Volume 13, Issue 10, October 1972
Index of content:

Kinetic Theory of Scattering by a Plasma Cylinder
View Description Hide DescriptionThe kinetic theory of scattering by a circular homogeneous isotropic plasma cyclinder is treated for plane wave incidence parallel to the axis of the cylinder. The relativistic form of the Vlasov equation is inverted, subject to the specular boundary condition, expressing the electronic distribution function in terms of the electric field intensity. After inverting Maxwell's equations and eliminating the distribution function, a set of Fredholm integral equations of the second kind are obtained for the angular Fourier components of the electric field. Since for low temperature the Neumann series converges, the low temperature solution is easily obtained. The first order temperature corrections are thus derived for the reflection coefficients associated with the Fourier components.

Lagrangian Density for Perfect Fluids in General Relativity
View Description Hide DescriptionA previously discussed variational principle for a perfect fluid in general relativity was restricted to irrotational, isentropic motions of the fluid. It is proven that these restrictions can be dropped, and the original variational principle can be generalized to general motions of the perfect fluid. The form of the basic Lagrangian density is unchanged by these generalizations. An Eulerian fluid description is used throughout. As a by‐product of our variational principle, the 4‐velocity is required to have the generalized Clebsch form.

Kramers‐Kronig Relations and Sum Rules
View Description Hide DescriptionThe concept of moments, which are integrals of positive or negative integral powers ω^{ n } weighted by real or imaginary parts of admittance functions, is here generalized so as to be applied to a wide category of admittance functions, including Lorentzian functions. The generalized moments are related to the derivatives or integrals of sum rules in a general sense. This analysis is based on differentiation and integration‐mapping of admittance functions and the associated Kramers‐Kronig relations. Some model calculations are also shown.

Steady‐State Solutions in the Two‐Group Theory of Neutron Diffusion
View Description Hide DescriptionFunctional analysis arguments are used to prove the existence of a unique solution to the integral form of the two‐group neutron‐transport equation for subcritical half‐spaces. The analytic properties of the solutions are discussed and used to prove that the partial indices of canonical solutions of the matrix Riemann problem, basic to H‐matrix or half‐range completeness considerations, are nonnegative.

A Perturbation Method for Two Synchronously Tuned, Coupled, Autonomous, Nonlinear Oscillators
View Description Hide DescriptionA perturbation procedure is developed for two synchronously tuned, coupled, autonomous, nonlinear oscillators. The procedure results in ordinary nonlinear differential equations for the slowly varying amplitude envelopes and the slowly varying phase difference of the two oscillators. A method of obtaining initial values is included as are two examples for coupled van der Pol and linear oscillators.

Symmetry in Einstein‐Maxwell Space‐Time
View Description Hide DescriptionBy using the complex null tetrad as basis for the tangent space, a Killing vector field (``symmetry'') is introduced into the system of Einstein's equations with Maxwell's equations. The two bivectors F _{μν} and K _{μ;ν} (the associated Killing bivector) are assumed to have a principal null direction in common. Killing's equations,Maxwell's equations, and Einstein's equations are then written down for the case where this special direction is also a principal null geodesic for the Weyl conformal tensor. A certain analog of the Goldberg‐Sachs theorem is proved. The static cases, plus a sizeable class of the static algebraically special cases are examined, to wit: where the special direction is also shear‐free. In particular, all such algebraically special spaces must be Petrov Type D as a result of a coupling of the principal null directions for F _{μν}. This algebraically special metric is derived as an example of the static classes and is a static generalization of the Reissner‐Nordström metric.

Form Factors for Any Spin and Charge Coupling
View Description Hide DescriptionA prescription for decomposition into form factors of the matrix elements of totally symmetric tensor currents between (s, 0) and (s′, 0) spinor states of the incoming and outgoing particles of spins s and s′ is presented. The case of the electromagnetic current is studied in some detail, and a procedure to modify the above prescription resulting in the grouping of the form factors into Dirac‐type and Pauli‐type classes is given. Several examples are used for illustration.

Neumann Series Solution for the Atom‐Rigid Rotor Collision
View Description Hide DescriptionThe solutions of the coupled differential equations arising in the quantum mechanical discussion of the collision of an atom with a rigid, rotating diatom are written as Neumann series, i.e., expanded in terms of spherical Bessel functions. The coefficients of these series are generated by a set of coupled recursion relations. The formalism is limited to potentials less singular than r ^{−2} at the origin.

A Geometrical Theory of the Electromagnetic Field and the Gauge Transformation
View Description Hide DescriptionA modification of Weyl's theory of the electromagnetic field is presented, such that the usual gravitational Lagrangian becomes invariant under the corresponding definition of gauge transformation. As another advantage the problem of the weight of the tensors in the construction of Lagrangians is eliminated.

Conditions for the Existence of the Generalized Wave Operators
View Description Hide DescriptionIf E _{0}(G) is the spectral projection operator associated with the free Hamiltonian H _{0}, corresponding to a bounded measurable subset G of , and E _{1}(G) is associated with the total Hamiltonian H = H _{0} + V, where the operator E _{1}(G)VE _{0}(G) is of trace class, it is proved that the element g = E _{0}(G)f belongs to the domain of the generalized wave operators Ω_{±} if and only if .A stronger version of this result is also proved, from the theory of time‐dependent scattering, and is applicable to scattering systems for which families {G _{1}}, {G _{2}} of measurablesets may be found such that E _{1}(G _{2})VE _{0}(G _{1}) is of trace class.

On the Connection between Levinson's Theorem and Singular Integral Equations
View Description Hide DescriptionLevinson's theorem is deduced from a general property of singular integral equations in the case in which both the unperturbed and the total Hamiltonian have a finite number of discrete eigenvalues. We also discuss the conditions of validity of the theorem.

On the Generalized Exchange Operators for SU(n)
View Description Hide DescriptionBy following the work of Biedenharn we have redefined the k‐particles generalized exchange operators (g.e.o.'s) and studied their properties. By a straightforward but cumbersome calculation we have derived the expression, in terms of the SU(n) Casimir operators, of the 2‐, 3‐, or 4‐particle g.e.o. acting on the A‐particle states which span an irreducible representation of the grown SU(n). A striking and interesting result is that the eigenvalues of each of these g.e.o.'s do not depend on the n in SU(n) but only on the Young pattern associated with the irreducible representation considered. For a given g.e.o., the eigenvalues corresponding to two conjugate Young patterns are the same except for the sign which depends on the parity of the g.e.o. considered. Three appendices deal with some related problems and, more specifically, Appendix C contains a method of obtaining the eigenvalues of Gel'fand invariants in a new and simple way.

Steady‐State Sound Propagation in Continuous, Statistically Isotropic Media
View Description Hide DescriptionA stochastic Eulerian‐Lagrangian procedure is applied to steady‐state sound propagation from a small, collimated acoustic source to an omnidirectional point receiver imbedded in an infinite, continuous, statistically isotropic medium. An analytic procedure is developed for obtaining the Lagrangian measure function B(x, ξ  s) from its characteristic function φ(k, ξ  s) for stochastic‐Fermat media. The results include a coefficient of intensity variation V that evinces a frequency‐dependent, phase‐dominance region and a frequency‐independent, amplitude‐dominance region. The methods employed in this study are new to the problem of sound propagation through continuous stochastic media and avoid three common difficulties: (1) range limitations due to cumulative phase effects, (2) discrete scattering assumptions, and (3) restriction to an Eulerian path.

Recurrence Formula for the Veneziano Model N‐Point Functions
View Description Hide DescriptionA recurrence formula is derived for a function which reduces to the Veneziano model (n + 3)‐point function. It is shown that the formula is equivalent to, but is more self‐contained than, the Hopkinson and Plahte formula in that it does not require the prescription for the parameters involved.

Linear Inequalities for Density Matrices. II
View Description Hide DescriptionA general method is described for finding linear inequalities relating physical properties of an ensemble which can exist only in a finite number of states. This algorithm is used to solve a few examples of the fixed‐N and variable‐N Slater hull problem and the fixed‐N Boson hull problem. Interpretations of the results are made in terms of boundary conditions for fermion and boson reduced density matrices, pair distributions of lattice vacancies, and pair distributions of spins in an Ising model.

Stationary Axially Symmetric Fields and the Kerr Metric
View Description Hide DescriptionIn this note the axially symmetric metric for stationary gravitational field, in a slightly general form, is discussed. The vacuum field equations for this metric are given. Specialization of this metric leads to a different form of field equations previously discussed in literature. In particular, the Kerr metric is given in a new form. A justification for interpreting the Kerr metric as an exterior solution corresponding to a spinning rod or a rotating spherical body is given.

On the Convergence of the Born Series for All Energies
View Description Hide DescriptionFormal solution of the Schrödinger equation for nonrelativistic scattering by a spherically symmetric static potential −μV(r) leads to a power series in the real parameter μ for the scattering amplitude (the Born series). It is shown that if and if −μV(r) is too weak to support a bound state, then the Born series converges at all energies. The method gives a lower bound for the radius of convergence of the Born series which is exact if V ⩾ 0.

Nonmetrical Specification of Space‐Time Sources
View Description Hide DescriptionThe role played by the energy‐momentum conservation law in general relativity is examined. It is noted that this law can be interpreted in two ways. It may be thought of as a condition determining the evolution of the energy‐momentum tensor density in time, or it may be thought of as a condition determining the metric. In the present paper, the second of these ways of thinking about the energy‐momentum conservation law is explored. Einstein's nonvacuum gravitational field equations (which imply the conservation law) are examined. It is shown that given any analytic symmetric contravariant energy‐momentum tensor density as a function of the space‐time coordinates, a solution to the gravitational field equations always exists. Furthermore, this solution is such that the law of conservation of energy‐momentum is satisfied. The proof uses a coordinate transformation method to exploit the covariance of the energy‐momentum conservation law. Riquier's existence theorem enters as an important part of the proof, and a general discussion of Riquier's existence theorem from a physical point of view is given. Both the geodesic nature of the trajectories of free particles and the unit magnitude of the velocity 4‐vector are discussed. An interpretation of the above‐described results is given.

Properties of Lorentz Covariant Analytic Functions
View Description Hide DescriptionA theorem is proved that asserts, roughly, that a function that is real Lorentz covariant anywhere is complex Lorentz covariant everywhere in its domain of regularity. It is also shown that the analytic continuation of a scattering function from a regularity domain in the physical region of a given process along all paths generated by complex Lorentz transformations leads to a function that is single‐valued in the neighborhood of all these paths. Applications are discussed. The results derived constitute necessary preliminaries to a discussion of the analytic structure of scattering functions given in other papers.

The Quantum Field Theory without Cutoffs. IV. Perturbations of the Hamiltonian
View Description Hide DescriptionWe introduce an inductive method to estimate the shift δE in the vacuum energy, caused by a perturbation δH of the Hamiltonian H. We prove that if δH equals the field bilinear form , then δE is finite. We show that the vacuum expectation values of products of fields (Wightman functions) exist and are tempered distributions. They determine, via the reconstruction theorem, essentially self‐adjoint field operators , for real test functions. We also bound the perturbation of the Hamiltonian by a polynomial. so long as is formally positive. In that case, and with is bounded by const(1 + diam supp h).