Volume 11, Issue 12, December 1970
Index of content:

Singularity Structure of Causal Distributions and Restricted Equal‐Time Limits
View Description Hide DescriptionWe study aspects of the space‐time singularitystructure of several classes of causal distributions, including the cases usually encountered in perturbation theory. Various definitions of restricted equaltime limit are considered which allow for the presence of highly singular Schwinger terms. It is shown that, with one definition, every causal distribution has a restricted equal‐time limit. A form of sum rule valid even in the presence of singular Schwinger terms is given.

Finding Hidden Symmetry by Numerical Methods
View Description Hide DescriptionGiven a Hermitian matrix that depends upon some parameter, one often wants to be able to find all the parameter independent symmetries of the matrix. A method for accomplishing this is given here. The only numerical procedure involved is the diagonalization of Hermitian matrices.

Ground State Functional of the Linearized Gravitational Field
View Description Hide DescriptionThe ground state functional of the linearized Einstein theory of gravitation is given as a functional of the gauge invariant Ricci tensor, and compared with the corresponding electromagnetic expression. The connection of the canonically quantized nonlinear theory of gravitation with the linearized theory is exhibited. Time is treated as a momentum variable rather than as a superspace coordinate, which leads to an ``extrinsic time representation'' h ^{TT} _{ ik }, h_{i}, t = −½Δ^{−1}π^{T}. The state functional of the linearized theory is shown to be the initial value of the state functional of the canonical theory on a constant extrinsic time hypersurface in the lowest order of a perturbation expansion. By means of the Einstein‐Schrödinger equation, this functional can be integrated off this initial hypersurface.

Energy‐Momentum Spectrum and Vacuum Expectation Values in Quantum Field Theory
View Description Hide DescriptionWe consider nonlinear boson self‐interactions with a periodic spatial cutoff. We prove that the energy‐momentum spectrum lies in the forward light cone. A momentum cutoff does not influence this result. For theories with finite‐field strength renormalization, we obtain bounds on the vacuum expectation values of products of the φ_{ t }'s and ∇φ's. These bounds are uniform in the volume (and possible momentum) cutoff.

Semiclassical Perturbation Theory of Molecular Collisions. III. Graphical Angular Momentum Methods and the nth Order
View Description Hide DescriptionA review of graphical angular momentum methods is presented. As an example, the graphical methods are applied to the coupling problem in the semiclassical first and second Born approximations with an electric multipole potential. The general nth‐order Born term is then considered. A simple expression is derived for the matrix elements of the anisotropic terms in the potential. The evaluation of the remaining integrals over time is discussed.

Einstein Tensor and 3‐Parameter Groups of Isometries with 2‐Dimensional Orbits
View Description Hide DescriptionThe algebraic classification of the Weyl and Ricci tensors and the relation between them in a Riemann space with an isometry group possessing a nontrivial isotropy group are reviewed. All metrics with Minkowski signature, invariant under a 3‐parameter isometry group with 2‐dimensional orbits having nondegenerate metrics, are constructed from the group properties and are shown to have Ricci tensors with a double eigenvalue, and the orbits are shown to be surfaces of constant curvature. The null orbits are shown to have a triply degenerate eigenvalue of the Ricci tensor. The various additionally degenerate metrics are classified in further detail, extending the work of Plebański and Stachel.

Representations of Local I‐Spin Charge Densities. I. Spaces Where Each I Occurs Only Once
View Description Hide DescriptionAn approach to the problem of representation of the algebra of currents that puts essential emphasis on the study of infinite‐parameter Lie algebras is proposed. As an example, a class of irreducible Hermitian representations of the commutation relations [V^{i} (φ_{1}), V^{j} (φ_{2})] = iε^{ ijk } V^{k} (φ_{1}φ_{2}), where the φ's are elements of a commutative algebra with identity, is derived. The dependence of the representations on the algebra {φ} is completely characterized by two functional equations that are explicitly solved, for {φ} an algebra of polynomials. States of well‐defined momentum and rotational properties are constructed using translational and rotational invariance and forming direct integral spaces. The representations so constructed are seen to belong to two distinct subclasses, distinguished by the vanishing or nonvanishing of a length parameter η. The subclass with η = 0 is unbounded in isospin and has the trivial momentum‐transfer structure characteristic of field‐theoretical point particles. On the other hand, the spaces characterized by η ≠ 0 are bounded in isospin and suited to describe particles with structure. A brief discussion on how to derive invariant form factors from the results here presented is included.

Spinor Treatment of Stationary Space‐Times
View Description Hide DescriptionA generalized SU(2) spinor calculus is established on the ``background space'' V _{3} of the stationary space‐time. The method of spin coefficients is developed in three dimensions. The stationary field equations can be put to a form which in V _{3} is analogous to the Newman‐Penrose equations. A V _{3} filling family of curves is determined by the gravitational field and is called the eigenray congruence. Stationary space‐times may be characterized by the geometric properties of eigenrays. The relation of this classification to the algebraic ones is discussed. The method of solving the equations obtainable for various classes is illustrated on the case of nonshearing geodetic eigenrays. Assuming asymptotic flatness, we obtain the Kerr metric.

Inverse Problem for a Cylindrical Plasma
View Description Hide DescriptionWe study the Maxwell equation for the electric potential inside a neutral nonuniform cylindrical plasma in an external oscillatingelectric field, from the point of view of the inverse problem. The logarithmic derivatives of the radial Fourier components of the electric potential at the edge of the cylinder are considered as experimental data. We obtain an explicit and exact representation of the electron density in terms of the high frequency behavior of the experimental data. The general analyticity properties in the complex‐frequency plane are also discussed.

Newman‐Penrose Constants and Their Invariant Transformations
View Description Hide DescriptionThe origin and significance of the Newman‐Penrose (N‐P) constants of the motion are examined from the point of view that constants of the motion generate invariant transformations. Here the calculation makes use of a generalization of Green's theorem to a situation applicable to the coupled Einstein‐Maxwell fields in general relativity. One finds strictly electromagnetic constants generated by an incoming electromagneticshock wave, with dipole symmetry, at future null infinity. The gravitational constants contain an admixture from the electromagnetic field. They are generated by an incident quadrupole gravitational shock wave at future null infinity . Both the electromagnetic dipole field and the gravitational quadrupole field behave like linearized fields at . All higher‐multipole fields do not uncouple from the nonlinear corrections induced by the self‐coupling of the gravitational field and its coupling with the electromagnetic field. It is shown that the gravitational constants are related to the rate of shear of the null rays near a result which suggests a connection with sources. Finally, the supertranslation invariance of the result is discussed.

Perturbation Theory for Damped Nonlinear Oscillations
View Description Hide DescriptionA perturbation theory has been worked out for the decay of autonomous, nonlinear oscillations in the case where there is large linear damping. The solution reduces to a solution obtained by Kryloff and Bogoliuboff for small damping and to the perturbationsolution for periodic oscillations for vanishing damping. The solution is applied to the decay of oscillations in Duffing's equation. In this case it shows good agreement with a solution obtained by numerical integration.

Half‐Range Expansion Theorems in Studies of Polarized Light
View Description Hide DescriptionThe established normal modes of the vector equation of transfer describing the transport of polarized light are used to construct solutions to typical half‐space problems. The half‐range completeness theorem required by this method is discussed in the context of systems of singular integral equations. Although the Riemann‐Hilbert problem encountered here is defined in terms of continuous rather than Hölder‐continuous functions, the existence of a canonical solution is established, and the developed properties of this canonical matrix are used to complete the proof of the necessary half‐range expansion theorem.

Diagrammatic Technique for Constructing Matrix Elements
View Description Hide DescriptionA diagrammatic technique is presented for computing the matrix elements of the generators of the unitary, orthogonal, symplectic, and symmetric groups (A_{n}, D_{n}, B_{n}, C_{n} , and S_{n} ) within any of their unitary irreducible representations. Examples are worked out.

Group Representations and Geometry
View Description Hide DescriptionWe confine our attention here to simply reducible groups and show how six of the seven points of a finite projective plane PG(2, 2) constitute a ``Pasch'' diagram representing a 6j symbol. The class of all equivalent symbols may thus be represented by the seventh point in the plane. Analyzing the symmetries of such configurations, we derive two theorems, the first of which is the geometrical analog of Regge's result while the second gives the geometrical analog of the multiplication of two 6j symbols. In these terms the analogs of Eqs. (I1), (I2), and (I3) of Appendix I of Irreducible Tensorial Sets by Fano and Racah are very simply expressed. In particular, the Biedenharn identity (I3) becomes a vector equation (mod 2), and the relation with Desargues' theorem is clarified. The advantage of this geometrical model is that the structure alone survives and all summations and complicated coefficients disappear.

Generalized Operators
View Description Hide DescriptionA formal expression T in creation and annihilation operators (e.g., the Hamiltonian for a field theory model) is generally not a densely defined bona fide Hilbert space operator but is usually a densely defined sesquilinear form; as such it is convenient to consider it as a linear map from a dense domain Φ_{−} of a Hilbert space Φ_{0} to a still larger space Φ_{+} of antilinear functionals on Φ_{−}; that is, We give here the basis of a mathematical structure theory of such generalized operators. The idea which we explore is that, associated with T, there is a (not necessarily unique) analytic family R _{λ} of generalized operators called the resolvent of T. Formally, R _{λ} = (λ − T)^{−1}, an equation to which we give more precise interpretations. The ambiguities in determining R _{λ} are associated with the arbitrary adjustments that are characteristic of renormalization programs. When appropriate conditions are met, we can construct from R _{λ} a new Hilbert space Ψ_{0} and a bona fide operator T_{R} (the renormalized T) which is related to T by a formal intertwining equationT_{R} Δ = ΔT, where Δ maps Φ_{−} into a space containing Ψ_{0}. Given several generalized operators, we outline a procedure by which a subset of these can be renormalized to bona fide operators while the rest are reinterpreted as new generalized operators in the new Hilbert space. These are the rudiments of a multiplicity theory. Numerous examples illustrate the methods; in particular, the Nθ sector of the Lee model with arbitrary cutoff (including none) is treated in detail.

S Matrix in the Heisenberg Representation
View Description Hide DescriptionThe elements of the S matrix are calculated directly from an operator formalism, using the method of Yang and Feldman. This method has the advantage of providing a simple and direct justification of the Feynman rules for gauge fields, which express the contribution to the S matrix from a diagram containing closed loops in terms of sums over lowest‐order physical amplitudes (tree amplitudes) in which all external lines are on the mass shell and have physical polarizations. This guarantees unitarity. A condensed notation due to DeWitt is used. First, the one‐ and two‐loop contributions to the amplitude for production of a single quantum, and the amplitudes for pair production and scattering of a single particle by a classical background field are calculated in the absence of an invariance group. Noncausal chains (loops of cylically connected advanced or retarded Green's functions) never appear at any stage of the calculation, thus giving the decomposition into sums over tree amplitudes directly. This result is then generalized in an obvious way to the case in which an invariance group is present. The amplitudes are expressed in terms of a noncovariant propagator which propagates only physical (transverse) quanta. Rewriting these expressions in terms of covariant propagators leads to the formal appearance of ``fictitious quanta,'' which compensate the nonphysical modes carried by these propagators. All results are in agreement with those obtained by other methods.