Volume 19, Issue 5, May 1978
Index of content:

High‐energy behavior of renormalized Feynman amplitudes
View Description Hide DescriptionThe high‐energy polynomiala n d logarithmic behavior of renormalized Feynman amplitudes, involving s u b t r a c t i o n s, is derived with total generality when s o m e or all of the external momentum components of the graphs in question become large in Euclidean space nonexceptionally. This is achieved by explicitly carrying out the subtractions of renormalization as dictated, for example, by the improved Dyson–Salam renormalization scheme directly in momentum‐space.

Spectral theory for the acoustic wave equations with generalized Neumann boundary conditions in exterior domain
View Description Hide DescriptionIn this paper we studied the spectral theory of the operator (either in the exterior domain or in the whole space) that is induced from problems in the n‐dimensional Euclidean space for the hyperbolic linear partial differential equations with the generalized Neumann boundary condition. The resulting theory provides a foundation for studying the wave operator and scattering operator involved in scattering theory and possibly also for studying the respective inverse problem.

The use of the symmetric group in the construction of multispinor Lagrangians
View Description Hide DescriptionThe construction of Lagrange functions for nth rank multispinor (Bargmann–Wigner) fields is developed using the symmetric group S _{ n }. Restrictions on the number of fields present in the Lagrangian and their couplings to one another are obtained by constructing differential operators which transform irreducibly under S _{ n }. The technique is illustrated by a brief discussion of the second and third rank cases and a more detailed fourth rank example. The Lagrangians thus obtained are precisely those specified by Lorentz invariance, but the method used greatly facilitates their construction.

Definition of polarization of a spin 1/2 particle in an external electromagnetic field
View Description Hide DescriptionThe 3‐vector polarization of a Dirac particle with Pauli anomalous moment, in external electric and magnetic fields, is shown to be given by a certain ratio of components of the wavefunction. The result is valid in the same approximation as leads to the Bargmann–Michel–Telegdi classical equations for the polarization.

A second order calculation of the adiabatic invariant of a charged particle spiraling in a longitudinal magnetic field
View Description Hide DescriptionThe change of the action integral of a charged particle spiraling in a slowly varying longitudinal magnetic field is investigated. The method used to solve this two‐dimensional problem is a generalization of Vandervoort’s analysis of a one‐dimensional system. It is based on a canonical transformation of the usual conjugate coordinates and momenta into a set of four variables of the action‐angle type. The new canonical equations are solved by a method of iteration, and the solution is used to calculate the change of the adiabatic invariant. Our results are analogous to those obtained with the approximations derived by Hertweck and Schlüter and by Chandrasekhar. We have compared our analytic results to those obtained by numerical integration of the equation of motion in the case of a magnetic solenoid with a slowly decreasing field.

The graded Lie groups SU(2,2/1) and OSp(1/4)
View Description Hide DescriptionWe study the graded Lie groups corresponding to the graded Lie algebras SU(2,2/1) and OSp(1/4). General finite group transformations are parametrized, and nonlinear representations are obtained on coset spaces. Jordan and traceless algebras are constructed which admit these groups as automorphism groups.

A Galileian formulation of spin. I. Clifford algebras and spin groups
View Description Hide DescriptionBy generalizing the concept of spin group to the case where the underlying orthogonal space is degenerate, the spin group associated with the homogeneous Galilei group is calculated. In so doing, the Galilei group and its spin group are clearly displayed as stability subgroups of the de Sitter group and its spin group. A notion of Clifford algebra contraction is introduced in the physical (Galilei) case and its relation to Lie algebra contraction is explored. Both the stated generalization of spin group to cases with degenerate bilinear form and the idea of Clifford algebra contraction appear to be new.

Solitonlike solutions of the elliptic sine–cosine equation by means of harmonic functions
View Description Hide DescriptionExact solutions of the elliptic sine–cosine equation ∂^{2}ψ/∂x ^{2}+∂^{2}ψ/∂y ^{2}=sin(ψ+g) are derived in two space dimensions with the aid of a new Bäcklund transformation and by exploiting the properties of the h a r m o n i c function g (x,y). Two generating formulas are developed which allow us to generate w i t h o u t a d d i t i o n a l q u a d r a t u r e s an infinite number of real solutions α and infinitely many imaginary solutionsiβ. Some α solutions behave like solitons and can be labeled by a topological quantum number. Which solutions are solitonlike and which are not depends decisively on the analytic structure of g and its domain of harmonicity in the R^{2} plane.

Stability of Weiss Ising model
View Description Hide DescriptionThe problems of stability and approach to equilibrium of the Weiss Ising model are studied. Our investigations are performed in the exact and linear response senses in order to compare both theories. The change of a metastable state of the Weiss Ising model is discussed under local perturbations.

The asymptotic reduction of products of representations of the universal covering group of SU(1,1)
View Description Hide DescriptionIrreducible representations are realized on nuclear spaces. Products of certain elements of two such spaces can be expanded into an almost everywhere convergent series that possesses rudimentary covariance properties. This unique expansion is therefore called ’’asymptotic reduction.’’ Products of elements of three such spaces possess a simultaneous expansion only in an asymptotic sense. We define recoupling coefficients for this reduction and give them explicitly.

Local supersymmetry in (2+1) dimensions. II. An action for a spinning membrane
View Description Hide DescriptionWe present a locally supersymmetric action for a spinning membrane. This is obtained by supersymmetrizing the induced volume element action which is reformulated in terms of a three‐dimensional field theory. We also discuss more complex actions which are possible due to the nontriviality of (super) gravity in three dimensions.

Nontranslationally covariant currents and symmetries of the S matrix
View Description Hide DescriptionGenerators of symmetries constructed from nontranslationally covariant currents are defined on scattering states and commute with the S matrix.

Electromagnetic solutions of Brans–Dicke theory of gravitation from Einstein theory
View Description Hide DescriptionA class of static and nonstatic solutions of the Brans–Dicke theory of gravitation is obtained in the presence of an electromagnetic field. The metric coefficients and fields (both scalar and electromagnetic) are supposed to be functions of any three independent variables. The major result of the paper may be stated as follows: ’’Corresponding to any diagonalizable solution of Einstein’s vacuum field equations in which fields and metric coefficients are functions of not more than three variables, we can generate a solution of the coupled Brans–Dicke Maxwell field equations with nonzero electromagnetic field.’’

Some constraints on finite energy solutions in non‐Abelian gauge theories
View Description Hide DescriptionWe exploit dilatational invariance and some inequalities for the stress tensor to derive constraints for finite energy solutions in sourceless non‐Abelian gauge theories. The results extend known no‐go theorems considerably and provide some hints on what nondissipative finite energy solutions could look like.

On the inverse problem of transport theory with azimuthal dependence
View Description Hide DescriptionThe infinite medium inverse problem with an azimuthally dependent plane source leads to integral moments of the intensity over all space and angle. A new relationship has been derived between the moments and the coefficients of the expansion of powers of ν in terms of the g ^{ m } _{ k }(ν) polynomials which arise in transport problems without azimuthal symmetry. This relationship has been used to obtain an improved method for determining the moments.

Asymptotic behavior of group integrals in the limit of infinite rank
View Description Hide DescriptionWe show that in the limit N→∞ integrals with respect to Haar measure of products of the elements of a matrix in SO(N) approach corresponding moments of a set of independent Gaussian random variables. Similar asymptotic forms are obtained for SU(N) and Sp(N). An application of these results to Wilson’s formulation of lattice gauge theory is briefly considered.

Borel summability and indeterminacy of the Stieltjes moment problem: Application to the anharmonic oscillators
View Description Hide DescriptionAn indeterminacy criterion is proven for the moment problem associated with the coefficients of a Borel summable power series of Stieltjes type which diverge faster than (2n) !. As an application we show that the Stieltjes type continued fraction corresponding to the Rayleigh–Schrödinger perturbation expansions for the energy eigenvalues of the anharmonic oscillators (x ^{2(m+1)} and in any finite number of dimensions) does not converge to the eigenvalues if m≳2. In particular, this implies the nonconvergence of the Padé approximants to the eigenvalues of p ^{2}+x ^{2}+λx ^{2(m+1)} if m≳2.

Analytic connection between configuration–interaction and coupled‐cluster solutions
View Description Hide DescriptionThe coupled‐cluster (CC) equations in the work of Coester, Kümmel, Čižek, Paldus, and others are inhomogeneous, nonlinear and algebraic in the cluster operators to be determined. If taken to all orders, they are equivalent to complete configuration–interaction (CI) equations, except for states orthogonal to the reference state Φ. However, if taken only to nth order, they are not equivalent to the nth order CI equations, and due to their nonlinear form, the existence and the number of the solutions is not guaranteed. Also, the reality of the associated energy values is not certain since these values do not arise as eigenvalues of a Hermitian operator. We show that the equations can be cast in the form of perturbed CI equations, with the ’’perturbations’’ being non‐Hermitian and nonlinear in the CI‐like coefficients to be calculated. In the case of a finite number of single‐particle states, we construct the solutions to the CC equations by analytic continuation from the CI solutions.Singularities peculiar to the method are identified and studied, and conditions for reality and the maximum multiplicity of solutions are given. In general, the energy will be real, and the number of solutions equals that of the associated CI problem. Singularities or instabilities in the coupled‐cluster equations can be traced to unphysical assumptions in the basis set Hamiltonian, or a poor description to highly excited states.

On the representation of linear operators in L ^{2} spaces by means of ’’generalized matrices’’
View Description Hide DescriptionThis paper concerns the representation of linear operators of L ^{2} spaces by means of ’’generalized matrices’’ as it is usual, following Dirac, in quantum mechanics and in electronics. The known possibility of representing (on a nuclear test‐function space) the bounded operators by means of distribution kernels is shown to extend to all the closable operators whose domain contains the test‐function space (hence to all the Hermitian operators whose domain contains the Schwartz space D of the infinitely differentiable functions with compact support). The representation of the adjoint operator is considered, and the possibility of representing the product of operators by means of a suitably defined ’’Volterra convolution’’ is studied. In particular it is shown that ‐algebras of unbounded operators (which are, for instance, generated by the canonical coordinates and momenta and the total energy of most quantum mechanical systems of n particles) may be represented isomorphically by means of distribution kernels, so that the Dirac’s rules on ’’generalized matrices’’ apply in the sense of distributions without further assumptions.

Entropy of an n‐system from its correlation with a k‐reservoir
View Description Hide DescriptionLet a random pure state vector be chosen in n k‐dimensional Hilbert space, and consider an n‐dimensional subsystem’s density matrix P. P will usually be close to the totally unpolarized mixed state if k is large. Specifically, the rms deviation of a probability from the mean value 1/n is [(1−1/n ^{2})/(k n +1)]^{1/2}. ’’Random’’ refers to unitarily invariant Haar measure.