Volume 17, Issue 1, January 1976
Index of content:

Unification of gauge and gravitation theories
View Description Hide DescriptionA new approach to local Gauge theories is presented. The theory is developed in a curved space–time, and therefore gravitation is not neglected. Besides the Yang–Mills A ^{α} _{μ} vector bosons associated with a symmetry group, scalar bosons g _{αβ} appear just as naturally. The Lagrangian describing the interaction of these fields is the Ricci scalar for an extended Riemannian geometry.

Generalized Källén–Pauli equation
View Description Hide DescriptionThe Källén–Pauli (KP) equation in the 2V particle model was solved. In addition to satisfying all the requirements of the integral equation, this solution is found to be reducing to that of the ordinary KP equation. Unlike earlier authors, we found that there is a resonance in the Vϑ sector also. The solution given here shows that the deductive method used in the case of the ordinary KP equation does not hold good in the present case. The uniqueness of the solution is yet to be proved.

On analytic nonlocal potentials. IV. A closed contour dispersion relation
View Description Hide DescriptionWe give a class of analytic nonlocal potentials which give rise to a closed contour s‐wave dispersion relation for sufficiently small coupling constant.

On conserved quantities in kinematic dynamo theory
View Description Hide DescriptionUsing a Lagrangian approach to the magnetic inductionequation in an infinite medium, we demonstrate that there exist seven conserved quantities which, by analogy with classical mechanics, we label as ’’energy,’’ ’’momentum,’’ and ’’angular momentum.’’ For prescribed fluid motions we spell out the detailed conservation equations. For a fluid motion which is turbulent we also give the average conserved quantities. In a pragmatic sense it is expected that these conservation laws will be of use in attempts to obtain numerically accurate solutions to the turbulent kinematic dynamo equations. Since the magnetic inductionequation is not self‐adjoint, numerical attempts to date have to impose some extraneous ad hoc ’’criteria of goodness’’ at any given level of numerical truncation. The conserved quantities given here provide an internal check of the accuracy of any numerical calculation without the necessity for arbitrarily imposed external criteria of accuracy. As such they should be a powerful tool in rapidly increasing the accuracy of numerical solutions to the kinematic dynamo equations. We also point out that the conserved quantities can be used to indicate the possibility of kinematic dynamo activity a h e a d of any detailed calculations.

Properties of massless relativistic fields under the conformal group
View Description Hide DescriptionUsing the ladder representations of SU(2,2), we derive explicit transformation laws for massless free fields with arbitrary helicities under global conformal transformations.

The generalized Wiener–Feynman path integrals
View Description Hide DescriptionThe generalized Wiener–Feynman path integrals are defined by the primitive mappings of the canonical Gaussian measure on a Hilbert space of real square integrable functions. The expressions of the covariance of the pro (pseudo) measures are found to be form covariant. The measures known in the literature by names as Wiener–Feynman and Uhlenbeck–Ornstein appear as special cases of no particular remark in our general definition. The connection of the primitive mapping with the general class of linear Cameron–Martin transformations is established.

A theorem on stress–energy tensors
View Description Hide DescriptionThe equality of the symmetrized Noether stress–energy tensor (Belinfante’s tensor) and the canonical stress–energy tensor (functional derivative of the Lagrangian density with respect to the metric) is established by methods based on the formalism of tetrads and Ricci rotation coefficients. The result holds for any Lagrangian which contains no derivatives of the fields higher than first order.

Killing vectors in empty space algebraically special metrics. II
View Description Hide DescriptionEmpty space algebraically special metrics possessing an expanding degenerate principal null vector and Killing vectors are investigated. Attention is centered on that class of Killing vector (called nonpreferred) which is necessarily spacelike in the asymptotic region. A detailed analysis of the relationship between the Petrov–Penrose classification and these Killing vectors is carried out.

Interpretation of Kato’s invariance principle in scattering theory
View Description Hide DescriptionA simple proof is given that Kato’s invariance principle holds for a class of generalized piecewise linear (GPL) functions, under the sole assumption of existence of the Mo/ller wave operators. The invariance principle for GPL functions is viewed as an expression of our freedom to change the scale of time and shift the zero point of energy. It is remarked that whenever scattering theory can be done, it can be done with bounded Hamiltonians. The motion of a particle is studied when its Hamiltonian is replaced by a GPL function of this Hamiltonian.

Weyl conform tensor of δ=2 Tomimatsu–Sato spinning mass gravitational field
View Description Hide DescriptionExtremely simple expressions are presented for the hitherto uncalculated invariants associated with the Weyl conform tensor of the δ=2 Tomimatsu–Sato solution of Einstein’s field equations.

Black holes in a magnetic universe
View Description Hide DescriptionWe present a general procedure for transforming asymptotically flat axially symmetric solutions of the Einstein–Maxwell equations into solutions resembling Melvin’s magnetic universe. Specific applications yield metrics associated with black holes in a magnetic universe. It is hoped that these solutions will be of interest to astrophysicists studying gravitational collapse in the presence of strong magnetic fields.

The spectrum of the Liouville–von Neumann operator
View Description Hide DescriptionWe relate the pure point spectrum, the singularly continuous, and the absolutely continuous part of the spectrum of the Liouville–von Neumann operator [H,⋅] to the respective parts of the spectrum of the Hamiltonian operator H. As a consequence of this result we obtain a theorem about the weak* limit of the time evolution W (t) of a normal state W for t → ∞.

On superpropagator for nonpolynomial Lagrangians with internal symmetry
View Description Hide DescriptionUsing the exponential shift lemma, a method to evaluate the superpropagator for scalar functions of a multiplet of fields is developed. As an application we obtain the generating function 〈T (Tr exp[λΦ (x)], Tr exp[λ′Φ (y)]) 〉_{0} for vacuum expectation values 〈T (TrΦ^{ N }(x)TrΦ^{ N }(y)) 〉_{0}, when Φ is a 3×3 matrix, which is sufficient to get the results of Ashmore and Delbourgo on the matrix superpropagator for the chiral symmetry case.

Some examples of symmetrical perturbation problems in several complex variables
View Description Hide DescriptionWe consider two examples of symmetrical perturbation problems in several complex variables and show that special singularities appear. We then discuss the spectral properties of the perturbed operators in connection with the involved symmetries.

Determination of nearest neighbor degeneracy on a one‐dimensional lattice
View Description Hide DescriptionA general method is proposed for the determination of the degeneracy of nearest neighbor pairs on a one‐dimensional lattice. Explicit results are given for the special cases involving two and three kinds of molecules.

A solution of the Korteweg–de Vries equation in a half‐space bounded by a wall
View Description Hide DescriptionWe give a solution of the Korteweg–de Vries equation in the half‐space 0<r<∞ with the boundary conditionV (0) =0. The boundary condition may be interpreted as the requirement that the plane which bounds the half‐space be a rigid wall. Aside from possible physical interest, this solution, which is obtained from one of the potentials for the radial Schrödinger equation which do not scatter, appears to indicate that the radial Schrödinger equation and the corresponding Gel’fand–Levitan equation play a role in the case of the half‐space bounded by a wall similar to that of the one‐dimensional Schrödinger equation (−∞<x<∞) and its corresponding Gel’fand–Levitan equation in the more usual full space treatment of the KdV equation. A possible interpretation of the solution presented in this paper is that it corresponds to the reflection of a wave by a wall, in which the incident wave is singular and the reflected wave is nonsingular but highly dispersive.

Multigroup neutron transport. I. Full range
View Description Hide DescriptionA functional analytic approach to the N‐group, isotropic scattering, particle transport problem is presented. A full‐range eigenfunction expansion is found in a particularly compact way, and the stage is set for the determination of the half‐range expansion, which is discussed in a companion paper. The method is an extension of the work of Larsen and Habetler for the one‐group case.

Multigroup neutron transport. II. Half range
View Description Hide DescriptionThis paper accompanies a preceding one in which a functional analytic method was used to obtain the full‐range expansion in multigoup neutron transport. In the present paper the analysis is extended to obtain the half‐range expansion. The method used is an extension of the work of Larsen and Habetler for the one‐group case. The results are given in terms of certain matrices which are solutions of coupled integral equations and which factor the dispersion matrix.

An upper bound on the energy gap in the (λφ^{4}+σφ^{2})_{2} model
View Description Hide DescriptionLet ΔE (λ,σ,m _{0}) be the energy gap in the infinite volume quantum field theory with bare mass m _{0} and interaction λφ^{4}+σφ^{2} in two space–time dimensions, obtained with either full‐Dirichlet or half‐Dirichlet boundary conditions. Then ΔE (λ,σ,m _{0})/m _{0}<exp[σ/(m ^{2} _{0} +3λ/Π)] for all λ≳0 and σ real. In particular ΔE (λ,0,m _{0}) <m _{0} for all λ≳0 and ΔE (λ,σ,m _{0})/m _{0}→0 as σ→−∞. For half‐Dirichlet boundary conditions one also has ΔE (λ,σ,m _{0})/m _{0}< (1+2σ/m ^{2} _{0})^{1/2} for σ⩽0,λ≳0. In each pure phase of a (λφ^{4})_{2}theory, let m _{phys} be the energy gap and 〈φ〉 the expectation of the field. Then m _{phys}/m _{0}<exp[2Π〈φ〉^{2}/(1+Πm ^{2} _{0}/3λ)].

Spectrum generating group of the symmetric top molecule
View Description Hide DescriptionThe spectrum generating group for the symmetric top is found and applied to derive spectra, energy values and selection rules. All the well‐known results are reproduced and some generalizations are obtained as a consequence of the group property, except for the Coriolis splitting for which an additional dependence upon the factor k/j is derived.