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Volume 8, Issue 2, February 1967

Induced and Spontaneous Emission
View Description Hide DescriptionThe problem of induced and spontaneous emission is investigated for an atomic two‐level system with incident beams of radiation which are either in a coherent state or in a stationary state (contain a definite number of photons). The treatment is fully quantum‐mechanical, and is confined to the case where the frequency spectrum of the incident beam is narrow compared to the natural linewidth of the system. It is shown that, under such conditions, the spontaneous emission for frequencies within the narrow band of the incident radiation is sharply reduced compared to the prediction of the natural lineshape. It is shown that a hole is burned in the natural lineshape within the narrow frequency band, thus effectively quenching the spontaneous emission at some frequency within the band. This effect is shown to occur both for the coherent and stationary beams. Quantities proportional to the induced and spontaneous probability amplitudes and the lifetimes are computed for times comparable to and long compared to the free lifetime of the state. An expression is found for the spectrum of the emergent radiation in terms of these quantities. Its physical meaning is briefly discussed. The density operator of the field for all times is given.

Discrete Series for the Universal Covering Group of the 3 + 2 de Sitter Group
View Description Hide DescriptionA classification is given of the irreducible unitary representations of the universal covering group of the 3 + 2 de Sitter group which contract to the usual physical representations of the Poincaré group. These representations include the discrete series for the 3 + 2 de Sitter group. The classification problem is reduced from one for the group to the corresponding one for the Lie algebra. The method used by Thomas for the representations of the 4 + 1 de Sitter group is then followed, except that a representation is reduced out with respect to the irreducible unitary representations of a noncompact 2 + 2 subalgebra. It is conjectured that each representation of this subalgebra occurs at most once. The action on the representation spaces of a basis for the Lie algebra is given. The contractions of the representations to those of the Poincaré, oscillator and the Galilei groups are briefly considered.

Structure of the Crossing Matrix for Arbitrary Internal Symmetry Groups. II. Matrices in SU (n)
View Description Hide DescriptionIt is shown that in SU(n) two distinct processes can give rise to identical crossing matrices in general. In SU(2) and SU(3), however, the crossing matrix corresponds to a unique scattering process, and thus the dynamical equations in the elastic channel are sufficient to discriminate between different processes. Some SU(n) crossing matrices, which generalize the work of Cook, Muturza, and Rashid, are given.

Note on the Robinson‐Trautman Solutions
View Description Hide DescriptionA procedure is given which enables one to construct all the type N Robinson‐Trautman solutions and an infinite class of type III solutions. Some approximate solutions, consistent with the Bondi‐Sachs radiation conditions for bounded source fields, are also given. These approximate solutions, obtained by perturbing the Schwarzschild solution, are Schwarzschild in the asymptotic future. It is also shown that there exists an infinity of exact solutions which are Schwarzschild in the asymptotic future.

Point‐Loop Renormalization in Regularized Field Theories. I
View Description Hide DescriptionThe relation between the augmented and the standard vacuum amplitudes is derived (for a finite‐valued amplitude for the point fermion‐antifermion loop) by diagrammatic methods. This relation is shown to consist of a mass renormalization, whereby the point fermion‐antifermion loop is absorbed into the fermion propagator, plus some additional factors, and the analytical properties of the amplitudes are greatly modified. The effects of the renormalization are the increase of the density of zeros, the increase of the exponential order in the variable z, and the change of the singularities in η from simple poles to essential singularities. The significance of these for massless fermions or bosons is briefly discussed.

Instantaneous Action‐at‐a‐Distance in Classical Relativistic Mechanics
View Description Hide DescriptionThe possibility of describing orbits in classical relativistic mechanics in instantaneous action‐at‐a‐distance fashion by second‐order differential equations (as in Newton's gravitational theory) is investigated with particular emphasis on the two‐body problem of classical relativistic electrodynamics. Differential conditions are stated to guarantee world‐line invariance and form‐invariance of the equations of motion under Lorentz transformation for such a description of an N‐particle system in three dimensions. A pair of integrodifferential equations for the equations of motion are derived to provide an explicit means of passing from a description via direct interaction along light cones to an instantaneous action‐at‐a‐distance description for a two‐body problem. These integrodifferential equations are applicable to the two‐body problem of classical electrodynamics with either retarded interactions and radiation damping or with half‐advanced plus half‐retarded interactions.

Functional Approach to Classical Non‐Equilibrium Statistical Mechanics
View Description Hide DescriptionA new approach to classical non‐equilibrium statistical mechanics is considered. The essential idea is to solve Bogoliubov's functional‐differential equation for the generating functional embracing all distribution functions for the particle system with the aid of the technique of functional integration. It becomes possible with a minor modification of his formalism. A general solution is given. The solution for our state functional is reduced to the Hopf characteristic functional for the Vlasov selfconsistent field when the interaction energy of particles is relatively small in each s‐body Hamiltonian H_{s} .

Series Representation Method for Obtaining the Emission Coefficient from the Integrated Intensity Distribution for Asymmetrical Light Sources
View Description Hide DescriptionA procedure for inverting the spectroscopicintegral equation which relates the emission coefficient to the integrated intensity distribution for an optically thin and asymmetrical light source is presented. In this procedure the emission coefficient is expanded in terms of a complete set of orthogonal polynomials which are ``invariant in form'' to a rotation of axes and use is made of the spectroscopicintegral equation to determine the unknown expansion coefficients in terms of known information for the integrated intensity distribution. In order to check its validity, the resultant series representation was summed exactly for a hypothetical example corresponding to an asymmetrical source whose emission coefficient possesses elliptical symmetry and diminishes with position in a Gaussian manner. Also the same hypothetical example was used to test the accuracy of the numerical procedure which was developed for summing the series representation in practical situations where the known information on the integrated intensity distribution is presented in the form of experimental data.

Relaxation to Equilibrium of a Dilute Electron Plasma
View Description Hide DescriptionThe relaxation of the velocity distribution to equilibrium in an electron plasma in which the dominant collisions are described by the Fokker‐Planck operator is studied. It is shown mathematically that the linearized collision operator possesses a continuous spectrum of eigenvalues which extends over the entire real interval from zero to infinity. Consequently the decay to equilibrium is not uniformly exponential. For large values of the time the decay to Maxwellian is shown to be of the order of the inverse power of the time variable. Moreover the rate of decay depends also on the initial perturbation.

Theory of the Linear Fokker‐Planck Collision Operator
View Description Hide DescriptionThe linearized Fokker‐Planck collision integral with coulomb interaction is expanded in terms of surface spherical harmonics. The radial part of the distribution function is shown to be governed by a set of decoupled differential‐integral equations. The differential operators are shown to be self‐adjoint while the integral operators are symmetric and completely continuous. The spectrum of the eigenvalues contains, on top of discrete points, a continuous part ranging from zero to minus infinity. The discrete part of the spectrum is obtained by requiring that the corresponding eigenvectors have a definite asymptotic behavior at large velocity. A variation principle is constructed for the computation of the discrete spectrum.

Relations between Field‐Plus Source and Fokker‐Type Action Principles
View Description Hide DescriptionThe problem of constructing a Fokker‐type action that will yield equations of motion for the sources of an original field‐plus‐source system is considered. It is shown that such an action can be constructed for a large class of systems by substituting for the fields appearing in the original field‐plus‐source action a solution of the field equations in terms of the source variables provided that certain conditions are satisfied. In general, these conditions will be satisfied only by half‐advanced, half‐retarded‐type solutions of the field equations and then only for a field‐plus‐source Lagrangian that differs from the usual one by a complete divergence. The problem of constructing a Fokker action for a system that possesses a gauge‐type covariance is complicated by the existence of differential identities that are satisfied by the left‐hand sides of the field equations and that arise as a consequence of the gauge covariance. These difficulties can be overcome by the introduction of gauge conditions on the field variables. The problem of finding solutions to the field equations suitably modified with the help of the gauge conditions that satisfy both the gauge conditions and the conditions for the construction of the Fokker action is discussed. As an application of the method, we have obtained the Fokker action of electrodynamics using a number of different gauge conditions, arriving thereby at several different expressions for this action. We have also considered the construction of a Fokker action when the field equations are solved by an approximation method. In such cases we have shown that to determine the Fokker action up to the nth order of the approximation it is only necessary to solve the field equations up to order ½(n + 2) or ½(n + 1) for n even or odd, respectively. The method is then applied to obtain the Darwin action of electrodynamics.

Maximal Analytic Extension of the Kerr Metric
View Description Hide DescriptionKruskal's transformation of the Schwarzschild metric is generalized to apply to the stationary, axially symmetric vacuum solution of Kerr, and is used to construct a maximal analytic extension of the latter. In the low angular momentum case, a^{2} < m^{2}, this extension consists of an infinite sequence Einstein‐Rosen bridges joined in time by successive pairs of horizons. The number of distinct asymptotically flat sheets in the extended space can be reduced to four by making suitable identifications. Several properties of the Kerr metric, including the behavior of geodesics lying in the equatorial plane, are examined in some detail. Completeness is demonstrated explicitly for a special class of geodesics, and inferred for all those that do not strike the ring singularity.

Difficulties in the Kinetic Theory of Dense Gases
View Description Hide DescriptionFor the determination of the transport coefficients of a dense gas, the long‐time behavior of the pair distribution functionF _{2} for small intermolecular distances is obtained from a density expansion in terms of the first distribution functionF _{1}. On the basis of the dynamics of small groups of particles, it is shown that this expansion contains divergences so that it cannot be used for (a) the computation of the long‐time behavior of F _{2} beyond O(n); (b) the demonstration of the decay of the initial state beyond O(n ^{2}). Similar divergences are encountered in the computation of the transport coefficients from time‐correlation functions. The nature of the divergences suggests (a) there is no kinetic stage in the approach of a dense gas to equilibrium, in the sense of Bogoliubov; (b) a weak logarithmic density dependence of the transport coefficients.

Microscopic Approach to Kinetic Theory
View Description Hide DescriptionA microscopic kinetic theory is developed for a plasma by the use of approximate equations of motion for the microscopic ``exact'' distribution function.These equations can be solved to obtain asymptotic expressions for f̂(r, v, t) that are then used to calculate correlation functions. These approximate equations are also used to obtain the approximate equations for the correlation functions and the principal results of the test‐particle approach. In particular, two sets of equations are constructed. One set describes a homogeneous, slowly varying system and yields the Balescu‐Lenard equation. The other set describes a homogeneous, slowly varying system that contains small, inhomogeneous and quickly varying perturbations.

Translational Invariance Properties of a Finite One‐Dimensional Hard‐Core Fluid
View Description Hide DescriptionA formalism is developed for expressing the n‐particle distribution functionsD_{n} (x _{1} ≤ x _{2} ≤ … ≤ x_{n} ) explicitly in terms of the configurational partition function for one‐dimensional fluids with hard‐core repulsive and nearest‐neighbor attractive forces. The translational invariance properties of the D_{n} functions are investigated for the case of no attractive forces when the system is finite. When the number density is less than half the close packing density, there exists a central region in which D _{1}(x) is constant and all the D_{n} functions, n ≥ 2, are functions of the (n−1) nearest‐neighbor separation distances. Several relevant theorems are proved and limiting cases are investigated.

Temperateness of the Absorptive Part of the Scattering Amplitude in the Proof of Dispersion Relations
View Description Hide DescriptionA proof is given within the context of the Bremermann, Oehme, and Taylor proof of dispersion relations that the absorptive part of the elastic scattering amplitude remains tempered throughout the relevant region of analytic continuation. The technique of this proof, which avoids the use of the Jost‐Lehmann‐Dyson representation, may be applicable to some field theories which do not assume local commutativity and for which such a representation cannot, therefore, be constructed.

Consistency Conditions on Models for High‐Energy Scattering
View Description Hide DescriptionFrom unitarity, analyticity in t, and analyticity in E, a variety of conditions can be derived for the high‐energy behavior of a crossing symmetric two‐body scattering amplitude F(E,t). These conditions are studied as consistency conditions for a class of models for high‐energy scattering based on smoothly varying functions. For this class of models they lead to: (1) the Froissart bound without making explicit use of Martin's enlargement of the Lehmann ellipse, (2) conditions on the phase of the amplitude and its derivative with respect to t in the forward direction. As an example of the use of the phase conditions, it is shown how the Lehmann ellipse can be enlarged to give analyticity in the circle t < R, where R is fixed. Although the method is different, this result is closely related to the general results of Martin, but with our smoothness assumptions a little more detail can be stated about the behavior of F(E,t).

Homogeneous Lichnerowicz Universes
View Description Hide DescriptionLichnerowicz has developed a general‐relativistic theory of an electrically charged fluid with infinite conductivity. No exact solutions of the corresponding Einstein‐Lichnerowicz equations have previously been given. The present paper establishes homogeneous solutions of these equations.

Twistor Algebra
View Description Hide DescriptionA new type of algebra for Minkowski space‐time is described, in terms of which it is possible to express any conformally covariant or Poincaré covariant operation. The elements of the algebra (twistors) are combined according to tensor‐type rules, but they differ from tensors or spinors in that they describe locational properties in addition to directional ones. The representation of a null line by a pair of two‐component spinors, one of which defines the direction of the line and the other, its moment about the origin, gives the simplest type of twistor, with four complex components. The rules for generating other types of twistor are then determined by the geometry. One‐index twistors define a four‐dimensional, four‐valued (``spinor'') representation of the (restricted) conformal group. For the Poincaré group a skew‐symmetric metric twistor is introduced. Twistor space defines a complex projective three‐space C, which gives an alternative picture equivalent to the Minkowski space‐time M (which must be completed by a null cone at infinity). Points in C represent null lines or ``complexified'' null lines in M; lines in C represent real or complex points in M (so M, when complexified, is the Klein representation of C. Conformal transformations of M, including space and time reversals (and complex conjugation) are discussed in detail in twistor terms. A theorem of Kerr is described which shows that the complex analytic surfaces in C define the shear‐free null congruences in the real space M. Twistors are used to derive new theorems about the real geometry of M. The general twistor description of physical fields is left to a later paper.