Volume 15, Issue 4, April 1974

Some useful properties of a theory of variable mass particles
View Description Hide DescriptionBy a simple extension of the canonical formalism, one can include mass and proper time as dynamical variables in mechanics. Such a theory allows one to treat particles with variable mass and also classically decaying particles. The theory has other properties which offer a fresh approach to classical dynamics. For example, the inertia of a system becomes an active concept, and the rest mass of an interacting particle changes to include its binding energy. Also, the proper time as measured by a clock on a decaying particle runs at a different rate from a clock on a stable particle. This effect causes a ``decay red shift'' to be present in classically decaying systems, over and above other red shift effects. The effect could be used to remove the acausal properties of ``pre‐acceleration'' in radiative reaction, and it is suggested that for rapidly decaying massive objects, like quasars, this effect might be responsible for part of their red shift.

Wavepackets for particles of indefinite mass
View Description Hide DescriptionUsing a formalism that includes particles with indefinite and variable masses, we can create a much richer set of states than in conventional quantum mechanics. For example, in a gravitational field the equivalence principle implies that the motion of a particle is independent of the mass. Thus it is possible to create physical states with large Δm but very small Δν and Δx, so that the trajectory of a particle is well determined, while the Δp · Δx uncertainty relation is satisfied. We also provide a quantitative description of wavepackets with arbitrary distributions in momentum and mass (or alternatively velocity and mass) and discuss a simple diffraction slit experiment as an example of their physical propagation. The Schrödinger equation is solved for a decaying particle at rest. It is also shown that Galilean invariance plays a natural and cohesive role within the variable mass formalism, unlike the case in conventional quantum mechanics where the existence of multiple mass states would violate Galilean invariance.

A property of asymptotically flat, static, vacuum space‐times
View Description Hide DescriptionIt is shown that a necessary condition for stationary, asymptotically flat, vacuum space‐times to be static is that the Weyl tensor be electric type to asymptotic order r ^{−7}, where r is an affine parameter along null geodesics.

Mayer's theory analysis of repulsive and weak, long range potentials
View Description Hide DescriptionThe irreducible integrals defined in the Mayer theory of real gases are derived for one‐dimensional systems having hard rod, weak long range, Curie‐Weiss and van der Waals potential functions. In two cases, independent knowledge of the exact equation of state of the gas permits the derivation of general star graph degeneracy relationships. These, in turn, are useful for the study of new potential functions; a penetrable repulsive force is shown to produce attractive force behavior in the gas state equation.

Conformal spinor fields in general relativity
View Description Hide DescriptionA spinor field which is covariant under the group of conformal motions in general relativity is defined. These spinor fields extend the concept of isospinors to the cases where conformal motions take place and coincide with Penrose's twistors in the case of flat space‐time. It is found that only certain special conformal spinor fields lead to physical entities. The example of Kruskal's conformal spinors is studied.

The Kerr congruence
View Description Hide DescriptionWe present the special relativistic Kerr congruence from two new points of view. The importance of the Kerr congruence stems from its role in the general relativistic rotating body problem. In the special relativistic limit, it is the simplest twisting generalization of the ordinary light cone. We first show that the Kerr congruence can be obtained from the light cone of complex Minkowski space, which induces a family of oblate spheroids in real Minkowski space. We then show that the rigid rotation of one of these spheroids with constant angular velocity also generates the Kerr congruence by shining comoving flashlights normal to the surface. In fact, the oblate spheroid is the unique surface which generates a shear‐free twisting null congruence in this manner. This result has direct generalization to the full Kerr geometry.

Iterative solution of the inverse Sturm‐Liouville problem
View Description Hide DescriptionAn iterative construction of the potential function entering in a regular Sturm‐Liouville problem is discussed. The given data is in the form of two spectra associated with distinct boundary conditions at one end point.

On the problem of diffraction
View Description Hide DescriptionIt is shown how the methods of Hadamard, developed for the Cauchy problem in free space, can be extended to include the presence of simple obstacles. The modifications necessary are illustrated by treating the problem of diffraction by a conducting half‐plane in two and three dimensions.

An approach to the study of quantum systems
View Description Hide DescriptionA perturbation method for quantum mechanical problems is presented in which the successive terms are obtained by algebraic recurrence relations without the use of any diagrams. Two examples, one an anharmonic oscillator with quartic self‐interaction, and one a spin with quadratic self‐interaction, are given to illustrate the use of this recurrence relationperturbation method.

Note on gauge invariance and conservation laws for a class of nonlinear partial differential equations
View Description Hide DescriptionRelations between gauge invariance and the conservation laws are discussed relative to the Korteweg‐deVries and related equations. The technique used here is conventional in field theory‐the canonical formalism related to the invariant variational problem. The function of constants of motion, when represented in terms of canonical variables, is to generate infinitesimal partial gauge transformations of any order, under which the variational problem is invariant. From this point of view, one example is presented together with the general theory.

Two‐particle approximation for the quantum third virial coefficient: Comparison of two approaches
View Description Hide DescriptionTwo distinct approaches have been followed to produce an approximate expression for the third virial coefficient of a quantum system in terms of the pertinent two‐particle quantities (standard T matrix elements and bound stateeigenvalues and eigenstates): One approach is based on the T approximation for the self‐energy in the theory of temperature Green functions, and the other uses the diagrams which represent the various terms in the multiple‐scattering expansion of the three‐particle T operator. It is shown that the result given by the former method is reproduced exactly by the contribution of a small number of selected diagrams. The Green function method requires the evaluation of the Kadanoff and Baym generalized T matrix to one order in the activity higher than its familiar low‐density limit. The common result can be written entirely in terms of the lowest order self‐energy and should give a reasonable approximation to the third virial coefficient for systems with short‐range forces.

Applications of the concept of strength of a system of partial differential equations
View Description Hide DescriptionThe concept of ``strength'' of a system of field equations was introduced by Einstein, and is of such generality that one can compare vastly different systems of field equations. We review here this concept in arbitrary number of dimensions and apply it to some of the well‐known equations of physics. We calculate the strength, in arbitrary dimensions, of massless Klein‐Gordon equations,Maxwell equations (in both potential and field formulation), and Einstein equations. We also determine the strength of massless Dirac equation and Weyl's neutrinoequation for the case of four dimensions. It turns out that the strength for all these equations is identical for space‐time dimensionality of four. Other possible applications of this concept are indicated.

Occupation statistics from exact recursion relations for occupation by dumbbells of a 2 × N array
View Description Hide DescriptionStarting from an exact recursion relation of the form , we present a recursion method for calculating the various moments These results are applied to obtain the occupation statistics for occupation of a 2 × N array by parallel and not necessarily parallel dumbbells.

Spacelike representations of the inhomogeneous Lorentz group in a Lorentz basis
View Description Hide DescriptionThe problem of introducing a Lorentz basis in a spacelike representation of the inhomogeneous Lorentz group in the spinless case, is solved by the methods of integral geometry. A new description of the representation associated with isotropic lines, which is peculiar to the spacelike case, is also given.

On the relaxation to quantum‐statistical equilibrium of the Wigner‐Weisskopf atom in a one‐dimensional radiation field. VI. Influence of the coupling function on the dynamics
View Description Hide DescriptionAn investigation is made of the effect on the dynamics of spontaneous emission of the Wigner‐Weisskopf atom in an infinite‐system limit of various choices of the coupling function or form factor describing the atom's interaction with the spectrum of the radiation field. This is carried out both for the exact solution to the problem of spontaneous emission, obtained in the earlier papers in the series, and for some approximate solutions, also previously considered, in particular one based on the Schrödinger equation of the problem and one based on the weak‐coupling Prigogine‐Résibois master equation. The details of the form factor are found, by numerical computation of the solutions, to be critical in determining the nonexponential parts of the solutions, and these parts are seen to be capable in some cases of dominating the exponential parts, which are given only by the values of the form factor near the resonance energy. The approximate solutions discussed are found to vary widely in their worth, and one, which yields the exact solution for the Wigner‐Weisskopf problem, is singled out as being of probable use in the statistical‐mechanical description of more complicated systems.

On the existence of solutions to S matrix models
View Description Hide DescriptionWe analyze the charged scalar Low equation and show that, in this case at least, the breakdown of the standard fixed point theorems occurs because of their inherent weakness rather than through some topological change in the structure of the family of solutions. We also show that unless we reformulate the equation the Schauder principle only handles the basic solution, that solution without CDD poles. We discuss methods, alternative to the N/D method, to handle the other solutions.

Energy transfer in radiation chemistry. I. Dynamics of the electron‐oscillators system
View Description Hide DescriptionSingle‐mode excitation events in radiation chemistry are investigated by considering a simple model: an electron (characterized by mass m and momentum ℏk) in interaction with a collection of harmonic oscillators. The Hamiltonian corresponding to this model is identified, and then used within the framework of a master equation constructed from the time‐dependent Schrödinger equation. The usual long‐wavelength approximation is employed, which, in the infinite‐system limit, necessitates an upper bound on the frequency spectrum of the oscillator bath. Once these constraints have been imposed, however, the time evolution of the system is determined exactly (i.e., no ``weak‐coupling'' approximation) in two limiting regimes‐the case of a low‐energy incident electron and that of a high‐energy incident electron. Our results are compared with those obtained using a weak‐coupling approximation, first within the framework of the master equation cited above and then in conjunction with the Prigogine‐Résibois master equation, and it is pointed out that for the system under study, the use of a weak‐coupling approximation in both equations leads to unphysical results. Our results are also compared with those obtained earlier by Van Hove and co‐workers on the electron‐random scatterers model, and an attempt is made to understand the role of the approximations introduced in both models. Finally, suggestions for future work are given; in particular, we discuss the generalization of our approach to deal with multiple‐mode excitation events in radiation chemistry.
 ERRATA


Errata: Ether flow through a drainhole: A particle model in general relativity
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Erratum: The evaluation of lattice sums. II. Number theoretic approach
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