Volume 17, Issue 2, February 1976
Index of content:

Functional calculus for the symmetric multigroup transport operator
View Description Hide DescriptionA rigorous treatment of the symmetric multigroup transport equation is given by developing the functional calculus for the transport operator. Von Neumann spectral theory is applied to nonorthogonal cyclic subspaces, and the isometries onto C (N) are explicitly evaluated.

The Case eigenfunction expansion for a conservative medium
View Description Hide DescriptionBy using the resolvent integration technique introduced by Larsen and Habetler, the one‐speed, isotropic scattering,neutron transport equation is treated in the infinite and semi‐infinite media. It is seen that the results previously obtained by Case’s ’’singular eigenfunction’’ approach are in agreement with those obtained by resolvent integration.

Inverse problem for complex ’’r ^{a}‐analytic’’ potentials of finite range
View Description Hide DescriptionWe present a simple solution to the inverse problem for the Schrödinger equation at fixed energy for complex ’’r ^{a}‐analytic’’ potentials of finite range. This is done via an interpolation formula for the regular radial solutions as functions of complex angular momentum. The interpolation formula is derived by Frobenius techniques and Cauchy’s theorem. As an application we study an inverse problem for a spherically symmetric cold plasma perturbed by a small oscillatory electric potential of fixed (finite) frequency.

A new topology for curved space–time which incorporates the causal, differential, and conformal structures
View Description Hide DescriptionA new topology is proposed for strongly causal space–times. Unlike the standard manifoldtopology (which merely characterizes continuity properties), the new topology determines the causal, differential, and conformal structures of space–time. The topology is more appealing, physical, and manageable than the topology previously proposed by Zeeman for Minkowski space. It thus seems that many calculations involving the above structures may be made purely topological.

Kerr black holes in a magnetic universe
View Description Hide DescriptionWe present exact expressions for the electromagnetic fields associated with arbitrarily charged Kerr–Newman black holes in a magnetic universe. In the particular case of charge Q=2 B _{0} J, where B _{0} is the magnetic field parameter and J=m a is the angular momentum, exact expressions for the gravitational field are also presented, while for arbitrarily charged black holes metrical corrections of order B _{0} are evaluated.

The master analytic function and the Lorentz group. I. Reduction of the representations of O(3,1) in O(2,1) basis
View Description Hide DescriptionThe reduction of the principal and supplementary series of representations of SL(2,C) in the SU(1,1) basis is carried out by using a basis function which formally resembles the coupled state of two angular momenta. The spectrum of the SU(1,1) representations contained in SL(2,C) and the transformation coefficients are obtained by expanding the SU(2) in terms of the SU(1,1) bases with the help of the Sommerfeld–Watson transformation. The orthogonality conditions for the principal and supplementary series are discussed. For the principal series this follows easily from the standard Sturm–Liouville theory of the second order differential equations. For the supplementary series the orthogonality condition is obtained from the fourth order differential equation satisfied by the Fourier transform of the basis function.

The master analytic function and the Lorentz group. II. The Clebsch–Gordan problem for O(2,1)
View Description Hide DescriptionThe Clebsch–Gordan coefficients of the noncompact group O(2,1) representing Lorentz transformations in three‐dimensional space–time are calculated in the compact O(2) basis. Considerable simplification is achieved by introducing a variable x and replacing all algebraic equations by differential equations. The coupled state appears in the theory as a solution of an ordinary differential equation reducible to the hypergeometric equation by a simple substitution. The coefficients in the Taylor–Laurent expansion of this solution in powers of x are shown to be identical with the Clebsch–Gordan coefficients. The inverse expansion, obtained by the use of certain identities for the hypergeometric function and the Sommerfeld–Watson transformation, yields the normalization factor and the values of j appearing in the reduction.

New identities on the Riemann tensor
View Description Hide DescriptionA set of new identities which involve second covariant derivatives and quadratic forms of the Riemann tensor are proved. These new identities can be thought of as integrability conditions derived from the equations that define the Rieman tensor in terms of the affine connections.

The semi‐Euclidean approach in statistical mechanics. I. Basic expansion steps and estimates
View Description Hide DescriptionThe semi‐Euclidean formulation, developed in constructive quantum field theory to handle boson–fermion models, is adapted to the statistical mechanics setting.

The semi‐Euclidean approach in statistical mechanics. II. The cluster expansion, a special example
View Description Hide DescriptionA form of the Glimm–Jaffe–Spencer cluster expansion, adapted to the statistical mechanics setting, is shown to converge for certain two‐body potential interactions. The theory treated corresponds to negatively charged fermions and positively charged bosons interacting by a modified Coulomb interaction—the 1/r potential, cutoff at high and low momenta, becoming (1/r)(e ^{−αr }−e ^{−γr }).

Stability of nonlinear parametric‐decay interactions in finite homogeneous plasma
View Description Hide DescriptionWe prove the existence and uniqueness of a stable steady state of stimulated backscattering from a bounded homogeneous lossless interaction region, with boundary conditions corresponding to a steady incident pump wave‐mode Poynting flux and zero‐flux input to the decay wave modes. In steady state, once the excitation of the interaction region exceeds a certain critical value, the boundary value problem is characterized by a finite number of eigenvalues, and associated nontrivial eigenfunctions equilibria of the system, corresponding to mutually distinct states of anomalous reflection of the pump wave. A stability analysis of these equilibria with respect to small phase and amplitude perturbations reveals that (i) in the vicinity of the nonfundamental equilibria the phase perturbations exhibit singularities, preventing phase locking from occurring, and (ii) in the vicinity of the fundamental equilibrium both the phase and amplitude perturbations asymptotically vanish. A WKBJ phase‐integral stability condition is derived to show that growing normal modes of the amplitude‐perturbation boundary‐value problem cannot propagate in the potential formed by the field of the depleted, spatially inhomogeneous, pump of the fundamental equilibrium.

Gurtin‐type properties associated with wave propagation in a superfluid
View Description Hide DescriptionVariational and reciprocity principles of the Gurtin type are established for the initial‐boundary value problem associated with wave propagation in a superfluid.

A geometric formulation of the Taylor theorem for curves in affine manifolds
View Description Hide DescriptionThe paper contains a theorem from differential geometry which exhibits the geometrical content of the classical Taylor expansion theorem applied to curves in a differentiable manifold. It also presents a review of a slightly modified version of the calculus of affine extensions, necessary for the proof of the theorem.

Multiple time scale analysis of an anharmonic crystal
View Description Hide DescriptionWe use the multiple time scale perturbation method to study the lattice dynamics of an anharmonic crystal. The Heisenberg equations of motion for the creation and annihilation operators are solved, and the frequency shift and the decay constants are found to the second order. We also discuss briefly how to apply our solution to calculate the correlation function.

Simple supersymmetries
View Description Hide DescriptionTwo infinite families of simple graded Lie algebras (GLA’s) over the complex numbers are described: the special linear algebras SL(m‖n) [whose Bose sector is the direct sum of a one‐dimensional algebra with the ordinary Lie algebra SL(m) ×SL(n)], and the orthosymplectic algebras OSp(2r‖s) [with Bose sector Sp(2r) ×O(s)]. The GLA’s of physics fit into these two families either directly or via Inönü–Wigner contraction. These algebras along with further exceptional GLA’s constitute all the classical GLA’s, i.e., all GLA’s with a nondegenerate metric (not necessarilly Killing) form. The existence of infinite families of hyperexceptional GLA’s, i.e., of GLA’s that are simple but not classical is pointed out.

Approach of the statistical theory of light scattering to the phenomenological theory
View Description Hide DescriptionA fully statistical treatment of the spectrum of light scattered by a simple fluid is given. The results are shown to be in close accord with the phenomenological theory of the same process.

Construction of quantum fields from Euclidean tensor fields
View Description Hide DescriptionWe define Euclidean tensor fields over S (R^{4}), from which we construct quantum tensor fields satisfying all the Wightman axioms except the uniqueness of the vacuum. By a process of reduction, it is possible to obtain, from some suitably chosen Euclidean tensor field, a quantum field satisfying all the Wightman axioms except the uniqueness of the vacuum and transforming according to any arbitrarily chosen one‐valued finite‐dimensional irreducible representation of the restricted Lorentz group L^{↑} _{+}. We give a Euclidean vector field and a Euclidean tensor field of rank two as examples, leading respectively to the real Proca Wightman field and the free electromagnetic Wightman field.

Superposition of states and the structure of quantum logics
View Description Hide DescriptionA derivation of the classical Hilbert space model of quantum theory is given (including superselection rules) based (i) on various axioms on the behavior of events and (ii) an explicit formula for the superposition of two pure states in terms of transition probabilities.

Renormalization group for a system of continuous spins on a lattice
View Description Hide DescriptionThe renormalization group for a field theory on a lattice, or equivalently, in the language of statistical mechanics, a system of continuous spins on a lattice, described by Wilson, is considered in detail. The conditions under which a class of transformations are renormalization group transformations are studied.

Asymptotic properties of generalized Chaba and Pathria lattice sums
View Description Hide DescriptionFor an arbitrary Bravais τ lattice in an m‐dimensional Euclidean space amd for 0<a<∞, we present an extension of the Chaba and Pathria method of evaluating the lattice sums Σ′_{τ}τ^{−2k } exp(−aτ^{2}) from integral k values to all positive k values. We use the extension to study the asymptotic properties of these sums as the parameter a approaches zero. The leading term is given by π^{ m/2} [Γ (k)(m/2−k) a ^{ m/2−k }]^{−1} for 0<2k<m, by π^{ m/2}Γ (k)^{−1} ln(1/a) for 2k=m, and by Σ′_{τ}τ^{−2k } for 2k≳m. Thus 2k=m gives a transition point from structure independence to structure dependence.