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Volume 4, Issue 7, July 1963

More Integral Representations for Scattering Amplitudes with Complex Singularities
View Description Hide DescriptionPrevious work is generalized in order to achieve a better understanding of the role of complex singularities in connection with integral representations. The most general conditions under which the box‐diagram contribution to a scattering amplitude satisfies a representation with real integration contours are derived. Explicit representations are derived in several special cases. It is frequently found possible to obtain representations that are essentially of the Mandelstam type, although the more general Bergman‐Oka‐Weil representation must be invoked in general. One example of a three‐dimensional representation is given, which exhibits the analytic structure in one of the external masses in addition to the kinematical variables. The significance of the ``physical sheet'' is discussed.

Acnodes and Cusps and the Mandelstam Representation
View Description Hide DescriptionComplex singularities on the physical sheet associated with acnodes (isolated real points) and cusps are found for a particular Feynman diagram. These singularities emerge before the appearance of anomalous thresholds, and cause a breakdown of the Mandelstam representation.

Singularities of the Four‐Point Function in Perturbation Theory with Variable Masses
View Description Hide DescriptionThe Landau surface for the four‐point function with one or more variable external masses is parameterized. The real singularities of the four‐point function with one external mass are explicitly determined. The location of the complex singularities of the function depend on the choice of the cuts for the three‐point function. Several suggestions are made as to how these could be taken.

Density Matrices Arising from Incomplete Measurements
View Description Hide DescriptionThe problem of how to associate a statistical ensemble of a quantum mechanical system with an incomplete set of measured ensemble averages is discussed. The view adhered to in this note is that the most chaotic ensemble consistent with the measured ensemble averages is a reasonable choice of ensemble. The properties of this ensemble are studied rigorously for the case when the quantum mechanical system is associated with a finite‐dimensional Hilbert space. The results are extended heuristically to some particular ensembles of many‐boson and many‐fermion systems.

N‐Dimensional Total Orbital Angular‐Momentum Operator
View Description Hide DescriptionA set of generalized polar coordinate systems are defined in N‐dimensional space. The total orbital angular‐momentum operator as defined by Louck is found to be a tensor invariant on the (N − 1)‐dimensional unit sphere; hence it is easily explicitly determined in any of the possible generalized polar coordinate systems. Commuting operators can be found and the eigenvalue problem solved in many coordinate systems. Two examples are given: (1) a coordinate system of 3M dimensions where M is an integer, and (2) a coordinate system of 8 dimensions.

A Remarkable Representation of the 3 + 2 de Sitter Group
View Description Hide DescriptionAmong the infinitesimal operators of the 3 + 2 de Sitter group, there are four independent cyclic ones, one of which is separate from the other three. A representation is obtained for which this one has integral eigenvalues while the other three have half‐odd eigenvalues, or vice versa. The representation is of a specially simple kind, with the wavefunctions involving only two variables.

Particle Charge Symmetries as R _{8} Subgroups
View Description Hide DescriptionIt is shown how the formalism of 8 × 8 Dirac matrices can be extended to include all the groups so far proposed as relevant to elementary‐particle charge symmetries: R _{4}, R _{7}, G _{2}, SU _{3}. These are all treated as subgroups of R _{8}, which appears to determine the eightfold structure of the particle families, even before the particle interactions are ``switched on''. Since these subgroups of R _{8} are incompatible, they will lead to a ``clash of symmetries'', as observed experimentally. It is pointed out that if the plausible association is made between real and charge‐space statistics in representations of R _{8}, the group SU _{3} satisfies charge‐conjugation invariance for 3‐boson interactions but not for two‐fermion‐one‐boson interactions.
An argument is given that the representations 8 and 8̄ of SU _{3} plus the representations (7 + 1) of G _{2} and R _{7} completely span the symmetries obeying the limited invariance implied by conservation of isotopic spin I and hypercharge Y.

Empty‐Space Generalization of the Schwarzschild Metric
View Description Hide DescriptionA new class of empty‐space metrics is obtained, one member of this class being a natural generalization of the Schwarzschild metric. This latter metric contains one arbitrary parameter in addition to the mass. The entire class is the set of metrics which are algebraically specialized (contain multiple‐principle null vectors) such that the propagation vector is not proportional to a gradient. These metrics belong to the Petrov class type I degenerate.

The Flatter Regions of Newman, Unti, and Tamburino's Generalized Schwarzschild Space
View Description Hide DescriptionThe ``generalized Schwarzschild'' metric discovered by Newman, Unti, and Tamburino, which is stationary and spherically symmetric, is investigated. We find that the orbit of a point under the group of time translations is a circle, rather than a line as in the Schwarzschild case. The time‐like hypersurfaces r = const which are left invariant by the group of motions are topologically three‐spheres S ^{3}, in contrast to the topologyS ^{2} × R (or S ^{2} × S ^{1}) for the r = const surfaces in the Schwarzschild case. In the Schwarzschild case, the intersection of a spacelike surface t = const and an r = const surface is a sphere S ^{2}. If σ is any spacelike hypersurface in the generalized metric, then its (two‐dimensional) intersection with an r = const surface is not any closed two‐dimensional manifold, that is, the generalized metric admits no reasonable spacelike surfaces. Thus, even though all curvature invariants vanish as r → ∞, in fact R _{μναβ} = O(1/r ^{3}) as in the Schwarzschild case, this metric is not asymptotically flat in the sense that coordinates can be introduced for which g _{μν} − η_{μν} = O(1/r). An apparent singularity in the metric at small values of r, which appears to be similar to the spurious Schwarzschild singularity at r = 2m, has not been studied. If this singularity should again be spurious, then the ``generalized Schwarzschild'' space would represent a terminal phase in the evolution of an entirely nonsingular cosmological model which, in other phases, contains closed spacelike hypersurfaces but no matter.

Initial‐Value Problem on Einstein‐Rosen Manifolds
View Description Hide DescriptionA class of solutions of the time‐symmetric initial‐value equations for gravitation and electromagnetism is obtained on a two‐sheeted manifold containing N Einstein‐Rosen bridges. The initial metric tensor and electric field are expressed in terms of a pair of harmonic functions, called ``metric potentials,'' which are required to be analytic and asymptotically flat and to satisfy certain match‐up conditions; these potentials are then determined by applying the method of inversion images. The particular case of two identical Einstein‐Rosen bridges is examined in detail, and is shown to correspond to a wormhole in an asymptotically flat (one‐sheeted) universe. A charge and ``renormalized mass'' are defined for each Einstein‐Rosen bridge, as well as an interaction energy (gravitational plus electrostatic potential energy) for the system as a whole. Expressions for these and other physical characteristics are evaluated explicitly in the two‐body case.

Undulatory Riemannian Spaces
View Description Hide DescriptionA demonstration is given that Riemannian spaces of very high curvature in submicroscopic domains do not contradict the existence of a macroscopic line element which is nearly Minkowskian. The signature of the microscopic line element is positive definite and the wave property of the metric in macroscopic domains comes about by a peculiar ``wave‐guide action'' of a strongly curved, two‐dimensional line element, in harmony with the particlelike behavior of the photon. The four‐dimensional lattice structure of the metrical vacuum field does not establish an absolute frame of reference and can be harmonized with the macroscopic validity of the Lorentz transformations.

On the Number of Self‐Avoiding Walks
View Description Hide DescriptionLet χ_{ n } be the number of self‐avoiding walks on the integral points in Euclidean d space and γ_{ n } the number of n‐stepped self‐avoiding polygons. It is shown that and tend to zero as n → ∞ where Asymptotic estimates for these differences are given. β is also characterized as the unique positive root of where the λ_{ k } are the number of certain k‐stepped self‐avoiding walks.

New Method for Solving Eigenvalue Problems
View Description Hide DescriptionA new method of solving eigenvalue problems is proposed. The secular equation is represented in the form of a dispersion relation. Three methods of solving the dispersion relation are given. Several examples including both nondegenerate cases and degenerate cases are presented. All these examples demonstrate that the present treatment is better than other conventional perturbation treatments. The transformation coefficients between the unperturbed wavefunctions and the perturbed ones are given in terms of the eigenvalues thus obtained. The wavefunctions of the perturbed states form a complete orthonormal set if those of the unperturbed one do.

Solution of the Cauchy Problem for Laplace's Equation in Axially Symmetric Systems
View Description Hide DescriptionThe determination of the electrodes for an electron gun reduces to solving the Cauchy problem for Laplace's equation. Using Green's Theorem, an analytic solution for the case of axial symmetry is obtained in terms of a line integral involving the product of the analytically continued values of the voltage and normal electric fields at the beam edge by an appropriate weighting function. Since an explicit solution is given in terms of line integrals, numerical results should be obtained far more accurately and quickly than those obtained by integrating Laplace's equation directly by finite‐difference methods.

Errata: An Approach to Gravitational Radiation by a Method of Spin Coefficients
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Errata: Cluster Sums for the Ising Model
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