Volume 7, Issue 7, July 1966
Index of content:

Space Groups and Selection Rules
View Description Hide DescriptionA rigorous method is presented for obtaining the Clebsch‐Gordan coefficients for the Kronecker products of induced representations of finite groups. The result obtained is applied to the theory of space groups, and the final formula for the coefficients has the advantage, as with other subgroup techniques, of being expressed in terms of the characters of small representations, thereby providing an additional proof independent of Lax, yet supporting his point of view the subgroup and full‐group procedures are equivalent, and at the same time the result clarifies subgroup treatments so far given. An example using P23 is given.

Kinetic Equation for the Electron‐Phonon Gas
View Description Hide DescriptionThe random phase approximation equations of motion for the electron‐phonon system are solved to obtain long‐time expressions for the basic operators that appear in the calculation of correlation functions. These operator expressions are used to obtain a kinetic equation for the electrons and an expression for the phonon spectral function. The kinetic equation has the same form as the Balescu‐Lenard equation. Expressions for the density autocorrelation function and the two‐particle correlation function are also obtained. The latter is shown to have the same form as the test‐particle result for a classical plasma.

Majorization of Feynman Graphs
View Description Hide DescriptionThe different methods of majorization of Feynman graphs are reviewed and their respective domains of application and relative merits are discussed. The initial steps of Wu's majorization method are discussed in a variational formalism. Effort is made to present the rather complex arguments involved in Nakanishi's ``path method'' in a systematic manner.

Some Collapsing Cylinders and their Exterior Vacuum Metrics in General Relativity
View Description Hide DescriptionA cylindrical form of the Friedman metric is used to obtain nonstatic infinite cylinders of incoherent fluid. The vacuum metrics exterior to the cylinders are determined from the hypothesis that the metric tensor and its first partial derivatives be continuous. This hypothesis is applied by requiring the continuity of the first and second fundamental forms of the boundaries of the cylinders. The exterior metrics are nonstatic and may be expressed in the Einstein‐Rosen form, and equations governing their behavior are derived. It is found that the exterior metric carries a flux of gravitational ``C energy,'' the direction of which in the Einstein‐Rosen frame is the same as that of the surface of the cylinders.

Covariant Conservation Laws for General Relativistic Exploding Matter
View Description Hide DescriptionIn this paper we attempt a general relativistic extension of some simple notions, e.g., total angular momentum, so far correctly defined only in the frame of special relativity. The mathematical aspect of the problem leads us to ask, given a tensor of vanishing divergence, how to construct an integral object, conserved in time and covariant.
A formalism based upon bitensors ensures the covariance, while the assumption of an exploding (or imploding) schema of matter seems to be the only available means to preserve conservation when the space‐time is curved. This can be formulated in a general theorem then applied to different physical situations. The total linear momentum occurs as a vector fixed at the point of explosion (or implosion). Its length turns out (at least when a pure matter schema is concerned) to be superior to the total mass of the fluid. The excess appears as the energy carried by the explosion. The case of a so called ``uncompressible'' holonomic fluid gives a quite analogous result. In order to find a nontrivial angular momentum we also consider, in the last section, the case of a fluid possessing an intrinsic spin density. Both linear and angular momenta are conserved and covariantly defined. Moreover, they reduce to the conventional ones when the curvature vanishes.

Electromagnetic Fields of Moving Dipoles and Multipoles
View Description Hide DescriptionA method of invariant Green's functions which is frequently used to find the fields of a moving charge is applied to evaluate the fields of dipoles and multipoles. Concise expressions are obtained from the integrals by successively integrating by parts. Two well‐known methods for finding the radiation field on the world line of a charge are repeated for a dipole. It is found that a nonrotating (Fermi‐Walker propagated) electric dipole has vanishing radiation field on the world line when it moves hyperbolically, as in the corresponding case for a charge. Radiation on the world lines of multipoles is also discussed (with particular reference to quadrupoles and octupoles), and the problem of evaluating the radiation reaction for a dipole is described within the context of the methods here given. It is further shown that a classical description of mass renormalization is possible to within the approximation of first‐order terms, but not beyond.

Non‐Markovian Model for the Approach to Equilibrium
View Description Hide DescriptionThis paper is intended to provide the axiomatic study of nonequilibrium quantum statistical mechanics with some simple and rigorously solvable models. The models considered here are obtained as generalizations of the Ising model. They illustrate and allow a rational discussion of the following concepts relevant to the theory of irreversible phenomena: coarse‐graining and time‐smoothing, ergodicity, recurrences, impossibility of a Markovian description of the approach to equilibrium for some physical systems, justification of the various random phase assumptions, properties of the interaction responsible for the approach to equilibrium, master equations, etc.

On the Poles of the S Matrix for Long‐Range Potentials
View Description Hide DescriptionThe analytic properties in the complex k plane of the S matrix for scattering by a screened Coulomb potential are studied. Particular attention is given to the limit as the screening radius tends to infinity, so as to show, in an explicit example, the effect of the tail of the potential on the properties of the analytically extended S matrix. It is shown that the pole configuration obtained in this way is different from that obtained in the usual description of the analytic properties of the Coulomb S matrix.

Kinks
View Description Hide DescriptionIn sufficiently nonlinear field theories there are extended objects whose number is strictly conserved because of continuity of the underlying field as a function of space. We call these kinks. Kinks provide a covariant description of extended but indestructible particles. We give the properties the field theory must possess in order for kinks to exist, and the circumstances under which kinks can have spin ½.

Solution of very Singular N/D Equations—An Approach to Nonrenormalizable Field Theory
View Description Hide DescriptionIn this paper, we find closed form solutions to unsubtracted and once‐subtracted N/Dequations, both nonrelativistic and relativistic, with pathological left‐hand absorptive parts that increase at large ν like α(−ν) ∼ −λν^{ m }, m not necessarily integral. The solutions allow dispersive, unitary and regulator‐free calculation in nonrenormalizable field theory. For example, N/Dequations with exchange of a higher spin particle (spin etc.) can be solved. In general, the technique allows the use of a large class of divergent graphs as input, that is, those whose left‐hand absorptive parts are finite, although asymptotically ill behaved. For example, in the Wtheory of leptonicweak interactions, one of the possible inputs is any number of ladder graphs cut so that all the bosons are on the mass shell. Many nonplanar graphs can be included as well. In much the same way, we can also calculate in the Fermi theory, (multiple exchanges in) theories of higher spin in general, and theories with derivative coupling.
There are an infinite number of solutions to these singular integral equations, each of which is characterized by a branch point at g ^{2} = 0, and a branch point of oscillatory nature at infinite ν. Out of these, we pick the solution which sums the iterative expansion of the equations as most meaningful. It is seen explicitly that the unitary requirement generates its own regulation in the form of rapid oscillations at large unphysical energies, thus eliminating the need for any regulator limiting process. The oscillations are associated with an infinite number of ghosts, but these stay very far from the physical region. In addition, the solutions violate unitarity in the cross channels, so we expect the program to be useful at most in and near the physical region. As shorter range forces are systematically included, there is some indication that the program may converge rapidly for small physical energies.

Wick Polynomials at a Fixed Time
View Description Hide DescriptionIn general, a Wick polynomial must be smeared with a test function depending on both time and space in order to yield an operator in Hilbert space. However, in space time of two dimensions, it is sufficient to smear in the space direction alone. This statement is proved by an application of Weinberg's asymptotic theorem. Operators formed in this manner are candidates for approximate interaction Hamiltonians.

Exact Invariants for a Class of Nonlinear Wave Equations
View Description Hide DescriptionA method is given for obtaining explicitly an infinite number of exact invariants for a physical system described by the coupled set of first‐order hyperbolic partial differential equations.Temporal and spatial invariants are constructed as integrals of temporally and spatially invariant densities T and X, over appropriate spatial and temporal intervals, respectively. For physical systems the energy and momentum densities are temporal invariant densities. These invariant densities are solutions of the hodograph transformedequations corresponding to (A1). For the case n = 1 every invariant density T satisfies an equation in conservation form: (T_{u}0) _{t} − (T_{u}1) _{x} = 0. The methods are applied to the equation,and a denumerable infinity of invariant densities, each expressible as a polynomial, are calculated in two equivalent cases: the first when (A2) (with α = 1) is expressed in zero diagonal form u_{t} = v_{x} , v_{t} = (1 + εu)u_{x} , where u = y_{x} and v = y_{t} ; and the second when (A2) is expressed in diagonal form , where and r and s, the Riemann invariants of (A2), are related to u and v. A theorem of Noether is used to construct from the invariant densities continuous transformation groups that leave the action functional invariant. Using the methods of Kruskal we derive the adiabatic invariant for the continuous system (A2) (α = 1) which has nearly periodic solutions. To order ε the adiabatic invariant is identical with one of the exact invariants and gives no new information about the system.

One‐Dimensional Impenetrable Bosons in Thermal Equilibrium
View Description Hide DescriptionThe simple relationship between the wavefunctions of a system of impenetrable one‐dimensional bosons and impenetrable one‐dimensional fermions is exploited to derive an expansion of the boson reduced density matrices in terms of the fermion reduced density matrices and vice versa. This expansion is independent of the statistical ensemble used, and of the interparticle potential (subject to the impenetrability condition). The special case of zero impenetrability radius with no other forces is treated in detail, using the grand canonical ensemble. This leads to a natural generalization of a formula previously found for zero temperature only, for the one‐particle reduced density matrix.

Classical Field Theory and Gravitation in a de Sitter World
View Description Hide DescriptionThe algebra and the calculus pertaining to a certain class of representations of the de Sitter group is developed. This permits us to formulate covariant field equations in de Sitter space, and, in particular, to construct quantum mechanical equations of motion associated with particles of given spin. The gauge principle is invoked and a spin‐two field emerges, which we identify with the gravitational field. Its coupling to sources is discussed and conservation laws are derived. The emerging nongeometric theory of gravitation is compared with both the Einstein‐Riemann type and other previously proposed nongeometrical theories.

Unitary Irreducible Representations of SL (3, R)
View Description Hide DescriptionIt is shown that to each finite‐dimensional single‐valued irreducible representation of SL(3, R) there corresponds an infinite‐dimensional representation which is unitary on any member of a certain one‐parameter family of Hilbert spaces. We set up an eigenfunction problem for the members of a three‐parameter family of Hilbert subspaces on which such a unitary representation is irreducible. The relatively simple but especially important three‐dimensional case is worked out completely. Unitary irreducible representations for the unimodular real linear groups SL(N, R) with N > 3 and their subgroups can be obtained by generalizing the formalism described here.

Small‐Angle X‐Ray Scattering from Rods and Platelets
View Description Hide DescriptionFor highly elongated rods or flat platelets, there is a range of scattering angles in which the intensity of small‐angle x‐ray scattering cannot be conveniently approximated either by techniques normally used at small scattering angles or by the asymptotic expansion which is applicable in the outer part of the scattering curve. Expressions for the scattered intensity from rods and platelets have therefore been developed which can be used both at these intermediate scattering angles and also in the outer portion of the small‐angle x‐ray scattering curve.

On the Chakrabarti Transformation
View Description Hide DescriptionIn this note, we intend to present an appropriate Hamiltonian formulation for the extension of the Chakrabarti transformation [A. Chakrabarti, J. Math. Phys. 4, 1215 (1963)] to spin‐0 and spin‐1 particles. The suggested formulation is derived from that of Duffin‐Kemmer and has been used previously [L. M. Garrido and P. Pascual, Nuovo Cimento 12, 181 (1959); L. M. Garrido and J. Sesma, Am. J. Phys. 30, 887 (1962); J. Sesma, J. Biel, and L. M. Garrido, Am. J. Phys. 32, 559 (1964)] to achieve a generalization of the Foldy‐Wouthuysen transformation.

Global Covariant Conservation Laws in Riemannian Spaces. I
View Description Hide DescriptionIt is demonstrated that in the theory of general relativityintegralconservation laws can be obtained from a vanishing covariant divergence of a tensor in a completely covariant form. This is achieved by introducing into tensorcalculus a new operation, namely, tensorintegration which is defined as an inverse of tensor differentiation. In particular, a one‐dimensional absolute integration of a tensor along a curve is defined as an inverse operation to the same type of differentiation. A representation of absolute integration is developed by a perturbation method as an infinite series where each term consists of ordinary integrations only. As an example of absolute integration, a vector field is obtained whose components can be employed as a coordinate frame having a very close resemblance to the Riemannian coordinates. Covariant integration is then introduced as an inverse of covariant differentiation; however, its usefulness is severely limited by the conditions of integrability which have to be satisfied to make covariant integration possible. A set of unspecified covariantly constant base vectors is used to explain the idea of a covariantly constant tensor and to express symbolically absolute integration in terms of the ordinary integration. The one‐dimensional absolute integration is then extended to higher dimensions in such a way that it is independent of the order of integrations. Finally, Gauss' theorem is proved for the absolute integration which enables one to convert a volume integral of a covariant divergence of a tensor into a corresponding surface integral of the same tensor.

Global Covariant Conservation Laws in Riemannian Spaces. II
View Description Hide DescriptionUsing the idea of tensor integration, the vector field developed in Part I of this report, and the full Bianchi identities, it is shown that in a general Riemannian space there are four global covariantly‐conserved tensors. The ranks of these tensors are three, four, five, and six. The traces of the first two of these tensors yield the generally covariant equivalent of the familiar linear and angular momentum. The remaining four traceless tensors describe, residually, the gravitational field. With each covariantly conserved tensor one can associate a number of independent invariants. Such invariants are conserved in the ordinary sense. Among these are two types of rest energies and two types of angular momentum magnitudes obtained from the trace and traceless tensors. Examples of global, conserved tensors are derived for a Schwarzschild metric with the electron mass, and a metric of a point electron. It is shown that the rest energy of the Coulomb field diverges as ln(1 /r) at the origin and the second rest energy, that is, the rest energy of the gravitational field diverges as ln r as r approaches infinity. When cutoffs are introduced at the Schwarzschild radius r _{0}, at the classical electron radius r _{1}, and at the radius of the visible universe r _{2}, the rest energy of the gravitational field contained in the shell of thickness r _{1} − r _{0} is approximately 100 times that of the electron rest energy. It is twice this value in the entire visible universe. Since the gravitational field is described by the traceless tensors and the former forms a heavy, compact cloud around the point particle, it is conjectured that the traceless tensors represent the internal degrees of freedom of the elementary particles.

Cylindrical Gravitational News
View Description Hide DescriptionThe concepts of news function and mass aspect are generalized to a class of cylindrically symmetric metrics containing both degrees of freedom of the gravitational field. It is proved that the mass/unit length always decreases if there is any cylindrical news. The asymptotic behavior of the Riemann tensor in the cylindrical case is analyzed and a peeling theorem proved for this case. An example is given to show that asymptotic conditions on the metric or the Riemann tensor which are analogous to the conditions used in the asymptotically spherical case do not exclude certain infinite incoming radiation trains in the cylindrical case. Pure incoming and outgoing solutions are defined for the cylindrical case, and their generalization to the asymptotically spherical case is suggested. An exactly conserved quantity is shown to exist which may be the cylindrical analog of the ten exactly conserved quantities recently discovered by Newman and Penrose.