Volume 6, Issue 6, June 1965
Index of content:

Unitarity and On‐Mass‐Shell Analyticity as a Basis for S‐Matrix Theories. I
View Description Hide DescriptionMany‐particle unitarity relations are analyzed and related to maximal analyticity conjectures with particular reference to continuations into unphysical sheets. Poles and cuts associated with unstable particles are treated in detail and generalized unitarity relations derived.

Unitarity and On‐Mass‐Shell Analyticity as a Basis for S‐Matrix Theories. II
View Description Hide DescriptionSets of postulates for S‐matrix theories are given and used to construct potentially complete dynamical theories for scattering of strongly interacting particles, based essentially on the unitarity relations and on‐mass‐shell analyticity of scattering amplitudes. In one formulation, the unitarity relations for stable particles are used to derive the corresponding relations for cuts associated with unstable particle states. In a second formulation, an attempt is made to treat stable and unstable particles on a more equal footing. Crossing symmetries and decomposition laws for single‐particle poles appear as consequences of the postulates.

Unitarity and On‐Mass‐Shell Analyticity as a Basis for S‐Matrix Theories. III
View Description Hide DescriptionThe singularity structure of many‐particle scattering amplitudes in momentum transfer variables is investigated in terms of the many‐particle unitarity conditions. Cauchy‐type kernels for analytic functions defined on complex rotation group in three dimensions are constructed and related to a theory of local representations of the rotation group, corresponding to complex angular momenta.

Long‐Time Behavior of the Electric Potential and Stability in the Linearized Vlasov Theory
View Description Hide DescriptionIn this paper we study in a mathematically rigorous manner how the electric potential, produced by small electronic charge densityoscillations of definite wavenumber vector k in a plasma, behaves in the long‐time limit and the connection between this behavior and the stability of a given steady, spatially uniform, distribution of the plasma electrons. Our work is based on the linearized Vlasov equation and on the associated Poisson equation. We formulate a very general initial‐value problem concerning this system of equations, writing the above electric potential at a given position vector r and time t as φ(t)e^{i} ^{ k·r } multiplied by a suitable constant, where φ(t) is independent of r. We establish the existence and uniqueness of solution of this problem by exploiting the fact that, in the linear theory, φ(t) obeys an inhomogeneous Volterra integral equation of convolution type, which is rigorously derived here. A detailed study of the asymptotic properties of the solutions of this equation for t → ∞ is made, including the establishment of necessary and sufficient conditions on the initial perturbations(perturbations of the steady electron distribution function at t = 0) for φ(t) to be of negative exponential order as t → ∞. As a byproduct of this asymptotic investigation, we give a precise discussion of the Landau damping of long wavelength plasma oscillations in an initially Maxwellian plasma, concluding that in this case φ(t) exhibits such damping for a broad range of initial perturbations and that the damping decrement is essentially that first computed by Landau. We introduce criteria of stability and instability based on the boundedness and unboundedness, respectively, in the limit t → ∞ of certain nonnegative quantities W_{p} (t), which are defined as suitable norms of the perturbed electron distribution function. New sufficient conditions for stability and instability are proved for extensive classes of initial distributions and initial perturbations. These results are compared with conclusions on stability and instability reached by Backus.

Projected Angular Momentum States
View Description Hide DescriptionKnown results regarding the projection of total spin states from product wavefunctions involving spin‐½ particles are generalized. It is shown that total spin states projected from product wavefunctions involving particles with arbitrary spin, but otherwise restricted to having either maximum or minimum z component of spin, are no more complicated in form than for the spin‐½ case.

Conserved Quantities Associated with Symmetry Transformations of Relativistic Free‐Particle Equations of Motion
View Description Hide DescriptionA general technique is presented for associating conservation laws with the symmetry transformations that leave invariant the relativistic equations of motion for a free particle. These transformations may be either continuous with the identity (such as infinitesimal transformations) or discontinuous (such as reflections). It is found that for each transformation there exist two classes of conservation laws. The number of separate laws within a class depends on the spin of the particle. The particular cases of the Dirac equation and Maxwell's equations are investigated in some detail. For the Dirac equation, conserved quantities involving discontinuous transformations and also matrix elements between particle and antiparticle states are obtained, in addition to the usual conservation laws. Application of the general method to Maxwell's equations yields not only the usual conserved quantities and Lipkin's ``zilch,'' but also twenty new gauge‐independent conserved quantities and other additional integrals associated with discontinuous transformations.

Spatially Inhomogeneous States of Many‐Body Systems
View Description Hide DescriptionTo treat many‐body systems in the presence of a static potential or problems of highly collective spatially inhomogeneous motions such as vortex lines, wavefunctions of a typehave been proposed. Here Φ is the exact ground state of the homogeneous system and g(x) is a one‐particle state introduced to describe the effect of the spatial inhomogeneity. However, to determine g(x) by the variation principle, one needs to know the spatial correlation functions of all orders for the homogeneous many‐body system. It is shown that the method of point transformations allows one to work with qualitatively similar but different states. The description of the system in the presence of a static impurity or of a state representing a vortex line in liquid helium requires a knowledge of only the average kinetic energy and x‐ray scattering factor for homogeneous liquid helium. Both of these are available from experiments. The treatment of a recoiling impurity atom, strictly speaking, requires a knowledge of the current correlation tensor for the ground state of the homogeneous many‐body system. This term vanishes in the Hartree limit of the theory for bosons.

Structure of Gravitational Sources
View Description Hide DescriptionThe purpose of this paper is to propose a definition of multipole structure of gravitational sources in terms of the characteristic initial data for asymptotic solutions of the field equations. This definition is based upon a detailed study of the corresponding data for the linearized equations and upon the close analogy between the Maxwell and the linearized gravitational fields.

Note on the Kerr Spinning‐Particle Metric
View Description Hide DescriptionIt is shown that by means of a complex coordinate transformation performed on the monopole or Schwarzschild metric one obtains a new metric (first discovered by Kerr). It has been suggested that this metric be interpreted as that arising from a spinning particle. We wish to suggest a more complicated interpretation, namely that the metric has certain characteristics that correspond to a ring of mass that is rotating about its axis of symmetry. The argument for this interpretation comes from three separate places: (1) the metric appears to have the appropriate multipole structure when analyzed in the manner discussed in the previous paper, (2) in a covariantly defined flat space associated with the metric, the Riemann tensor has a circular singularity, (3) there exists a closely analogous solution of Maxwell's equations that has characteristics of a field due to a rotating ring of charge.

Metric of a Rotating, Charged Mass
View Description Hide DescriptionA new solution of the Einstein‐Maxwell equations is presented. This solution has certain characteristics that correspond to a rotating ring of mass and charge.

Unified Dirac‐Von Neumann Formulation of Quantum Mechanics. I. Mathematical Theory
View Description Hide DescriptionIn this paper the results from various areas of mathematical research which are necessary for a consistent unification of the Dirac and von Neumann formulations of quantum mechanics are collected and presented as a single synthesis. For this purpose, direct integral decompositions of Hilbert space must be introduced into Dirac's formulation of spectral theory and representation theory; true unit vectors in the direct integral decomposition spaces replace unnormalizable vectors of infinite length. It then becomes clear that families of modified Dirac projection operators are simply related to the Radon‐Nikodym derivative of von Neumann spectral measures. In terms of these mathematical preliminaries a second paper will present the more physical aspects of the resulting unified formulation of quantum mechanics.

Transformation from a Linear Momentum to an Angular Momentum Basis for Particles of Zero Mass and Finite Spin
View Description Hide DescriptionThe infinitesimal generators of the inhomogeneous Lorentz group have been given in a basis in which the components of the linear momentum operators are diagonal and in another basis in which the square of the angular momentum is diagonal for all unitary irreducible ray representations of the group. In the present paper we show how the two bases are related for representations corresponding to zero mass and any (finite) spin. It will be shown how this relation enables one to integrate the infinitesimal generators in the angular momentum basis and thereby permits one to show how the angular momentum of a particle changes under the inhomogeneous Lorentz group. In particular, we study the way that the angular momentum of a massless particle of any spin appears in translated and moving frames of reference.

Calculation of Some Homology Groups Relevant to Sixth‐Order Feynman Diagrams
View Description Hide DescriptionA homology group that determines an upper bound for the number of linearly independent analytic functions connected with the sixth‐order ladder diagram is here computed. The formalism is that of Fotiadi, Froissart, Lascoux, and Pham. Calculations use the standard methods of homology theory. We find that there are at most 127 such functions in general.

Solution of the Källén‐Pauli Equation
View Description Hide DescriptionWe give the relevant solution to the integral equation which Källén and Pauli have derived for elasticVΘ‐scattering in the Lee model. As an application of this result the exact Tamm‐Dancoff solution for the entire VΘ‐sector is obtained. This includes the amplitudes for VΘ‐ and N2Θ elastic scattering and VΘ‐N2Θ production, as well as their extensions off the mass shell.

Integrals of the Second‐Order Linear Differential Equation
View Description Hide DescriptionIn this study we describe procedures for the numerical solution of the second‐order linear differential equation which have either continuous or discontinuous coefficients. Our motivation is a well‐known technique in the theory of inhomogeneous transmission lines: the treatment of a continuously varying line by considering it to be comprised of various sections of uniform lines. Although this procedure is very suggestive physically, there are difficulties with its applications to second‐order equations which do not describe wave propagation. First, the language of the circuit engineer is such that a pair of first‐order equations describing some analogs of the complex quantities voltage and current seem to be required. Whereas these equations appear naturally in transmission line theory, we show that it is an unnecessary burden to find their counterparts when the problem is but to solve a second‐order equation. The second objection to this approach is that it is not apparent that a piecewise constant partition of the coefficients in a differential equation will yield the rigorous solution if the subdivision is carried out to an arbitrary degree. Indeed, we show that the limit of the quantization scheme can not yield the rigorous solution. On the other hand, we are led to a well‐defined technique which generates the solution by a method suggested by the procedures used in the discrete case. This enables the second‐order equation to be solved rigorously by iteration with no complications, and in a form ideally suited for computer programs.

Mathematical Study of the Cutoff Procedure for Divergent Integrals in Bound‐State Problems
View Description Hide DescriptionWe consider the class of conveniently normalized functions having a given rate of decrease at infinity on the real axis. The lower bound of the length of the supports of the Fourier transforms of such a given class of functions is evaluated. This could be considered as a refined form of the uncertainty relation. The condition, which is necessary and sufficient for this lower bound to be finite is indicated. In the physically important case when this lower bound is infinite, a length, which we call the ``natural cutoff of this class of functions'' is nevertheless defined, it characterizes the interval outside of which the value of the Fourier transform of a function of the given class may be negligible. A construction which gives the cutoff, or the natural cutoff, is given; it is valid for practically any given rate of decrease. We carry out the calculation explicitly in the case of a decrease of the functions characterized by F(x) ≤ A e ^{−μx} (in this case the support of the Fourier transform is necessarily infinite). A canonical function is built such that the abscissa for which it begins to become ``small'' should be reasonably considered to be the natural cutoff of the Fourier transform of all the functions satisfying this inequality. The same method can be used for the class of functions satisfying an inequality of the form F(x) ≤ A exp [−C(x)], with ∫_{1} ^{∞} C(u) u ^{−3} du < ∞.

On the Quantum Field Theories Leading to the Corben Equations
View Description Hide DescriptionFormulations are presented for the second quantized versions of the field theories which lead to Corben's equations of motion. It is demonstrated that an indefinite metric is required to guarantee positive energies for all the particles, but that otherwise the theories are physically unambiguous. The number and properties of the resulting particles are studied and compared with the conclusions from previous work. Alternative formulations are also discussed.

Momentum Transfer Cross‐Section Theorem
View Description Hide DescriptionThe momentum transfer cross section is expressed in terms of a matrix element of grad V, where V is the potential which need not be spherically symmetric. The result may be useful for estimating the momentum transfer cross section in circumstances where the usual expansion in partial waves is inconvenient or inapplicable as, e.g., when V is noncentral.

Note on Bég's Approach To Peratization
View Description Hide DescriptionIt is shown that Bég's elegant formulation of the peratization result of Feinberg and Pais can be rigorously justified.

Disconnected Groups as Higher Symmetry Groups
View Description Hide DescriptionDisconnected groups are investigated to see whether they can be used as higher symmetry groups. Some disconnected groups are given which satisfy certain minimal physical conditions. Two disconnected groups are analyzed in detail using little group techniques, and it is shown how a doubling of certain multiplets results, so that particles and their antiparticles are in the same multiplets—rather than being in separate, though equivalent multiplets as in the usual SU _{2} or SU _{3} schemes.