Volume 8, Issue 7, July 1967
Index of content:

Approach to Gravitational Radiation Scattering
View Description Hide DescriptionA method is presented for studying asymptotically flat spaces possessing both incoming and outgoing gravitational radiation at infinity. The method uses multipole expansions and the invariance of general relativity under time reversal; calculations are facilitated by a small‐parameter perturbation approach. Some calculations are carried out to second order to show the practicability of the method.

Canonical Form Factors and Current Commutation Relations
View Description Hide DescriptionStarting with the vector (or axial vector) currentsj ^{μ} and the momentum operators P ^{ν}, we define the canonical operator j̃, such that instead of the four covariantly transforming components of j, we have a scalar j̃ ^{0} and three other components j̃ undergoing Wigner rotations under Lorentz transformations. We first give a construction of j̃ explicitly in terms of j and P. But, since the transformation properties are not quite the most convenient ones, a subsequent generalized definition, leading to a convenient canonical parametrization of the matrix elements of j̃, is introduced. We then study the physical significances of the canonical form factors thus obtained. For vector currentsj̃^{v} the transformation properties correspond to a separation of the physical charge (j̃ ^{0}) and magnetic (j̃) form factors in any frame (and not only in Breit frame as for j). For nonconserved axial currents we relate the matrix elements of j̃ ^{0A } with mass‐difference effects and express the partial conservation condition in terms of the canonical form factors. We then study in detail the application of our formalism to the limiting case of infinite momentum and small momentum‐exchange, as often introduced in the study of current algebras. Next we give explicitly the canonical form factors for photoproduction processes. In the last section we study the possibility of constructing a canonical spin operator directly in terms of the vector and axial vector charges and the consequences for the ``inner orbital'' contribution to be added to obtain the total spin of a composite particle. Some useful formulas are collected in Appendixes A and B.

Remarks on Relativistic Statistical Mechanics. II. Hierarchies for the Reduced Densities
View Description Hide DescriptionIn Paper I the basis of relativistic statistical mechanics was discussed and notions such as phase space, Gibbs ensemble, distribution functions were defined. Paper II deals with hierarchies of equations for the reduced densities. Extensive use is made of the beautiful methods of Klimontovich. In this way ``classical mesic interactions'' are dealt with neglecting ``radiation'' effects (i.e., ``classical'' emission of mesons). It is shown how the ``renormalization of mass'' affects the hierarchies obtained. Electromagnetic interactions are dealt with: (a) neglecting radiation effects, (b) including radiation effects. The latter case is treated on the basis of the Lorentz‐Dirac equations and with the help of suitable modifications of the formalism. In this way a new approach to radiation phenomena (for instance, in plasmas) is obtained. Finally, as a matter of illustration, several well‐known relativistic kinetic equations are rederived in a slightly improved manner (i.e., Vlasov and Landau equations).

Time Behavior of a Reactor and Ergodic Theory of Semigroups
View Description Hide DescriptionThe solution of the time‐dependent neutron transport equation is a semigroup of linear transformations acting on a Banach space. There are some ergodic theorems that can be used to describe the asymptotic behavior of the solution under very general conditions on the semigroup. The results are compared with Wing's famous approach.

Upper Bounds to Eigenvalues of the One‐Dimensional Sturm‐Liouville Equation
View Description Hide DescriptionThis work develops a straightforward technique for giving an upper bound to any eigenvalue of the one dimensional Sturm‐Liouville problem. It is shown that any trial function that fulfills the proper boundary condition of the problem and possesses the same number of nodes as an exact eigenfunction of the problem can provide an upper bound to that eigenfunction's eigenvalue. Application of the above technique is made to provide a one‐sided bound to quantum mechanical scattering phase shifts.

Logarithmic Density Behavior of a Nonequilibrium Boltzmann Gas
View Description Hide DescriptionWe consider the temporal evolution of the BBGKY hierarchy in the Boltzmann approximation for spatially homogeneous nonequilibrium situations, in the absence of initial correlations. For times of the order of the mean free time or greater, the single particle function f _{1} is found to be of the formwith (n is the density, r _{0} the range of binary interaction) and , , and of order unity. For times less than the mean free time, with t in units of the duration of a binary interaction, f _{1} is of the form.In both cases the same formally higher‐order binary correlation functions are neglected.

New Approach to the Ising ModeI. II
View Description Hide DescriptionOnsager's results for the partition and correlation functions for the Ising model on a two‐dimensional rectangular lattice are rederived using a Green's function technique. The definition of the Green's function is based on a recently published casting of the Ising model into a many‐body fermion problem. The relation between this approach and other methods used for solving the Ising problem are indicated.

Relation Between ``Outer'' and ``Inner'' Multiplicity for SU(2) and SU(3)
View Description Hide DescriptionThe Clebsch‐Gordan series for SU(3) is given in terms of irreducible representations of SU(3) such that the ``outer'' multiplicity of the Clebsch‐Gordan series is related to the ``inner'' multiplicity of irreducible representations.

Unitary Representations of the Lorentz Group and Particle Dynamics
View Description Hide DescriptionWe examine the quantum dynamics of particles with arbitrary spin implied by explicit use of the infinite dimensional unitary representation of the Lorentz group, introduced by Majorana. Comparison with the classical theory of spinning particles shows that this unitary representation leads to a quantum mechanical analog of the relativistic pure gyroscope.

Electromagnetic Behavior of Superconductors with Impurity Scattering
View Description Hide DescriptionA general equation for the electromagnetic current J of a superconductor responding to a frequency‐dependent external electromagnetic field and under the influence of impurity scattering and temperature has been derived in momentum space using the Green's function method. This general equation can be regarded in two ways. (a) It is the Fourier transform, from coordinate space to momentum space, of the equation of Mattis and Bardeen [D. C. Mattis and J. Bardeen, Phys. Rev. 111, 412 (1958), Eq. (3.3)]. (b) It summarizes, in one equation, the previous works of Abrikosov et al. on the electromagnetic behavior of superconductors. In addition to these general properties, this equation also shows, in particular, that for a superconducting alloy with a few percent of impurity concentration, the kernel for the current is the same as that of a Pippard pure superconductor, with , where v is the Fermi velocity and k is the momentum exchange during collision replaced by τ_{ tr }, the transport collision time, valid for all temperatures up to T_{c} , the transition temperature. Expressions of the current in closed form for both superconducting alloys and normal metal with impurity are also given for vk ∼ 1/τ.

A Unifying Principle in Statistical Mechanics
View Description Hide DescriptionA fundamental problem of statistical mechanics is to obtain simplified descriptions of complex systems. A general principle is presented for obtaining equations of motion for such descriptions. The principle involves maximizing an appropriate entropy functional. It also involves the particle dynamics through the Liouville equation. Various special cases are presented in which the principle yields the Vlasov equation, the Boltzmann equation, Euler's hydrodynamicequations, a generalization of Grad's ten‐moment approximation, the Gibbs distribution (i.e., equilibrium statistical mechanics), and Onsager's equations of irreversible thermodynamics. The principle also yields, trivially, the Liouville equation and Hamilton's equations of classical mechanics. Some of these results have been derived elsewhere by very similar procedures, but apparently the generality of the principle has been unrecognized. In terms of the general principle, the origin of irreversibility in the various equations of motion is easily seen, and the relation between the numerous definitions of entropy is clarified. No a priori justification of the principle itself is given.

Integral Representations of Invariant States on B* Algebras
View Description Hide DescriptionLet be a B* algebra with a group G of automorphisms and K be the set of G‐invariant states on . We discuss conditions under which a G‐invariant state has a unique integral representation in terms of extremal points of K, i.e., extremal invariant states.

Null Electromagnetic Field in the Form of Spherical Radiation
View Description Hide DescriptionAn example is given of a differential from which leads, through Einstein's gravitational equations, to an energy tensor representing a null electromagnetic field in the form of spherical radiation.

Stress‐Tensor Commutators and Schwinger Terms
View Description Hide DescriptionWe investigate, in local field theory, general properties of commutators involving Poincaré generators or stress‐tensor components, particularly those of local commutators among the latter. The spectral representation of the vacuum stress commutator is given, and shown to require the existence of singular ``Schwinger terms'' at equal times, similar to those present in current commutators. These terms are analyzed and related to the metric dependence of the stress tensor in the presence of a prescribed of a prescribed gravitational field and some general results concerning this dependence presented. The resolution of the Schwinger paradox for the T ^{μν} commutators is discussed together with some of its implications, such as ``nonclassical'' metric dependence of T ^{μν}. A further paradox concerning the vacuum self‐stress—whether the stress tensor or its vacuum‐subtracted value should enter in the commutators—is related to the covariance of the theory, and partially resolved within this framework.

Note on the Kerr Metric and Rotating Masses
View Description Hide DescriptionKerr's metric is often said to describe the geometry exterior to a body whose mass and rotation are measured by Kerr's parameters m and a, respectively, even though no interior solution is known. In this paper we give an interior solution valid in the limit when the rotation parameter a is sufficiently small so that terms of higher power than the first are negligible, but the mass parameter m is allowed to be large. This is accomplished by bringing Kerr's exterior metric into the form of the metric for a slowly rotating mass shell. Also, the connection is found between Kerr's parameters and the physical parameters characterizing the rotating body.

Particlelike Solutions of a Class of Nonlinear Field Equations
View Description Hide DescriptionA procedure for generating integral relations satisfied by particlelike solutions of the class of nonlinear field equations Δφ = F ^{1}(φ) is proposed and used to develop the first few such relations. The relations are used to reduce the expression for the ``energy'' associated with the system; as an example, the case F ^{1}(φ) = φ − φ^{3} is treated. It is shown generally that the variational bound to any of the possible energy values will approach the exact value from above if the trial functions are chosen to satisfy one of the integral relations, which is satisfied identically by the exact solutions.

Quantum Correlation Function for the Noninteracting Particle System
View Description Hide DescriptionA method for determing the quantum correlation functions of the noninteracting particle system in thermal equilibrium is developed. It is designed to reduce the labor involved in treating the large number of permutation operators of the symmetric group that occurs. An alternate form of the n‐particle correlation function is obtained in order to simplify computation. The London‐Placzek formula is derived as a check. The error in Kirkwood's superposition approximation for this system is investigated, and the exact relationship between the two‐ and three‐particle correlation functions is found. Finally, a method determining the pair correlation function in the Hartree‐Fock approximation of a pair‐interacting particle system at a finite temperature is presented.

Characteristic Hypersurfaces in General Relativity. I
View Description Hide DescriptionLightlike (null) hypersurfaces are treated by means of an intrinsic Ricci rotation coefficient technique. This provides an effective way of dealing with the various types of geometry on a null hypersurface. The formalism is used to examine inner affinities, differential invariants, local features such as asymptotic and shear directions and geodesic lines, and to give a short description of null hypersurfaces in flat space‐time. Applications to gravitational radiation theory and cosmology are briefly mentioned.

Decomposition of Tensors of the Classical Groups
View Description Hide DescriptionA tensor symmetrization procedure obtained in a recent publication [Phys. Rev. Letters 16, 1058 (1966)] is shown to support rather than disprove Weyl's tensor symmetrization theorem. This ``extended'' symmetrization procedure differs from Weyl's approach in that to construct a subspace irreducible under GL(n, c) one starts with a set of formal states (symmetrized tensors with formal index values) spanning an irreducible representation of the permutation group rather than starting with a single formal state. Extended symmetrization is often more useful than Weyl's approach because the states obtained are highly organized and because it also yields an efficient independent state selection method for the symmetrization procedures using modified Young symmetrizers and Wigner projection operators. The state organization obtained makes it possible to show that the nonorthogonality which is present for bases obtained with Young symmetrizers can be easily removed. The state organization also makes it possible to simplify the task of recoupling symmetrized tensor representations to gain a simply‐coupled form. This form enlarges the class of Clebsch‐Gordan and recoupling coefficients which can be evaluated by tensor methods. Group matrices and Lie group generator matrix elements are also obtained by tensor methods. Extended symmetrization using unitary representation Wigner projection operators based on unitary representations is shown to result in orthogonal states although usually not the orthogonal states desired. The usual Young symmetrizers are shown to often be more useful than modified Young symmetrizers or Wigner projection operators.