Volume 8, Issue 4, April 1967
Index of content:

Energy Tensor of the Null Electromagnetic Field
View Description Hide DescriptionThe differential conditions are obtained which must be satisfied by the energy tensor of a null, source‐free electromagnetic field. At the same time a method is indicated of determining the electromagnetic field when the energy tensor satisfying the necessary conditions is given.

Relation of the O(2, 1) Partial‐Wave Expansion to the Regge Representation
View Description Hide DescriptionThe general two‐particle scattering amplitude is expanded in terms of partial waves corresponding to the crossed channel little group, O(2, 1). Under the assumption of square integrability over the group manifold, the invariance of the S matrix under the complex Lorentz group, which follows from the Bargmann‐Hall‐Wightmann theorem, enables this expansion to be identified with the Regge representation in the crossed channel, whenever no dynamical singularities occur to the right of Re j = −½. The identification requires the assumption of the fixed tdispersion relation necessary for the definition of the Regge representation.

Tensor Operators and Mass Formula in the Minimal Extension of U _{3} by Charge Conjugation
View Description Hide DescriptionThe transformation property of the mass (or mass square) M is specified in the extended group. The most general mass formula, , is derived where a(α), b_{i} (α) are constants depending on a charge‐even set α, specified in the text, and Q _{2} is the second invariant operator of SU _{3}.

Evaluation of Phase‐Space Integrals
View Description Hide DescriptionThe relativistic phase‐space integral over the submanifold defined by total momentum zero and fixed total energy is reduced to a single contour integration. The number of particles involved and their masses are arbitrary. It is shown that the contour integration may be readily approximated by the saddle‐point technique and yields a result which is easily handled by a computer. In the nonrelativistic and extreme relativistic limits, this method leads to expressions for the phase space which may be obtained from the exact results for these cases by replacing Γ‐function factors by the Stirling approximation.

Canonical Definition of Wigner Coefficients in U_{n}
View Description Hide DescriptionTwo general results applicable to the problem of a canonical definition of the Wigner coefficient in U_{n} are demonstrated: (1) the existence of a canonical imbedding of U_{n} × U_{n} into U_{n} ^{2} and (2) a general factorization lemma for operators defined in the boson calculus. Using these results, a resolution of the multiplicity problem for U _{3} is demonstrated, in which all degenerate operators are shown to split completely upon projection into U _{2}.

Null Tetrad Approach to Motions in Empty Space‐Time
View Description Hide DescriptionThe integrability conditions of conformal motions are written in the null tetrad formalism of Newman and Penrose. The maximum order of the group of conformal motions admitted by nonflat empty space‐times of given Petrov type is shown to be at most one greater than the maximum order of the group of Killing motions. The symmetries of those empty space‐times which possess hypersurface orthogonal geodesic rays with nonvanishing divergence are determined. Among these space‐times is one of type III which admits a group of Killing motions of order three. This provides a counter example to a result of Petrov which states that the maximum order of the group of Killing motions for such space‐times is two.

Induced Representations of Strong Coupling Groups
View Description Hide DescriptionMackey's method of induced representations is applied to the strong coupling group G = K·T, where K is compact and T is Abelian, to obtain the general irreducible representations. The form of the meson‐isobar‐isobar couplings is obtained by reducing these, with the use of the Peter‐Weyl theorem, to irreducible representations of the compact subgroup K. The results are applied to the cases: pseudoscalar octet mesons,K = SU _{2} ⊗ SU _{3}, T = T _{24}, and SU _{6} 35‐plet mesons,K = SU _{6}, T = T _{35}. Explicit representations are obtained which are consistent with mass formulas and in the respective cases.

Symmetry Properties of the 3j Symbols for SU(3)
View Description Hide DescriptionFor an arbitrary compact group it is in general not possible to choose the 3j symbol (j _{1} j _{2} j _{3})_{ r, m 1 m 2 m 3 } such that its absolute value is invariant under every permutation of the j's and of the corresponding m's. Still, it is commonly assumed that for SU(3) such a choice is possible. In this paper it is shown that this assumption is indeed justified.

Bootstrap Prediction of Symmetry for a Soluble Static Model
View Description Hide DescriptionThe Huang‐Low bootstrap criterion of self‐consistency, in the form of Levinson's theorem, is imposed on the exact solutions of the two‐channel Low equation with an arbitrary crossing matrix. It is found that this condition, together with a number of dynamical conditions are sufficient to restrict the continuous crossing‐matrix parameter to discrete values corresponding to an SU _{2} symmetry.

Group Embedding of Space‐Time and Internal Symmetry Groups
View Description Hide DescriptionAn embedding of SU(2) and an internal symmetry group G into a larger group G̃ containing SU(2) and G as subgroups is constructed for all G possessing a generalized spin‐½ quark model. The starting point is a set of three mathematical conditions for the embedding group G̃ which are derived from physically plausible assumptions. By group theoretical techniques due to Dynkin and Malcev, it is shown that the embedding group is already uniquely determined by the proposed conditions, with only one set of groups G for which two solutions are obtained. The results for G̃ are given explicitly. Identifying SU(2) as covering of the rotation group the spin extension G̃ is enlarged to an embedding G̃_{h} of the homogeneous Lorentz groupL_{h} and G. It is shown that Ḡ_{h} can also be obtained without using the spin extension. The minimal translation group which can be attached to Ḡ_{h} is calculated. The results are also taken over to the Budini‐Fronsdal identification of SU(2).

Determination of Weight Factors in Linked‐Cluster Expansions for Lattice Systems
View Description Hide DescriptionIn a previous paper, it was shown that any function φ(G), defined for a general linear graph G and having the extensive property, can be expanded in terms of the lattice constants of connected subgraphs of G. In this paper, a graphical interpretation of the weight factors occurring in this expansion is given. The usefulness of the expansion in deriving series expansions for properties associated with crystal lattices is discussed with particular reference to percolation problems, dilute ferromagnets, and lattice gases. A result in the theory of linear graphs, recently proved by Rushbrooke in a paper concerned with dilute ferromagnets, is rederived.

CTP Invariance of the S‐Matrix in a Theory of Local Observables
View Description Hide DescriptionIn a theory of local observables as proposed by Haag and Araki, the assumptions which make possible a collision theory also guarantee the CTP invariance of the S‐matrix.

Lie Algebra Extensions of the Poincaré Algebra
View Description Hide DescriptionThe ``linear'' counterpart of the problem of analytic group extensions of the Poincaré group is presented in terms of the considerably simpler (but less general) analysis of Lie algebra extensions of the Poincaré algebraP. After easily proving with this technique that every C kernel (P, θ) has an extension and that every such extension is inessential, the problem of analyzing the central extensions of P is carried out with the well‐expected result that every such extension is trivial. But contrary to some claims, we exhibit an example which explicitly shows an essential noncentral extension of P.

Peculiarities of the Eight‐Dimensional Space
View Description Hide DescriptionA mathematical review of the peculiar properties of the space of eight dimensions is presented with the view of possible applications in the study of symmetries of elementary particles. This paper, written for physicists, is self‐contained in that it does not require any previous knowledge of the subject nor any advanced mathematics.

Topology in General Relativity
View Description Hide DescriptionA number of theorems and definitions which are useful in the global analysis of relativistic world models are presented. It is shown in particular that, under certain conditions, changes in the topology of spacelike sections can occur if and only if the model is acausal. Two new covering manifolds, embodying certain properties of the universal covering manifold, are defined, and their application to general relativity is discussed.

Double Representations of Space Groups
View Description Hide DescriptionIt is shown that, for calculating characters of double representations of space groups, the same methods can be used as for single representations. Two methods are reviewed: the ray representation method and the induction method. Examples are presented for both methods.

Expansion Theorem for the Linearized Fokker‐Planck Equation
View Description Hide DescriptionThe linearized Fokker‐Planck kinetic equation for each component of a homogeneous, nondegenerate, fully ionized plasma is separated by means of a spherical harmonic expansion into an infinite set of singular intergo‐differential equations. Each equation is shown to generate a continuous set of eigen‐functions, for which asymptotic high‐speed forms are found. By extending the theory of singular differential equations an expansion formula is developed, which is shown to be complete with respect to functions square integrable in velocity space.

Real Spinor Fields
View Description Hide DescriptionThe Dirac equation is expressed entirely in terms of geometrical quantities by providing a geometrical interpretation for the (−1)^{½} which appears explicitly in the Dirac equation. In the modification of the Dirac electron theory which ensues, the (−1)^{½} appears as the generator of rotations in the spacelike plane orthogonal to the plane containing the electron current and spin vectors. This amounts to a further ``relativistic'' constraint on the spinor theory and so may be expected to have physical consequences. It does not, however, conflict with well‐substantiated features of the Dirac theory.

Spin and Isospin
View Description Hide DescriptionSpinor fields can be classified according to the invariants of their derivatives. It is suggested that different invariants describe different interactions of elementary particles. Then the classification of spinor fields becomes a classification of elementary particles and their interactions. A geometric interpretation of isospin is suggested and is used in a model of nucleoninteractions. This theory entails an intimate connection between spin and isospin which appears to have some experimental support. Among other things, it connects the pseudoscalar and isospin properties of the pion and accounts for the polarization of the third axis in isospace by the electromagnetic interactions.

One‐Dimensional Three‐Body Problem
View Description Hide DescriptionThe three‐body problem in one dimension is examined to determine for what class of interactions the Schrödinger equation may be solved by separation of variables.