Volume 20, Issue 3, March 1979
Index of content:

Particle–boson interactions and the weak coupling limit
View Description Hide DescriptionWe consider the interaction of a finite number of nonrelativistic particles with a positive or zero mass quantum field. We show that in the weak coupling limit the quantum field gives rise to an effective interaction between the particles of a Yukawa or Coulomb type, as well as, in some cases, a mass renormalization. In a simple exactly soluble model we investigate the higher order radiation processes.

Microscopic theory of superfluid Fermi systems. I. Binary expansion
View Description Hide DescriptionThis is the first of two papers in which the binary expansion is used to study the properties of hard core fermi systems with broken gauge symmetry. In this paper, which is primarily formal, symmetry‐breaking source terms are introduced into the generating function for thermodynamic quantities, and an exact expression is obtained for the generating function in terms of single‐particle and two‐particle sources and the reaction matrices for the two‐body problem. In a subsequent paper, the theory is applied to a dilute hard sphere fluid.

Microscopic theory of superfluid Fermi systems. II. Application to low density hard sphere system
View Description Hide DescriptionThis is the second of two papers in which the binary expansion is used to study properties of hard core Fermi systems with broken gauge symmetry. In this paper, we make contact with the phenomenology of superfluid Fermi systems commonly used in field theory. This requires, however, that each scattering event is approximated by its forward scattering value, an approximation which, at best is valid at very low densities. Within this approximation we show that a hard sphere Fermi fluid can exhibit superfluidity.

Classification and construction of finite dimensional irreducible representations of the graded algebras; application to the (Sp(2n); 2n) algebra
View Description Hide DescriptionA method, which enables us to construct the finite‐dimensional representations of the graded Lie algebras [explicitly the GLA (Sp(2N); 2N)] on the irreducible tensors is suggested. Those tensors are constructed with the aid of the specifically symmetrized products of vectors from the fundamental [i.e., (2N+1) ‐dimensional] representation space of the graded Lie algebra. The tensors, on which it is possible to represent the graded algebra (Sp(2N); 2N) irreducibly, represent a generalization of tensors which are known from the general representation theory of the symplectic Lie algebra Sp(2N). The knowledge of the irreducible tensors of the algebra Sp(2N) gives us then the possibility of solving the problems of classification as well as construction of the irreducible tensors of the graded algebra (Sp(2N); 2N). For illustration, by using the suggested method of tensors the irreducible representations of the simplest graded algebra, i.e., of the algebra (Sp(2); 2) are constructed.

Off‐energy‐shell results for scattering by a nonlocal potential. I
View Description Hide DescriptionThe importance of analyzing the off‐shell effects due to scattering by nonlocal potentials is emphasized. Analytical expressions for l‐wave off‐shell wavefunctions associated with Jost (irregular) and physical (outgoing wave) boundary conditions are derived for an N‐term separable potential by using a differential equation approach. The half‐off‐shell and fully off‐shell T matrices are expressed in terms of appropriate Jost functions, Fredholm determinants, and transforms of the form factors of the potential. The general results presented are then used to construct exact expressions for T matrices for Tabakin, Beregi, and Mongan potentials. In limiting cases, each of our results can be seen to yield the off‐shell T matrix for the one‐term Yamaguchi potential calculated by other techniques.

Off‐energy‐shell results for scattering by a nonlocal potential. II
View Description Hide DescriptionThe wavefunction approach to off‐shell scattering on nonlocal potentials is used to obtain half‐off‐shell and off‐shell K matrices in terms of Jost functions and the Fredholm determinant associated with the off‐shell principal value wavefunction. The results are employed to construct exact expressions for K matrices for Tabakin, Beregi, and Mongan case IV potentials.

Lie algebras associated with motion in axisymmetric electromagnetic fields
View Description Hide DescriptionThe existence and analytical form of a vector constant of the classical relativistic planar motion of a point charge in an arbitrary time‐independent axisymmetric electromagnetic field are established. The components of the vector are utilized, in conjunction with the angular momentum, to construct realizations of the Lie algebras of the Euclidean group E(2), of the special unitary group SU(2), and of the Ladder operators of the harmonic oscillator. The charge is assumed to move in an externally prescribed field. The formulation is gauge invariant.

Extension theorems for operator‐valued measures
View Description Hide DescriptionBenioff [J. Math. Phys. 13 (1972)] has shown that every consistent, countable family of positive, normalized operator‐valued (PNOV) measures μ_{ n } over R ^{ n } can be extended to a PNOV measure over R ^{∞}. In this paper we show that the same result holds for arbitrary, consistent families of PNOV measures over complete, separable metric spaces. Further we show that, while there may be no extension if the topological conditions are relaxed, it is always possible to construct a related family of PNOV measure spaces which: (1) Are measures theoretically indistinguishable from the original spaces; (2) Have an extending PNOV measure. These results use developments in the theory of algebraic models of measures as initiated by Dinculeanu and Foias [Ill. J. Math. 12 (1968)] and applied to stochastic processes by Schreiber, Sun, and Bharucha‐Reid [Trans. AMS 158 (1973)].

Representations of a para‐Bose algebra using only a single Bose field
View Description Hide DescriptionRepresentations of a para‐Bose algebra are given entirely in terms of the operators of a single Bose field. Incidentally some types of boson representations of a nonsemisimple graded Lie algebra of the three‐dimensional Lorentz group are obtained.

Computation of a class of 3j‐coefficients
View Description Hide DescriptionWe give an analytical procedure to calculate 3j‐coefficients (^{ l } _{ m } ^{ l } _{−m } ^{2j } _{0}) for arbitrary integer l.

Semiclassical perturbation expansion of the multichannel scattering matrix
View Description Hide DescriptionA semiclassical perturbation method for the inelastic S matrix is described. The channels are transformed into a set of the eigenstates and it is assumed that the transition between them is small. The perturbation is the matrix which diagonalizes the coupling matrix. It is shown that such a series is independent of h/; hence the limit h/ →0 of the S matrix can be calculated. A special case of the weak coupling is also discussed.

On the diagonalization of the general quadratic Hamiltonian for coupled harmonic oscillators
View Description Hide DescriptionIt is shown that the general quadratic Hamiltonian for coupled harmonic oscillators can be diagonalized, provided the matrix of the quadratic form is positive definite. This condition is also necessary if the frequencies of the resulting uncoupled oscillators are to be positive. The construction of a diagonalizing matrix follows the usual procedure as in the Hermitian case; the only difference being a change of the metric from I= (^{1} _{1}) to J= (^{1} _{−1}).

Conformal Killing horizons
View Description Hide DescriptionWe introduce the concept of a conformal stationary limit surface as the boundary where a conformal Killing vector admitted by a spacetime becomes null. This hypersurface is an infinite frequency shift surface for conformal Killing observers and sources. We derive the local conditions under which it can be an event horizon in the sense that it be a null geodesic hypersurface. In particular, we show that it is null and geodesic if and only if the rotation of the conformal Killing congruence has a vanishing norm on this hypersurface. Finally, we discuss the surface gravity associated with such a conformal Killing horizon.

Linearized analysis of inhomogeneous plasma equilibria: General theory
View Description Hide DescriptionA generalized framework is presented for analyzing the linearized equations for perturbations of inhomogeneous plasma equilibria in which there is a collisionless species, some properties of the solutions of the linearized equations are described, and a basis is provided for numerical computations of the linearized properties of such equilibria. It is useful to expland the perturbation potentials in eigenfunctions of the field operator which appears in the linearized equations, and to define a dispersion matrix whose analytical properties determine the nature of the solutions of the initial‐value problem. It is also useful to introduce auxiliary functions to replace the usual perturbation distribution functions, and to expand the auxiliary functions in eigenfunctions of the equilibrium Liouville operators. By introducing the auxiliary functions, great freedom is achieved in the choice of the field operator which appears in the linearized equations. This freedom can be useful in some problems to define expansion functions for the potentials that are particularly suitable for studying specific normal modes.

Invariants in enveloping algebras under the action of Lie algebras of derivations
View Description Hide DescriptionAn upper bound on the number of algebraically independent invariants in an enveloping algebraU under the action of a Lie algebraG _{0} of derivations is obtained. We are able to determine the exact number of invariants for the case [G _{0},G _{0}]=G _{0}. This generalizes previous results about Casimir invariants.

Constrained Hamiltonian formulation for interacting fields: Stable particlelike solutions
View Description Hide DescriptionIt is observed that singularity‐free localized particlelike solutions to certain essentially nonlinear classical fieldequations are dynamically stable in a constrained free‐field Hamiltonian formulation. Being rather novel, the variational principle for such a theory pertains to an arbitrary pair of neighboring solutions to the field equations, with δφ specified as the difference between neighboring solutions.

Linearization stability of gravitational and gauge fields
View Description Hide DescriptionConditions are given for the linearization stability of the Yang–Mills and the Einstein–Yang–Mills equations on a spacetime with a compact Cauchy surface. There are sufficient conditions on the Cauchy surface, and necessary and sufficient conditions on the spacetime; the latter are identified with global infinitesimal symmetries of the principal fiber bundle associated with the Yang–Mills (gauge) field. For each system a splitting theorem for the initial data is given and the Cauchy problem is discussed.

Conservation laws for the classical Maxwell–Dirac and Klein–Gordon–Dirac equations
View Description Hide DescriptionSolutions to the classical coupled Maxwell–Dirac and Klein–Gordon–Dirac equations in a space–time of dimension four are considered. These equations are invariant under the 15‐dimensional conformal group, in the case of zero mass. The resulting conservation laws are explicitly exhibited in terms of the Cauchy data at a fixed time in a form suitable for analysis by the techniques of partial differential equations.

The Cauchy problem in general relativity. III. On locally imbedding a family of null hypersurfaces
View Description Hide DescriptionThis paper is concerned with the problem of locally imbedding a null hypersurface in a Riemannian manifold. More precisely, on a one‐parameter family of null hypersurfaces, rigged by an arbitrary null vector field, in a four‐dimensional space–time manifold, a particular symmetric affine connection is used to derive the corresponding generalized Gauss–Codazzi equations. In addition, expressions are obtained for the projections of the Ricci tensor, which are relevant to the characteristic initial‐value problem of general relativity.

Geometric derivation of the kinetic energy in collective models
View Description Hide DescriptionA separation of the many‐particle kinetic energy into collective and intrinsic components is shown to result simply from a general form of the Laplace–Beltrami operator. The geometric structure of the decomposition is thereby clearly exhibited and the intricate computations previously necessary are eliminated.