Volume 19, Issue 3, March 1978
Index of content:

On the stationary Einstein–Maxwell equations
View Description Hide DescriptionThe ansatz Φ=Φ (E, ε̄) for a solution of the stationary Einstein–Maxwell equations is analyzed. The possible forms of this function are listed and it is shown that one obtains from every solution of the vacuum Ernst equations an at most two‐parameter solution of the Einstein–Maxwell equations.

Evaluation of simple Feynman graphs
View Description Hide DescriptionA number of one‐loop graphs with arbitrary external momenta and internal masses contributing to the perturbation expansion of a Euclidean φ^{4}theory are evaluated exactly in three and two dimensions. The final expressions are simple closed forms involving elementary functions only. A method for handling the multidimensional angular integrations that arise in calculations of massless QED or φ^{4} in four dimensions is also discussed.

The geometry of the gravitational field at spacelike infinity
View Description Hide DescriptionThe asymptotic structure of gravitational fields at large spacelike separation from sources is studied. Limits of spacetime fields are discussed in terms of a three‐dimensional boundary manifold representing spacelike infinity. The boundary is endowed with the metric of a timelike unit hyperboloid. With sufficiently stringent conditions on the asymptotic spacetime geometry, the total energy–momentum and angular momentum emerge as integrals over any cross section of the hyperboloid at infinity. It is possible to identify physically relevant weaker conditions under which the energy–momentum, but not the angular momentum, is well defined. Under still weaker conditions, the energy–momentum also loses its meaning even though the spacetime admits a Minkowskian asymptote.

On the Palatini method of variation
View Description Hide DescriptionThe Palatini method of variation is compared with the Hilbert method for symmetric metrics and affine connections. It is found that the two methods are in general inequivalent. The Hilbert method is recommended as being more general.

Global thermodynamical stability and correlation inequalities
View Description Hide DescriptionUsing spatially homogeneous dissipative perturbations, we derive a correlation inequality for states satisfying the variational principle for infinitely extended quantum lattice systems.

Tachyonic scalar waves in the Schwarzschild space–time
View Description Hide DescriptionThe scalar wave equation of a tachyon is investigated in the background of Schwarzschild geometry. The scalar field is split up into partial waves of all integral momentum states and the space development of each partial wave is studied as it approaches the singularity. The problem is mainly considered in the light of the assumption that the tachyon mass‐parameter is comparable to the mass of a atomic particle while the black hole mass is comparable to that of an average star. The reflection and transmission properties of these partial waves at the effective potential barrier, arising partly from their angular momentum and partly from the curvature of space–time are discussed. It is found that in the radial case (l=0) the criteria for the bounce are different from the purely classical behavior of spacelike geodetic trajectories.

Slavnov–’t Hooft identities in Mandelstam’s formalism
View Description Hide DescriptionIn Mandelstam’s gauge‐independent quantization formalism, Slavnov identities are shown to originate from the fact that auxiliary Green’s functions in various gauges (Feynman gauge, Landau gauge, etc.) satisfy the same fundamental equation. Furthermore, we have the practically important result that the infinite number of Slavnov identities are obtained in concise and concrete form. With the help of this form, we are able to easily derive various ’t Hooft identities, which assure us of the gauge independence and the unitarity of the S matrix.

On independent sets of basis functions for irreducible representations of finite groups
View Description Hide DescriptionTwo theorems on bases of irreducible representations of finite groups are compared. It is stressed that their validity depends upon the functional sets for which they are formulated. The first theorem, which states that there are as many linearly independent (modulo the identity representation) sets of basis functions as is the dimension of the representation, is shown to hold only if the considered functional set constitutes a field. Otherwise, more such sets are necessary as shows the second theorem (extended Noether’s theorem), which is limited to polynomialalgebra. The second theorem seems to be more apt for explicit construction of functional bases.

Scattering theory for Stark Hamiltonians involving long‐range potentials
View Description Hide DescriptionA time‐dependent and stationary scattering theory is developed for operators of the form H=H _{0}+V, H _{0}=−Δ+E⋅x with V a long‐range potential having the asymptotic form V (x) =O (‖x‖^{−l }) as ‖x‖→∞, 0<l⩽1/2.

Matrix element expansion of a spin wavefunction
View Description Hide DescriptionAn expansion of the wavefunction for a free, massive particle with spin is obtained in terms of the matrix elements for the unitary, irreducible representations of SL(2,c) by a method based on the theory of induced representations. It is further shown that this expansion is equivalent to the Shapiro integral transformation for the wavefunction.

The interaction of the gravitational and electromagnetic fields
View Description Hide DescriptionVarious sets of conditions are presented which a reasonable theory might be expected to satisfy, in attempting to explain the interaction of the gravitational and electromagnetic fields. It is shown that each of these sets leads inevitably to the Einstein—Maxwell field equations. Attention is also drawn to the fact that these equations must be modified if it is furthermore demanded that Maxwell’sequation in flat space—time be an exact solution of them.

Multidimensional wave radiation from a source with Gaussian time variation and Gaussian‐approximated distribution about a spherical sheet
View Description Hide DescriptionThis paper concerns the wave field of a source with the title‐indicated space–time function which, additionally, possesses an arbitrary directional variation. The multivariate solution obtained comprises an estimated error plus peak‐induced spherical harmonics that are hyperconically confined, i.e., bounded by diverging and converging spherical fronts. Such fronts are not necessarily singular. Compliance with the radiation principle ensues, through contour integration, from Cauchy initial conditions. For an odd number of spatial dimensions, an inner zone created after a focusing phenomenon exhibits an analogy with a Petrowsky’s lacuna. Naturally, the wave field varies with direction, but only because its source does so. Spherically as well as axially symmetric cases constitute major corollaries. Asymptotic developments, evolving ultimately into steady limits, are also deducible. An indirect application is illustrated for magnetoacoustic flow parallel to a magnetic field; on induction by a cylindrical Gaussian‐approximated current distribution, weak effects appear everywhere during the steady state and are superposed upon strong stationary waveeffects bounded by cone sheets which project either (i) downstream for a supersonic–super‐Alfvénic flow, or (ii) upstream for a restricted subsonic–sub‐Alfvénic flow. Finally, the main results are directly applied to elastic wave propagation from a two‐component Gaussian body force concentrated about a spherical base; a spherically symmetric radial component generates a strong irrotational wave field normally involving an instantaneous point singularity; an axisymmetric azimuthal component generates a strong solenoidal wave field.

Invariant properties of n‐point functions and n‐point functionals connected with the translational invariance of the formal measure
View Description Hide DescriptionWe discuss invariant properties of the generating functionals resulting from the translationally invariant formal measure used to define these functionals. The functionals considered depend on functions defined on five‐dimensional space, and we relate to them n‐point functionals and n‐point functions. We derive equations for the above quantities, and we consider the connection with four‐dimensional n‐point quantities.

A geometric theory of charge and mass
View Description Hide DescriptionA geometric model of a charge is constructed by defining several geometries on the same spacetimemanifold. A Riemannian geometry describes the vacuum. On the same spacetime, two Weyl geometries are constructed for the charge description. The geometries are constrained by a variational principle. Energy conservation requires the equality of active and passive mass. Chargeless particles have essentially no mass. The treatment of radiation relies on the approximate nature of the wave equation. Variable mass terms in the wave equation cause the 2S–2P levels in hydrogen to separate by 30 000 Mhz. This unobserved transition together with the lack of spin sets a limit to the correspondence of the model to real electrons.

Large time behavior of the superfluorescent decay
View Description Hide DescriptionThe large time behavior of the superfluorescent decay is investigated by the inverse scattering technique.

Homogeneous and isotropic world models in the Yang–Mills dynamics of gravity. The structure of the adiabats
View Description Hide DescriptionThe time evolution of homogeneous and isotropic matter distributions is analyzed for the restricted Yang–Mills curvature dynamics of gravity. This theory of gravity is a tidal dynamics for which relativistic matter in detailed balancing cannot produce tidal forces. It defines a dynamical system on the curvature plane spanned by the two components of the Riemann curvature of Robertson–Walker space–times; the essential features of the cosmological solutions are presented by means of their phase portraits in the curvature plane. In the asymptotic limit (S→∞) the phase portrait, which in general depends on the equation of state and on the change of the entropy per particle, is structurally stable under the transition from Einstein’s dynamics to the Yang–Mills dynamics for any realistic equation of state. The phase portraits are explicitly constructed for the equation of statep=nρ, 0≦n≦1, and constant entropy per particle. A criterion for the existence of regular trajectories is given for the full Yang–Mills dynamics including entropy production. Finally, we discuss the relations between the observational parameters.

Integral representations of particular integrals of a class of inhomogeneous linear ordinary differential equations
View Description Hide DescriptionA method of obtaining integral representations of particular integrals of a class of inhomogeneous second‐order linear ordinary differential equations is presented. The integrands of the representations are exp[F (z;t)], where F (z;t) =z ^{2} a (t)+z b (t)+c (t), z is the independent variable of the differential equation, and a, b, and c are initially unspecified functions of the variable of integration, t. The lower limit of the contours of integration is zero. The upper limits of integration and the contours along which the integral is taken are initially unspecified. In the general class of inhomogeneous differential equations considered, the coefficients of the dependent variable and its derivatives are polynomials in z with complex constants and the homogeneous term is exp(k _{2} z ^{2}+k _{1} z+k _{0}), where the k _{ n } are complex constants. One imposes the conditions that the application of the homogeneous operator to the assumed form of integral representation give the integral of −∂F/∂t, that exp[F (z;0)] be equal to the inhomogeneous term of the differential equation, and that the limit of expF as t approaches the upper limit along the contour of integration be zero. By equating coefficients of different powers of z separately to zero, one obtains a set of coupled equations for a, b, and c. The basic class of inhomogeneous differential equations to which the method is applicable is determined by requiring that a, b, and c be algebraic or elementary transcendental functions of t. The class of equations to which the method is applicable is extended to include inhomogeneous terms of the form z ^{ n }exp(k _{2} z ^{2}+k _{1} z+k _{0}), where n is a positive integer, by repeated differentiation of integral representations of members of the basic class with respect to k _{1}, treated as a parameter. More general inhomogeneous terms may be treated by superposition and, in appropriate cases, by approximation in terms of sets of orthogonal functions.

Quantization of spinor fields
View Description Hide DescriptionInfluenced by Klauder’s investigations on the same subject, we study the question of correspondence principle for Dirac fields, looking for its formulation without use of Grassman algebras. We prove that with each Fermi operator (the series with respect to asymptotic free fields): Ω (ψ,ψ̄): one can associate the functional Ω^{ C }(ψ^{ C }, ψ̄^{ C }) with respect to classical spinor fields. Here the projector 1_{ F } and the Hilbert (Fock) space F_{ F }=1_{ F }F_{ B } are given such that the identity 1_{ F }: Ω^{ C }(ψ^{ B }, ψ̄^{ B }): 1_{ F }F_{ F } = :Ω (ψ, ψ̄):F_{ F } defines the mediating boson level, where coherent state expectation values of operator expressions are in order: 〈:Ω^{ C }(ψ^{ B }, ψ̄^{ B }):〉=Ω^{ C }(ψ^{ C }, ψ̄^{ C }). For proofs we employ functional differentiation (resp. integration) methods, especially in connection with the use of functional representations of the CCR and CAR algebras.

The local von Neumann algebras for the massless scalar free field and the free electromagnetic field
View Description Hide DescriptionThe properties of the von Neumann algebras of local observables for the free scalar field of zero mass are studied. The local algebras possess a lattice structure, and the duality condition is satisfied. The problem of duality for the free electromagnetic field is discussed.

Type N gravitational field with twist. II
View Description Hide DescriptionA full derivation is given of that one parameter family of type N twisting gravitational fields which was previously reported by the author. Explicit forms of this solution are obtained, and the coordinate ranges are specified for all possible cases. The general problem of the search for other type N twisting gravitational fields is also discussed; the differential equations required for that search are derived, and the extension of these equations to type (3,1) is given without derivation.