Volume 16, Issue 8, August 1975
Index of content:

The three‐body time‐delay operator
View Description Hide DescriptionUsing Faddeev’s form of time‐dependent scattering theory, we give an abstract definition of time delay valid for multichannel scattering. For the three‐body scattering problem we find an explicit relation, that is valid on the energy shell, between the time‐delay operator and the S operators and their energy derivatives.

Irreducible Cartesian tensor expansions of scalar fields
View Description Hide DescriptionIt is shown how a scalar function V (‖R + Σ^{ n } _{ i=1} a_{ i }‖) of a sum of n + 1 vectors can be expanded as a multiple Cartesian tensor series in the vectors a_{ i }. This expansion is a rearrangement of the multiple Taylor series expansion of such a function. In order to prove the fundamental theorem, Eq. (3.1) below, generalized Cartesian Legendre polynomials are defined. The theorem is applied to the eigenfunctions of the Laplace operator and to inverse powers. The expansions of the latter type of function leads to forms involving generalized hypergeometric functions in several variables. As a special case, the Cartesian form of the multipole expansion of the electrostatic potential between two linear molecules is derived. A number of sum rules for hypergeometric functions and addition formulas for (standard and modified) spherical Bessel functions are proved using a reduction property of the generalized Legendre polynomials. The case of the expansion of a tensorial function is also briefly discussed.

Decompositions of gravitational perturbations
View Description Hide DescriptionWe consider the Brill–Deser decomposition of the perturbations of a flat spacetime with compact Cauchy hypersurfaces. We propose a generalization of the Brill–Deser splitting which may be applied to the perturbations of arbitrary vacuumspacetimes with compact Cauchy slices. We split the space of perturbations of any allowed Cauchy data set into three subspaces which, with suitable inner product, are mutually orthogonal. Two of these subspaces comprise the solution set of the perturbed constraint equations, and one of these two subspaces represents pure gauge perturbations. Some possible applications of these splittings to the study of the vacuumperturbation equations and to the linearization stability problem for the Einstein equations are briefly discussed.

Onsager symmetry and the diagonalizability of the hydrodynamic matrix
View Description Hide DescriptionThe diagonalizability of the hydrodynamic matrix in the case that one of the variables is odd under time reversal is investigated. The implications for a normal modeanalysis and for the spectral density elements are considered.

On the density of the Breit–Wigner functions
View Description Hide DescriptionIt is shown, for certain sequences {λ_{ i }} in the complex plane, that linear combinations of the Breit–Wigner functions {B _{ i }} approximate, in the mean square, any function in L ^{2} (0,∞). Implications and numerical use of this result are discussed.

A nonlinear system of Euler–Lagrange equations. Reduction to the Korteweg–de Vries equation and periodic solutions
View Description Hide DescriptionThe Euler–Lagrange equations, which correspond to a variational
The Euler–Lagrange equations, which correspond to a variational principle with a Lagrange function depending on arbitrary functions and their first order derivatives, are shown to be reducible to the and their first order derivatives, are shown to be reducible to the Korteweg–de Vries equation under a small—but finite—amplitude approximation. Closed form approximation. Closed form periodic solutions to the Euler–Lagrange equations are found for a particular case, and the modulational stability of these solutions is discussed. Equations for waves in cold plasma are discussed as examples.

A general setting for Casimir invariants
View Description Hide DescriptionThis paper contains a general description of the theory of invariants under the adjoint action of a given finite‐dimensional complex Lie algebraG, with special emphasis on polynomial and rational invariants. The familiar ’’Casimir’’ invariants are identified with the polynomial invariants in the enveloping algebraU (G). More general structures (quotient fields) are required in order to investigate rational invariants. Some useful criteria for G having only polynomial or rational invariants are given. Moreover, in most of the physically relevant Lie algebras the exact computation of the maximal number of algebraically independent invariants turns out to be very easy. It reduces to finding the rank of a finite matrix. We apply the general method to some typical examples.

One‐dimensional excited state reduced Coulomb Green’s function
View Description Hide DescriptionThe nth excited state reduced Coulomb Green’s function in coordinate space for the one‐dimensional Kepler problem is investigated, and a closed expression for this function is obtained.

Continuous subgroups of the fundamental groups of physics. I. General method and the Poincaré group
View Description Hide DescriptionWe present a general method for reducing the problem of finding all continuous subgroups of a given Lie groupG with a nontrivial invariant subgroup N, to that of classifying the subgroups of N and the subgroups of the factor group G/N. The method is applied to classify all continuous subgroups of the Poincaré group (PG) and of the Lorentz group extended by dilatations [the homogeneous similitude group (HSG)]. Lists of representatives of each conjugacy class of subalgebras of the Lie algebras of the groups PG and HSG are given in the form of tables.

Continuous subgroups of the fundamental groups of physics. II. The similitude group
View Description Hide DescriptionAll subalgebras of the similitude algebra (the algebra of the Poincaré group extended by dilatations) are classified into conjugacy classes under transformations of the similitude group. Use is made of the classification of all subalgebras of the Poincaré algebra, carried out in a previous article. The results are presented in tables listing representatives of each class and their basic properties.

Structural equations for Killing tensors of order two. II
View Description Hide DescriptionIn a preceding paper, a new form of the structural equations for any Killing tensor of order two were derived; these equations constitute a system analogous to the Killing vector equations ∇_{α} K _{β} = ω_{αβ} = −ω_{βα} and ∇_{γ} ω_{αβ} = R _{αβγδ} K ^{δ}. The first integrability condition for the Killing tensor structural equations is now derived. Our structural equations and the integrability condition have forms which can readily be expressed in terms of a null tetrad to furnish a Killing tensor parallel of the Newman–Penrose equations; this is briefly described. The integrability condition implies the new result, for any given space–time, that the dimension of the set of second order Killing tensors attains its maximum possible value of 50 only if the space–time is of constant curvature. Potential applications of the structural equations are discussed.

Variational method and nonlinear oscillations and waves
View Description Hide DescriptionA direct variational method is developed for studying the asymptotic behavior of a wide class of nonlinear oscillation and wave problems. From some judiciously chosen trial solutions with adjustable parameters, equations governing the change of amplitudes and phases are derived and solved. The method is simple in concept and straightforward in application. Different aspects of the method are illustrated by applications to various examples: the oscillation of a pendulum with changing length; the motion of a charged particle in a strong magnetic field; the linear and nonlinear Klein–Gordon equations; and the linear and nonlinear Korteweg–de Vries equations.

Representations of the three‐dimensional rotation group by the direct method
View Description Hide DescriptionThe irreducible representations of the three‐dimensional rotation group are obtained directly from the irreducible representations of its infinitesimal generators (the spin matrices), parametrized in terms of the rotation angle and the direction of the rotation axis. Expressions are given for the rotation operator exp(iψn ⋅ S) in terms of two different bases of 2j+1 elements for spin j. The results are related to the spectral decomposition of the rotation operator and expressions obtained for spin projection operators along any spatial direction for arbitrary spin.

The symmetrical energy–momentum tensor derived by parametrization
View Description Hide DescriptionWith reference to a paper by Goedecke in this journal attention is drawn to the fact that already in his original paper on the subject Rosenfeld proved the equality of the results of the two general procedures of symmetrizing energy–momentum tensors, i.e., the procedure of Belinfante (1939), utilizing the angular–momentum tensor, and the procedure of Rosenfeld (1940), taking the Lorentz metric limit of the manifestly symmetrical energy–momentum tensor of Riemannian space. Since Rosenfeld’s presentation of his procedure may give the misleading impression that it has something to do with curved spaces, general relativity, or gravitational theory, we show in the present paper how his scheme can be recast in a form, where one merely takes resort to an infinitesimal transformation of the ordinary Lorentz coordinates to arbitrary curvilinear coordinates, describing the same original Lorentz space of zero curvature. This transformation, of course, means a parametrization of the variational principle, and the analysis can thus be performed by means of a generalization of the theory of parameter‐invariant variational principles. An expression for the symmetrized energy–momentum tensor is given, which is equivalent to that given by Rosenfeld, and in which the transformation functions are seen to vanish identically. The procedure is thus seen to be not so much a limiting process as a transformation to curvilinear coordinates, construction of a symmetrical energy–momentum tensor, and a transformation back again.

Heisenberg subgroups of semisimple Lie groups
View Description Hide DescriptionThe restriction of a unitary representation of a semisimple Lie group to a Heisenberg subgroup H _{ n } is shown to be quasiequivalent to the regular representation of H _{ n }. Spectral properties of elements of the Heisenberg subgroup are described. Conditions under which an element of a semisimple Lie algebra may be embedded in a Heisenberg algebra are found.

Boundedness below for fermion model theories. I
View Description Hide DescriptionThe spatially cut‐off Hamiltonians for the models (ψ̄ψ̄φ)_{2} and (ψ̄ψ̄φ^{ N } + φ^{2M })_{2} with M ≳ N are bounded below uniformly in a momentum cutoff, by using the semi‐Euclidean formulation.

Decay of correlations for infinite‐range interactions
View Description Hide DescriptionStrong cluster properties are proved at low activity and in various other situations for classical systems with infinite‐range interactions. The decay of the correlations is exponential, resp. like an inverse power of the distance, if the potential decreases itself exponentially, resp. like an inverse power of the distance. The results allow us to extend to the case of exponentially decreasing potentials the equivalence theorem between strong cluster properties and analyticity with respect to the activity, previously proved for finite‐range interactions.

The conditional entropy in the microcanonical ensemble
View Description Hide DescriptionThe existence of the configurational microcanonical conditional entropy in classical statistical mechanics is proved in the thermodynamic limit for a class of long‐range multiparticle observables. This result generalizes a theorem of Lanford for finite range observables.

Foundations of a quantum probability theory
View Description Hide DescriptionStatistical physical theories are frequently formulated in terms of probabilistic structures founded on a ’’logic of experimentally verifiable propositions.’’ It is argued that to each experimentally verifiable proposition there corresponds an experimental procedure which, in general, alters the state of the system, and is completely characterized by a ’’measurement transformation’’ or ’’operation.’’ An analysis of the relations among these experimental procedures leads us to a ’’logic of operations’’ which is quite different from the ’’lattice theoretic logics’’ that are often considered (albeit inadequate empirical justification), as models for the calculus of experimentally verifiable propositions of quantum theory. It is seen that the quantum probability theory based on the logic of operations provides the proper mathematical framework for discussing the statistics of successive observations in quantum theory. We also indicate how a theory of quantum stochastic processes can be formulated in a way similar to the Kolmogorov formulation of the classical theory.

Elementary systems in general relativity
View Description Hide DescriptionThe problem of definition of elementary systems in general relativity is analyzed and some current theories are compared. A definition of dynamically free local elementary systems in general relativity is proposed, taking into account the influence of the gravitational field arising from the intrinsic dependence of the symmetry groups on the field.