Volume 6, Issue 1, January 1965
Index of content:

Spherically Symmetric Gravitational Fields
View Description Hide DescriptionIt is shown that the Schwarzschild solution is the only spherically symmetric solution of the Einstein vacuum field equations, even when the differentiability of the metric is weakened to the extent of permitting solutions which are C ^{0}, piecewise C ^{1}. Petrov's purported counterexample is analyzed and shown to be essentially equivalent to Schwarzschild's example.

Cluster Properties of Multiparticle Systems
View Description Hide DescriptionThe object of investigation is a system of N particles in nonrelativistic quantum mechanics. The particles interact via two‐ or many‐body potentials, for which a sufficient condition is that they be square integrable in the relative coordinates of the interacting particles. Cluster properties are derived for the time translation operator, for the wave operators, for the transition probabilities, and for the S operator.

Relation between the Onsager and Pfaffian Methods for Solving the Ising Problem. I. The Rectangular Lattice
View Description Hide DescriptionAn algebraic proof is given showing the equivalence of the Pfaffian and Onsager methods of solution of the Ising problem for the two‐dimensional rectangular lattice with free edge conditions. With cyclic conditions on the rows and columns and with helical edge conditions it is shown how the two solutions differ. The relation between the appearance of crossed long‐range bonds and the appearance of unwanted negative signs in the Pfaffian method is shown explicitly for this particular lattice.

Improved Self‐Consistent Configuration Interaction
View Description Hide DescriptionUsing general coupling operators the configuration interaction method is reformulated. The new formulation proves that the configuration interaction procedure leads to self‐consistent results for the set of configurations chosen and makes it possible to include a self‐consistent contribution of the continuum in practical calculations.

Covariant Higher‐Spin Equations
View Description Hide DescriptionCovariant wave equations are derived for nonzero‐mass particles of arbitrary spin, with wave‐functions that involve no redundant components. The Dirac equation is seen to be the first of these, for spin ½; the Proca vector mesonequations for spin 1 are the next set. The derivation is based on synthesizing higher‐spin particles from particles of lower spin.

Position and Intrinsic Spin Operators in Quantum Theory
View Description Hide DescriptionThe problem of defining position and intrinsic spin operators in terms of the generators of the inhomogeneous Lorentz group is considered. These operators are taken to have the properties usually attributed to them in nonrelativistic quantum theory: their commutation relations must have the commonly accepted form. These commutation relations are then shown to define the intrinsic spin uniquely. The position operator is shown to be also essentially uniquely determined. Explicit forms of the spin and position operators for some special representations are exhibited. The relation of these operators to the spin and position operators of the spin‐½ Diractheory in the Foldy‐Wouthuysen representation is considered. An Appendix gives a heuristic derivation of the spin operator.

On the Number of Bound States of a Given Interaction
View Description Hide DescriptionA method for deriving bounds to the number of bound states of a given interaction is built up. The class of interactions for which the method works includes the nonlocal interactions besides the local ones. Problems with many channels and spin‐dependent interactions can also be treated. Some bounds are explicitly given.

Perturbation Theory of Microwave Interaction with Gyroelectric Plasmas
View Description Hide DescriptionAs a means of determining the effects of a uniform but arbitrarily directed magnetic field on cylindrical and spherical wave propagation in a cold, homogeneous plasma, we regard the magnetic field as a small perturbation. Assuming an expansion for the electric and magnetic fields in powers of the parameter iω_{g}/ω, where ω_{g} is the static gyrofrequency of the electron, we solve for the linear terms. This solution is carried out under the assumption that the fields are known for the limit of vanishing static magnetic field.
The first‐order theory is then applied to cylindrical and spherical systems. When the approximate solution for the axially magnetized column is compared with the exact result, agreement is obtained provided that the static magnetic field is weak, as expected.

Summation of Partial Waves
View Description Hide DescriptionFor high‐energy potential scattering, when the partial wave expansion converges too slowly to be directly useful, a technique is proposed for extracting the differential cross section more effectively from the phase shifts by means of a weight function and orthogonal polynomial approach.

Hilbert Spaces of Type S
View Description Hide DescriptionCertain Hilbert spaces of generalized functions are examined. They contain the space of tempered distributions, are invariant with respect to the Fourier transformation and contain functions that increase rapidly at infinity.

Continuous‐Representation Theory. V. Construction of a Class of Scalar Boson Field Continuous Representations
View Description Hide DescriptionA large class of continuous representations of separable Hilbert spaces is constructed with the aid of representations of the canonical commutation relations (CCR) for a scalar boson field φ(f) and its canonical conjugate π(g). A representation of the CCR for a scalar boson field consists of two operator‐valued functionsV[g] and W[f], defined for all f, g in Schwartz's space S of real test functions, where V[g] and W[f] are unitary operators defined on some separable Hilbert spaceH, and which satisfy the commutation relations V[g]W[f] = e ^{−i(f,g }) W[f]V[g]. These unitary operators are related to the field and its momenta by V[g] = e ^{−iπ(g)}, W[f] = e ^{ iφ(f)}. We explicitly construct a family of such representations with the help of von Neumann's theory of infinite direct products of Hilbert spaces, the pertinent parts of which are reviewed. A continuous representation H of the Hilbert spaceC is composed of a linear vector space of complex, bounded, continuous functionals defined on S × S. These functionals are defined for all . In this definition, Φ_{0} is a fixed unit vector in H. The properties of the functions in C depend on the choice of the representation of the CCR and on the choice of Φ_{0}. When C is constructed with the aid of an irreducible representation of the CCR, an inner product can be defined for all pairs of functionals in C by an intuitively meaningful, rigorously defined integral in the sense of Friedrichs and Shapiro. With this inner product, C is a complete Hilbert space congruent with H. As in all continuous representations, a reproducing kernel exists and determines the functions in the continuous representation. One such space is closely related to a space of analytic functionals introduced by Segal and Bargmann. The representation of various operators as kernels and as functional derivatives is discussed. Finally, the construction of a vast number of unitary invariants for a representation of the CCR is used to establish the unitary inequivalence of uncountably many of the representations that we construct.

The Structure of Space and the Formalism of Relativistic Quantum Theory. I
View Description Hide DescriptionThe structure of a Euclidean space can be approached, with an unlimited accuracy, by a part of a maximally ordered finite linear space. Accordingly, all the physical theories based on the space‐time continuum can also be considered in such a finite space‐time. The finiteness of the underlying space makes also some new kinds of theories possible. Among them is a purely group theoretical formalism of relativistic quantum theory, including a free‐particle theory as well as a group formalism of interaction of particles. The free‐particle formalism of a finite space‐time is considered here (Part I). An essential difference in comparison with the formalism of continuous space‐time is that there is, as a consequence of the relations of Euclidicity to be imposed on observable 4‐vectors, a nontrivial spectrum of momentum, mass, and energy in a finite geometry.

Invariant Approach to the Geometry of Spaces in General Relativity
View Description Hide DescriptionA procedure is described for obtaining a complete, invariant classification of the local, analytic geometries and matter fields in general relativity by a finite number of algebraic steps. The approach is based on an extension of the classification scheme to include differential invariants of all orders and to provide maximally determined standard frames of vectors at each point. It is further shown that the resultant invariant functions can be replaced, in a finite number of algebraic steps, by special invariant functions which, while still uniquely representative of the geometry, can be assigned arbitrarily to produce all possible local, analytic solutions to the Einstein equations, in this representation. It is suggested that this type and special function scheme, obtainable from ideal geometric measurements in a finite number of steps, could be useful in general relativity. Unfortunately, due to the extensive algebra involved, this scheme has not yet been explicitly calculated, even for empty spaces.

Solution of Einstein's Field Equations for a Rotating, Stationary, and Dust‐Filled Universe
View Description Hide DescriptionThe solution of Einstein's field equations,, for a line element of the form is found. The density, ρ, may be a function of position, and the cosmological constant λ is not necessary in order to have a finite density. The solution reduces to that of Gödel if the variable α is constant. If the requirement for an empty universe is made (R_{ij} = 0), the solution is conformally flat. The characteristics of the conformal curvature tensor are also obtained.

Method for Treating Fluctuation of Dynamical Variables
View Description Hide DescriptionAn upper bound, on the probability that a fluctuation of a given size will occur in given time interval, has been derived. The bound is useful because it is small for some cases of interest. The fluctuations of the kinetic energy in a canonical ensemble have been considered as an example.

Scalar Invariants of a Rotational System in a Lie Algebra
View Description Hide DescriptionWe consider matrix equations of the form d W/dz = [S, W], where S is a matrix function of z that is embedded in a given Lie algebraL‐i.e., it is a curve in L. If the initial condition on W is in L, then W describes a curve in L. If the Killing form is used as a metric on L, then the behavior of the system is a pure rotation about an axis that is a function of z. A set of scalar invariants of such a system is obtained. These invariants form a set of conservation laws that the system obeys regardless of the detailed behavior of S(z).
If S describes a curve in some L _{1}, which is a semisimple subalgebra of the algebraL in which the whole system is embedded, then we can split the initial condition into two parts, one of which is in L _{1} and generates a solution in L _{1}, the other in a subspace that is orthogonal to L _{1} and that generates a curve that remains in the subspace. We can, then, obtain conservation laws that apply separately to the two parts.
The results have application to quantum mechanics since the density matrix obeys this type of equation. They also have application to coupled mode theory if we use, instead of the vector the corresponding power density spectrum matrix or the like.

Formal Solution of a Nonhomogeneous Differential Equation with a Double Transition Point
View Description Hide DescriptionA formal solution of a fairly general nonhomogeneous, linear, second‐order differential equation with a large parameter and a double transition point is presented. This equation arises, for instance, in the quasilinear theory of pressurized membrane shells of revolution. The fact that the solution which is of foremost importance in practical applications converges at infinity makes it convenient to use a direct approach, avoiding any transcendental transformation. The solution is described by means of influence functions which arise from a formal inductive process. The more important influence functions are tabulated. The results of two approximate asymptotic procedures are compared with the exact solution.

Complete High‐Energy Behavior for Certain Planar Graphs
View Description Hide DescriptionThe results obtained by Polkinghorne for the set of ladder diagrams is generalized to a certain set of planar graphs. The leading asymptotic term behaves as s ^{−1}(ln s)^{ p }, and then the complete set of terms s ^{−1}(ln s)^{ m } is summed over m. The final result allows the writing of an equation for the Regge trajectory function.

Unitary Symmetry of Oscillators and the Talmi Transformation
View Description Hide DescriptionThe Hamiltonian of an isotropic harmonic oscillator is invariant under unitary transformations in three dimensions. This well‐known invariance is exploited in a treatment of the Talmi transformation, viz., the transformation of two‐particle oscillator functions to center‐of‐mass and relative coordinates. A simple and transparent form of this transformation in terms of rotation matrices and Wigner coefficients of SU _{3} is given. The calculation of these Wigner coefficients is described and the problem of degeneracies discussed.

Existence of Proper Modes of Helicon Oscillations
View Description Hide DescriptionIn this paper it is shown that the class of electromagnetic problems for which the operator i(∂/∂t) (where t denotes time) is self‐adjoint extends beyond problems involving only insulators and perfect conductors. The class includes problems in which the perfect conductor is generalized to a medium with antisymmetric resistivitytensor. The latter medium approximates media in which helicon waves can propagate. Helicon waves are known to propagate in good conductors in a strong magnetic fieldB _{0}; it will be found that two necessary conditions for self‐adjointness of the operator i(∂/∂t) are that the sample carrying helicons must not have a finite portion parallel to B _{0}, and it must be surrounded by a reflecting surface that prevents energy from escaping.