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Volume 9, Issue 11, November 1968

Spinor Structure of Space‐Times in General Relativity. I
View Description Hide DescriptionIn order to define spinor fields on a space‐time M, it is necessary first to endow M with some further structure in addition to its Lorentz metric. This is the spinor structure. The definition and the elementary implications of the existence of a spinor structure are discussed. It is proved that a necessary and sufficient condition for a noncompact space‐time M to admit a spinor structure is that M have a global field of orthonormal tetrads.

New Solutions of the Einstein‐Maxwell Equations from Old
View Description Hide DescriptionMethods are discussed with which one may derive theorems which allow one to generate new solutions of the Einstein‐Maxwell equations from old ones. The old solutions used to generate new ones must admit at least one nonnull Killing vector and may be required to satisfy other conditions, depending on the theorem derived. Examples of derivable theorems are shown; these theorems are used in turn to show how generation of new solutions is accomplished. Examples of the latter are shown, such as generation of Brill or electrified NUT space from the Schwarzschild solution, generation of a new twisted Melvin universe from flat space, and generation of a new generalization of the Ozsvath‐Schücking metric. Possible physical interpretations, uses, and extensions of this type of theorem are discussed.

Solution of the Equations of Radiative Transfer in a Free‐Electron Atmosphere
View Description Hide DescriptionThe radiative‐transfer equations are solved for an electron‐scattering stellar atmosphere as formulated by Chandrasekhar. The solution employs a transformation of the integro‐differential form of the transfer equations into singular integral equations for the angular intensities of the radiation field. The Milne problem is solved to illustrate the method. In addition, the relationship is found between the above method of solution and Case's normal‐mode expansion method. This leads to an alternate procedure for finding the normal‐mode expansion coefficients. As an example of the method, the constant distributed source problem is solved for a half‐space medium.

Connection between Spin, Statistics, and Kinks
View Description Hide DescriptionSufficiently nonlinear classical fields admit modes called kinks, whose number is strictly conserved in virtue of boundary conditions and continuity of the field as a function of space and time. In a quantum theory of such fields, with canonical commutation (not anticommutation) relations, kinks and their conservation still persist, and even if the intrinsic angular momentum is an integer, a rotating kink can have half‐odd angular momentum, if double‐valued state functionals are admitted. We formulate a natural concept of exchange appropriate for kinks. The principal result is that for fields with integer‐valued intrinsic angular momentum, the observed relation between spin and (exchange) statistics follows from continuity alone, parastatistics being excluded. It is likely that in the theories with even (odd) exchange statistics, suitable creation operators will commute (anticommute). We show that, while the rotational spectrum of a kink will in general possess both integer and half‐odd spin states, in fields with integer‐valued intrinsic angular momentum only one of these two possibilities will ever be observed for each kind of kink, and that there is a nonzero ``particle number'' (strictly conserved, additive, scalar quantum number) attached to half‐odd‐spin kinks of each kind. It then follows that a boson and a fermion kink will always differ in at least one particle number, as well as in spin, and that, in particular, every fermion kink will have some nonzero particle number. These results are consistent with the hypothesis that the spinor fields usually employed to describe half‐odd‐spin quanta are not fundamental, but are useful ``point‐limit'' approximations to operators creating or annihilating excitations in a nonlinear field of particular kinds of kinks in particular internal states.

Principle of Compensation of Dangerous Diagrams for Boson Systems. I. Maximum Overlap
View Description Hide DescriptionThe principle of compensation of dangerous diagrams (PCDD), used by Bogoliubov to determine the coefficients in the canonical transformation to quasiparticles in boson systems, is obtained by maximizing the overlap between the true ground‐state vector and the quasiparticle vacuum state. The zero‐momentum state is treated exactly, which implies that the sum of all diagrams leading from the vacuum to a one‐quasiparticle state must be zero, in addition to the diagrams leading from the vacuum to the two quasiparticle state. Other criteria, such as the diagonalization of the quasiparticle reaction operator up to terms cubic in the operators, and the absence of one and two quasiparticle contributions to the true ground‐state wavefunction, are also shown to lead to the PCDD. A generalization of the Hartree procedure of minimizing the ground‐state energy is obtained by replacing the bare quasiparticle interaction with the quasiparticle reaction operator, and is shown to be equivalent to the PCDD. Finally, a perturbation expansion of the PCDD is obtained, and the reducibility of diagrams is discussed.

Principle of Compensation of Dangerous Diagrams for Boson Systems. II. Minimum Quasiparticle Number
View Description Hide DescriptionThe principle of compensation of dangerous diagrams (PCDD) postulated by Bogoliubov to determine the coefficients in his canonical transformation in boson systems is obtained from four criteria: (1) the number of quasiparticles in the true ground state is a minimum; (2) the ``best'' approximation to the true density matrix and pair amplitude is made; (3) the expectation value of an arbitrary operator is diagonalized up to terms cubic in the quasiparticle operators; (4) the most convenient starting point for dressing the quasiparticles is used. QuasiparticleGreen's functions are introduced to obtain a diagrammatic expansion of the PCDD. The reducibility of the diagrams is also discussed.

Relativistic Stochastic Processes
View Description Hide DescriptionRelativistic stochastic processes in μ‐space are defined and studied in a completely (and manifestly) covariant manner, without particularizing the time variable. It is shown that a number of usual definitions such as ``Gaussian process,'' etc., cannot be given a fully invariant meaning. Markovian processes are also studied. We find anew, as a particular case, results already obtained by Łopuszaǹski [Acta Phys., Polon. 12, 87 (1953)] in the case of Markovian processes in Minkowski space‐time. Several suggestions are made to generalize these results.

Invariants in the Motion of a Charged Particle in a Spatially Modulated Magnetic Field
View Description Hide DescriptionIn this paper we study the effect of a spatial modulation in the magnetic field on invariants such as the magnetic moment. In particular, we investigate whether an invariant still exists when the wavelength of the modulation is comparable to the gyro radius of the particle. In an axially symmetric magnetic field, with a square‐wave modulation, the orbit equations reduce to algebraic relations convenient for numerical study. We find from such studies that orbits are of two types: (a) regular orbits which generate an invariant; (b) orbits which are quasi‐ergodic. We have also calculated an invariant by perturbation theory with the depth of modulation of the field as a small parameter. For this we developed a modified form of perturbation theory which overcomes the difficulty of infinities arising at resonance between the perturbation and the cyclotron period. This difficulty in fact corresponds to a change in topology of the invariant curves. The invariant calculated from this theory shows very good agreement with the numerically computed orbits of type (a). The transition to quasi‐ergodic behavior cannot be predicted analytically, but some indication of it may exist in the complex topology of the invariant curves in the ergodic regions.

Kinematic Singularities of Helicity and Transversity Amplitudes and Asymptotic Regge‐Pole Contributions
View Description Hide DescriptionThe crossing relation for transversity amplitudes is used in a simple proof of the kinematic‐singularity structure of helicity amplitudes, both in the general‐mass and in various particular‐mass cases. A comparison with earlier results is made. The kinematic properties of helicity and transversity amplitudes are employed in the derivation of an asymptotic expression for the contribution of Regge poles to the amplitude.

Spin‐Orbit Interactions of the Configuration l ^{ n−1} l′l″
View Description Hide DescriptionMatrix elements for the spin‐orbit interactions of the configuration l ^{ n−1} l′l″ in L‐S coupling are expressed as linear combinations of radial integrals.

Local Field Theory and Isospin Invariance. II. Free Field Theory of Arbitrary Spin Particles
View Description Hide DescriptionThe properties of free field theories of arbitrary spin and isospin particles are investigated. Self‐conjugate isofermion (I = ½, , , ⋯) field theories of arbitrary spin S are shown to be nonlocal, with commutators (or anticommutators) failing to vanish outside the light cone. Special attention is given to discrete transformations C, I, and S. For self‐conjugate multiplets the parity,charge conjugation, and time‐reversal phase factors η_{ C }, η_{ P }, η_{ T } are not arbitrary but obey , , , where the phase η_{α} = ξ(−1)^{ I+α} (ξ = 1, α = −I, ⋯, +I) arises when the complex‐conjugate representations of SU(2) are transformed to the standard basis. Composite products of C, I, and S are discussed, with general phases and attention to the dependence on order of the operators. The six possible CIS operations Θ_{ i } are analyzed. For pair‐conjugate multiplets, the operator is, in general, a gauge transformation, with phase depending on the C, I, S phases and the order in which C, I, and S enter into Θ_{ i }. By postulating that be independent of the order of C, I, and S, we derive the Yang‐Tiomno parity factors (±1, ±i) for pair‐conjugate fermions. For self‐conjugate multiplets, however, the phase restrictions previously stated lead to a unique order‐independent result . The usual local field theory result lacks the factor (−1)^{2I }. The latter differs from unity only for the case 2I = odd integer, in which case we are in fact dealing with a nonlocal field theory. The technical details of the theory involve construction of local fields from helicity wavefunctions and helicity particle operators. Fields of spin greater than one are described by the Rarita‐Schwinger formalism. An appendix treats in detail the form and properties of the high‐spin wavefunctions in the helicity basis.

Variational Approach to Multiple Scattering
View Description Hide DescriptionUsing a variation technique, two coupled equations are derived whose transition matrix is shown to describe multiple scattering. The resulting index‐of‐refraction equation differs from that obtained in previous theories by our inclusion of the effect of the scatterer on the target and certain constants of approximation.

Narrow Resonance Saturation of Sum Rules: A Nonuniqueness Theorem
View Description Hide DescriptionIt is shown that, if the superconvergence relations appertaining to a two‐particle scattering process are saturated by an infinite tower of resonances of mass m_{J} , where J is the spin, then m_{J} must increase less quickly than J in all cases. With this observation, it is shown that the complex of all the superconvergence relations for all the processes a(J _{1}) + a(J _{2}) → a(J _{3}) + a(J _{4}), where a(J) is the spin‐J member of the same tower of particles, possesses an infinity of solutions for the coupling constants. It is concluded that the superconvergence relations are incomplete, and need to be supplemented by some new physical principle, if they are to be of any practical use. A possible exception to this rule is when all but a finite number of residues are constrained to be positive, with no negative coefficient from the isospin crossing matrices. In this case, a given solution may be unique if m_{J} increases not more quickly than J ^{½}.

One‐ and Two‐Dimensional Green's Functions for Electromagnetic Waves in Moving Simple Media
View Description Hide DescriptionThis paper considers the one‐ and two‐dimensional Green's functions associated with electromagnetic radiation in a moving medium. The medium is assumed to be lossless, to have constant permittivity and permeability, and to move with constant velocity with respect to a given inertial coordinate system. For each source (line source or plane source) both the time‐dependent and the harmonic Green's functions are found. The results are obtained by means of special transformations which convert the equations to be solved into standard forms. The significance of the results, in terms of one‐ and two‐dimensional Cerenkov radiation, is discussed.

Diffusion Length and Criticality Problems in Two‐ and Three‐Dimensional, One‐Speed Neutron Transport Theory. I. Rectangular Coordinates
View Description Hide DescriptionSeveral problems in linear transport theory have been solved involving more than one dimension. We have considered the diffusion‐length problem in a slab system and have studied how the flux in the transverse direction varies as a function of distance from the source. The solution is found to consist of an asymptotic term, composed of a finite number of harmonics, plus a transient which is nonseparable in the x and z coordinates. When the slab is sufficiently thin only the transient term survives and the concept of a diffusion length loses its value. A method for overcoming this difficulty is presented. In another problem, the thickness of the slab is allowed to become semi‐infinite and the effect of decay in the z direction on the emergent angular distribution and surface flux are assessed. By approximating the flux in the y and z directions by a function of the form exp {iB_{y}y + iB_{z}z} and solving the resulting one‐dimensional problem in the x direction exactly, it has been possible to obtain a statement of the critical conditions in a bare rectangular parallelepiped system. The application of this method to the diffusion length problem in a system with a rectangular cross section is discussed. Finally, by comparison with the exact solution for the slab, we estimate the accuracy of a reduced Boltzmann equation deduced by the author in a previous publication.

Diffusion Length and Criticality Problems in Two‐ and Three‐Dimensional, One‐Speed Neutron Transport Theory. II. Circular Cylindrical Coordinates
View Description Hide DescriptionAn integral equation is derived which describes the diffusion of neutrons from a plane source perpendicular to the axis of an infinite circular cylinder. The equation is solved by a Fourier transform in the axial direction and by the subsequent solution of a singular integral equation for the radial distribution; this is accomplished as a result of a certain replication property possessed by the original integral equation. We have inverted the Fourier transform to obtain the simultaneous r − z variation of the neutron density in the cylinder. In addition, a formula is given from which the angular distribution of neutrons may be obtained. By approximating the density along the axis of the prism with the asymptotic distribution, it has been possible to assess the effect of axial leakage on the cylindrical critical problem. A critical equation is derived and the appropriate extrapolated endpoints to be used in this are defined.

Strong‐Coupling Limit in Potential Theory. II
View Description Hide DescriptionThe analytic properties of the Jost function in the coupling constant g, derived in an earlier article for a restricted class of potentials, are rederived generally for potentials V(r), for which V(r) can be bounded by a monotonically decreasing potential V̄(r), such that In particular, potentials with nonlinear exponential tails are studied as a function of the energy and coupling constant. Earlier results of Sartori on the energy dependence of the Jost function, derived only for special cases of such potentials in the Born approximation, are demonstrated to be true generally for large classes of such potentials.

Derivation of Nonrelativistic Sum Rules from the Causality Condition of Wigner and Van Kampen
View Description Hide DescriptionNonrelativistic sum rules previously obtained for each phase shift in the framework of elastic scattering by a local central potential of finite radius are shown to follow from the causality principle of Wigner and Van Kampen, under the assumptions that the phase shift satisfies the Levinson theorem and that, at high energies, the integrability condition holds. In fact, it is shown that under these assumptions, any interaction of finite radius which is causal in the sense of Wigner and Van Kampen is equivalent to a local potential of the same radius. First we recall the sum rules and mention some of their applications. We give a brief survey of their proof in potential scattering, which is based essentially on the analytic and asymptotic properties of the Jost function (or the S matrix). Then we show that, under the assumptions mentioned above on the phase shifts, the properties of the R matrix derived by Wigner and Van Kampen from causality lead to the same analytic and asymptotic properties of the Jost function (defined now directly from the S matrix) as in potential scattering, providing, therefore, sufficient information for the direct derivation of the sum rules. Finally, using the Gel'fand‐Levitan and Marchenko integral equations of the inverse‐scattering problem, we show that, in fact, this information is sufficient to entail that the causal interaction of Wigner and Van Kampen is equivalent to a local potential of the same radius.

Asymptotic Behavior of Certain Nonlinear Boundary‐Value Problems
View Description Hide DescriptionThe asymptotic properties of a class of nonlinear boundary‐value problems are studied. For large values of a parameter, the differential equation is of the singular‐perturbation type, and its solution is constructed by means of matched asymptotic expansions. In two special cases, very simple approximate analytic solutions are obtained, and their accuracy is illustrated by showing their good agreement with the exact numerical solution of the problem.

Microscopic Approach to Kinetic Theory: Inhomogeneous Systems
View Description Hide DescriptionThe microscopic linearized Vlasov equation is solved in terms of a generalized inverse dielectric function ε^{−1}(r, r′; t, t′) and the initial phase‐space density fluctuation. This expression is then used to calculate the density autocorrelation function and to obtain a generalized kinetic equation for plasmas and gravitational gases. It is shown that many of the results for the inhomogeneous system have a close similarity to the corresponding results for homogeneous systems. In particular, the test‐particle theory is exhibited and an expression is obtained for the phase‐space density of the polarization cloud associated with a test particle.