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Volume 9, Issue 2, February 1968

Dirac Algebra and the Six‐Dimensional Lorentz Group
View Description Hide DescriptionWe establish the relationship between the Diracalgebra and the six‐dimensional Lorentz group. By considering in an appropriate way the fifteen excentrical basis elements of the Diracalgebra as the components of an antisymmetrical tensor in six dimensions, the commutation as well as the anti‐commutation rules of the algebra can be written in a six‐covariant way. The group of automorphisms of the real Diracalgebra turns out to be isomorphic to the proper six‐dimensional Lorentz group. However, this result is very sensitive to the specific choice of Lorentz metric.

Representations and Classes in Groups of Finite Order
View Description Hide DescriptionThere are two main results in this paper. First, it is shown that we can develop a theory of classes in close analogy to the usual theory of representations. We can introduce concepts, such as reducible and irreducible classes, sum and product of classes, reduction of a class when going from a group to a subgroup, etc. Second, it is shown that it is possible to associate a ``magic square'' to each group. It is related to the numbers of pairs of commuting elements between classes and it can be used immediately to find the structure of the ``tensor operators'' of the group.

Structure of the Poincaré Generators
View Description Hide DescriptionThe Poincaré generators for an open system augmented by the interaction parts of the full Poincaré generators are shown to satisfy a closed set of coupled differential equations having a form which is independent of the nature of the interaction parts. The differential equations are formulated in the hyperplane formalism, the differentiation being with respect to the hyperplane parameters. The general solutions of the equations are studied, yielding relations among the augmented generators that must be preserved in the limit of zero interaction, i.e., for a closed system. Introducing a hyperplane‐dependent Hamiltonian density in a manner not implying local field theory, the obtained relations are shown to yield expressions for all the generators of a closed system in terms of the Hamiltonian density and its derivatives alone.

Note on the WKB Method
View Description Hide DescriptionIn the phase integral WKB method, a solutionu _{1}(z) of a second‐order linear differential equation is represented in terms of its logarithmic derivative iy(z) which satisfies a simple nonlinear first‐order equation. This representation does not in general lead directly to an independent second solution of the original equation. However, if y(z) is expressed in the form q(z) + iq′(z)/2q(z), where q(z) satisfies a nonlinear second‐order equation, then q(z) can be used to determine a second solution to the original equation. These two solutions remain linearly independent throughout their domain of definition. It is shown that q(z) is given by the sum of alternate terms in the well‐known asymptotic expansion of y(z). Any two linearly independent solutions of the original equation, normalized so that the Wronskian is −2i, give q(z) in the form (u _{1} u _{2})^{−1}.

Unitary Representations of the Affine Group
View Description Hide DescriptionThe unitary representations of the affine group, or the group of linear transformations without reflections on the real line, have been found previously by Gel'fand and Naimark. The present paper gives an alternate proof, and presents several properties of the representations which will be used in a later application of this group to continuous representations of Hilbert space. The development follows closely that used by von Neumann to prove the uniqueness of the Schrödinger operators.

Clebsch‐Gordan Coefficients for Space Groups
View Description Hide DescriptionIt is shown that to find Clebsch‐Gordan coefficients of space groups (both single and double), the representations of the groups of k alone are required. This is another example demonstrating the well‐accepted fact that in applications of space groups it is sufficient to know the representations of the groups of k. Final formulas are derived that enable the calculation of the Clebsch‐Gordan coefficients from the representations of the groups of k. As an example the spin‐orbit coupling in solids is considered.

Remarks on the Nature of Relativistic Particle Orbits
View Description Hide DescriptionInstantaneous action‐at‐a‐distance relativistic particle dynamics, embraced in Newtonian‐like equations of motion χ̈_{ i } = F_{i} (χ_{ i }, χ̇_{ i }) with suitable F's, is examined in once‐integrated or ``kinematical'' form χ̇_{ i } = f_{i} (χ_{ i }, V_{i} ) with V_{i} a set of first integrals transforming as velocities. The Lorentz covariance requirements on f_{i} are worked out and illustrative examples are given, including a family of many‐valued ones. A general meaning for integrals of χ̈_{ i } = F_{i} being in involution is adduced, and general counterparts to some well‐known theorems in Hamiltonian dynamics are obtained accordingly. A novel elementary proof of the zero‐interaction theorem is appended.

Particle Motion and Interaction in Nonlinear Field Theories
View Description Hide DescriptionA variational method is given for determining the motion and interaction of particles associated with fields governed by nonlinear differential equations. For field equations derived from Lagrangian densities of the type g ^{ικ}(∂θ/∂x ^{ι}) (∂θ/∂x ^{κ}) + f(θ), one obtains an attractive inverse‐square law of force between like particles, provided f(θ) vanishes more rapidly than (constant) θ^{4} for θ → 0.

Infinite‐Spin Ising Model in One Dimension
View Description Hide DescriptionThe partition function z, the pair correlation function ρ, and the zero‐field susceptibility χ for the one‐dimensional Ising model with infinite spin, are expressed in terms of the eigenvalues and eigenfunctions of an integral equation. The eigenfunctions of the integral equation are shown to be the oblate spheroidal wavefunctions, and, from known asymptotic expansions, high‐ and low‐temperature expansions are given for z, ρ, and χ. It is shown that the low‐temperature behavior of z, ρ, and χ differs qualitatively from the corresponding behavior for all finite spin.

2V Sector of the Lee Model
View Description Hide DescriptionThe Lehmann‐Symanzik‐Zimmermann (LSZ) formalism is further used to analyze the 2V sector of the Lee model with boson sources at zero separation. Unlike previous LSZ investigations of the model, it is found that the solutions to two singular integral equations solve the entire sector. Two scattering amplitudes, a production amplitude, and an equation for the determination of the 2V potential energy are obtained. Similarities and differences between this sector and the V + θ subspace are pointed out.

High‐Energy Scattering for Yukawa Potentials
View Description Hide DescriptionA reexamination is made of nonrelativistic scattering by a superposition of Yukawa potentials in the frame of a high‐energy perturbation method developed previously by one of the authors. It is found that for Yukawa potentials expandable in ascending powers of r, the S matrix may be written in a remarkably simple form. A high‐energy asymptotic expansion is obtained for the phase shift. The residue function at the Regge poles is found to have the same general form as for a Coulomb potential and is formally independent of any high‐energy approximation.

Asymptotic Theory of Cerenkov Radiation in Inhomogeneous Media
View Description Hide DescriptionThe problem of Cerenkov radiation in infinite inhomogeneous media is considered. The mathematical description of this phenomenon is given by the integro‐differential system of equations for the electromagnetic field in a dispersive medium. The leading term of the asymptotic expansion of the electromagnetic field is obtained by applying an expansion procedure called the ``ray method.'' In this method all the functions that appear in the expansion satisfy ordinary differential equations along certain space‐time curves called rays. The source which gives rise to the radiation is taken to be quite general. In fact, it is shown that any multipole moving along an arbitrary trajectory is a special case of the general source considered. From the expansion of the fields an expression for the total energy of the radiation is determined. Then, as an example, the case of plane‐stratified media is treated in detail.

Einstein Tensor and Spherical Symmetry
View Description Hide DescriptionThe classification of symmetric second‐rank tensors in Minkowski space and its application to the Einstein tensor is reviewed. It is shown that, for spherically symmetric metrics, the Einstein tensor always has a spacelike double eigenvector; and the possible types of Einstein tensor that this degeneracy allows are discussed. A complete classification of all spherically symmetric metrics with two double eigenvalues is given. A study of the timelike eigencongruence, in the case when one timelike and two spacelike eigenvectors exist, is carried out. Canonical forms for the metric, the Einstein tensor, and the Weyl tensor (which is always of type D) are given for each of the various possible types.

Intrinsic Vector and Tensor Techniques in Minkowski Space with Applications to Special Relativity
View Description Hide DescriptionThis paper describes an abstract formalism for tensoranalysis in Minkowski space which entails considerable notational simplicity and calculational advantages, as evidenced when compared with the usual component techniques. The need to express tensorequations in component form is eliminated, and manipulations become formally the same as those in Euclidean spece. The method is based on an extension to Minkowski space of the intrinsic concepts of vectors, differential operators, and polyadics in three‐dimensional Euclidean spece. Several examples from special relativity have been selected to illustrate the advantages of the formalism. In the first example, expressions in dyadic form for the Euler‐Lagrange equations and canonical energy‐momentum tensor are obtained and specialized to the electromagnetic field. From the electromagnetic‐field dyadic, invariants and other useful relations are derived easily and economically. The dyadic form of the field equations is also shown to be particularly amenable for a derivation of the Dirac‐like form of Maxwell's equations with the base elements of the Pauli algebra emerging in a most natural way. Further illustration of the practical utility of the method is given by considering several properties of the restricted homogeneous Lorentz transformations. Various dyadic expressions for these are obtained, and a detailed derivation of their eigenvalues and eigenvectors is given. By combining some of the results from the discussions of Lorentz transformations and the Dirac‐like form of Maxwell's equations, it is shown how an isomorphism between the three‐dimensional complex orthogonal group and the Lorentz group can be established in a simpler and different manner from other approaches appearing in the literature.

Semiclassical and Quantum Descriptions
View Description Hide DescriptionIn the semiclassical descriptions, it is usual to describe a quantum‐mechanical system in a classical language with (i) a correspondence between classical functions and operators of quantum mechanics and (ii) with a real, but not necessarily positive, probability density function in phase space corresponding to a particular quantum‐mechanical state. The general forms of such semiclassical descriptions is discussed. The conditions for the two descriptions to be equivalent are also examined.

Analytic Functionals in Quantum Field Theory and the Regularization of Ultraviolet Divergences
View Description Hide DescriptionThe commutator and propagator distributions associated with free‐particle fields are represented as boundary values of analytic kernels in complex space‐time. A distribution product of such quantities is then defined as a boundary value of the product of the analytic kernels of its various factors, with the proviso that the product kernel be understood in the sense of Hadamard's ``finite part'' in those cases when it would otherwise be nonintegrable. For the case of quantum electrodynamics, the momentum‐space representation of the products of causal propagators are derived. It is demonstrated by explicit calculation that the second‐order scattering matrix elements thus obtained are identical to those usually obtained by means of Pauli‐Villars regularization and renormalization.

Coherent Soft‐Photon States and Infrared Divergences. I. Classical Currents
View Description Hide DescriptionAs a first step toward a treatment of soft‐photon processes which is free of infrared divergences and avoids the necessity of introducing a fictitious photon mass, the specification of asymptotic photon states belonging to non‐Fock representations is discussed. As in the work of Chung, a basis consisting of generalized coherent states is used, but in contrast to his work, these states are rigorously defined in terms of von Neumann's infinite tensor product. It is shown that the states must be given an additional label which serves to distinguish various ``weakly equivalent'' vectors, and which corresponds formally to an infinite phase factor. A nonseparable Hilbert space is defined (as a subspace of the infinite tensor‐product space) which may be regarded as the space of all possible asymptotic photon states. The interaction of the electromagnetic field with a prescribed classical current distribution is discussed, and it is shown that a unitary S operator, all of whose matrix elements are finite, may be defined on .

Evolution of the Probability Distributions and of the Correlation Functions in a Bogoliubov Plasma
View Description Hide DescriptionRecently we developed a matrix formulation of nonequilibrium statistical mechanics which we applied to dilute gases and to small momentum transfer interactions (φ_{0}/kT ≪ 1). We now show that the matrix formulation of the nonequilibrium equations can be extended to the Bogoliubov gas The asymptotic behavior is calculated for times long compared with the plasma frequency for: (i) the probability distribution functions, and (ii) the Mayer correlation functions. The collision integrals that determine the higher‐order kinetic equations for the Bogoliubov gas are thereby constructed. The calculations performed directly with the nonlinearly coupled Mayer functions are shown to be equivalent to those performed with the linearly coupled probability distribution functions. In lowest order our theory coincides with a result obtained previously by Bogoliubov.

Another Interpretation of the Optical Scalars
View Description Hide DescriptionIn this paper an interpretation of the optical scalars θ and σ (the expansion and shear of an irrotational null congruence) is given in terms of the principal curvatures of a two‐dimensional subspace of the instantaneous rest frame of an arbitrary observer. The two space is defined as the intersection of the observer's rest frame and the particular null hypersurface the observer is intersecting.