representations of the complex Lie algebrasl(2) is presented in this article. This study generalizes a former series of works of W. Miller. Though the list is not complete, it gives hints as to the construction of a very vast family of representations.

On nonlinear transformations in vector spaces. Colineation and conformal groups
View Description Hide DescriptionThe sets of special colineations and conformal transformations of a pseudo‐orthogonal vector space can be given only a Lie groupoid structure, rather than a Lie group structure because of the non‐vanishing denominators. With every such groupoid one can associate a unique Lie group, which, however, no longer consists of transformations. For one‐parameter subgroups one can define infinitesimal transformations and a bilinear composition, called Lie bracket, which reduces for linear transformations to the commutator. In the special cases of colineations and conformal transformations on pseudo‐orthogonal vector spaces of arbitrary finite dimension and signatures, covering homomorphisms onto matrix groups are given together with the corresponding Lie algebra isomorphisms.

Faddeev‐like equations with multibody forces
View Description Hide DescriptionEquations for a system of four particles interacting through two‐, three‐, and four‐body potentials are derived using a generalization of Faddeev's technique for the three‐body problem. Two alternative approaches are condsidered and simple equations for iterative solutions are written down. The method is generalized to the N‐body problem with multibody forces.

A model unified field theory
View Description Hide DescriptionThe geometry of space is probed by using a quantum mechanical wavefunction, which allows the introduction of the ordinary curvature tensor, and also provides room for a vector which is identified with the electromagnetic vector potential. The coordinates of space are taken as complex. Thus the electromagnetic field is geometrized and its presence is found to affect the gravitational field equations.