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Some aspects of short-range correlations in nuclei
.4NNALS
OF PHYSICS:
a8,
Abstracts
560-502
(1964)
of Papers
to Appear
in Future
Issues
Electrostatic Oscillations in Inhomogeneous Cold Plasmas. E. M. BARSTON, Microwave Laboratory, W. W. Hansen Laboratories of Physics, Stanford University, Stanford, California. A theoretical investigation of small-amplitude electrostatic oscillations in cold inhomogeneous plasmas is presented. It is demonstrated that for those systems possessing continuous nonconstant unperturbed electron-ion densities, a Fourier or normal-mode analysis in time leads in general to singular modes and a continuous frequency spectrum. Wellbehaved solutions to the linearized partial differential equations describing the system are obtained by taking a Fourier integral superposition (inverse Fourier transform) of the singular modes over their continuous spectrum. The analysis leads, in particular, to the following results : 1. A finite plasma with an everywhere continuous and nowhere constant plasma frequency CO,(X) admits of only singular modes and an entirely continuous spectrum. No dispersion relation exists. 2. If, for a collision-free plasma, wpp(x) is continuous in a region R, that region contributes to the spectrum precisely those values of w such that I.O~ = wpz(x) for some x in R. 3. A finite plasma with an everywhere continuous op2(x) will resonate in the presence of an applied field, oscillating with the frequency W, precisely at those points x of the plasma where up’(x) = w2. 4. The introduction of jump discontinuities into op2(x) is necessary for the existence of well-behaved modes and a discrete spectrum (dispersion relation), if ~,~is nowhere constant. If up%(x) jumps discontinuously from a value W? to a value 02~ > W? upon crossing a surface S, and if (~‘2, ~22) and the range of W,“(X) have no points in common, then a well-behaved mode will exist with a frequency w such that ml2 < w2 < wz2. Unitarity
Corrections and Dispersion Modijications of One-Particle Exchange Theories. J. MORAVCSIK, Lawrence Radiation Laboratory, University of California, Livermore, California. Amplitudes for elementary particle reactions calculated from a one-particle exchange theory can be modified to satisfy unitarity and to conform to the constraints of a dispersion relation. Unitarity corrections are discussed for uncoupled and coupled phases, and expressions are given for the pure unitarity corrections to phases. Dispersion modifications are discussed, and it is concluded that the schemes so far proposed for such modifications are either inconsistent with the unsubtracted dispersion relations or are ambiguous in their definition of the pole contributions. The conclusion is reached that it is not possible to determine experimentally from a dispersion relation the parameters of a one-particle exchange model. A prescription is given for obtaining the modest amount of information that can in fact be gleaned from the use of such dispersion relations. The specific and quantitative summary of conclusions is given. MICHAEL
Some Aspects of Short-Range Institute for Theoretical Short-range correlations
Correlations in Nuclei. J. DA PROVIDENCIA AND Physics, University of Copenhagen, Coperhagen, are introduced into the nuclear wave function 500
C. M. SHAKIN, Denmark. by a unitary
ABSTRACTS
501
operator whose form was originally suggested by Villars. The modification of electricdipole transition rates as calculated in the shell model is discussed and is found to be small for the correlation we have assumed. Further, the effective interaction to be used for shellmodel calculations is considered in the context of this theory, and specific suggestions as to the nature of this interaction are put forth. Finally, it is shown how a modified HartreeFock procedure can be constructed for a finite system having short-range correlations. On Quasi-Free PoEyloeal Fields and Fields oj Infinite Spin. R. F. STREATER, Department of Physics, Imperial College, London. A method is given for constructing polylocal fields (depending on many space-time variables) similar to the generalized free field in one variable. The fields satisfy all the axioms given in a previous paper. A special case of the polylocal field may be said to correspond to a local field of infinite spin. The one-particle state of such a theory is infinitely degenerate unless the “one-particle unitarity condition” holds. In certain cases the existence of localized von Neumann algebras (Haag-Araki algebras) can be proved. The “spacelike” asymptotic condition in the form suggested by Haag and Ruelle holds, and so we may expect the interpretation of the theory in terms of asymptotic states. Octuplet Transformation Properties of Weak Interactions. S. COLEMAN, S. L. GLASHOM-, Harvard University, Cambridge, Massachusetts, AND B. W. LEE, University of Pennsylvania, Philadelphia, Pennsylvania. The transformation properties of the weak interaction Lagrangian under the group SU(3) are discussed. Selection rules (statements about the representation of the symmetry group according to which the Lagrangian transforms) and conservation laws (statements about members of the symmetry group which leave the Lagrangian invariant) are considered for leptonic and nonleptonic weak interactions. The In&ite Number of Particles Limit of a Solvable Model. G. T. SCHAPPERT, AFCRL. The energy spectrum of a simple many-body problem, which is solvable for any finite number of particles, is solved in the infinite particle number limit. The model consists of N fermions, in two N-fold degenerate levels, interacting through a monopole-monopole force of strength V. The Hamiltonian can be written in terms of ordinary angular momentum operators, and for a given N, the finite matrix representation can in principle be diagonalized. In this paper, the limit N + a, but NV finite is investigated from the point of view of diagonalizing the resulting infinite matrix. The complete energy spectrum is obtained, and the first exited state agrees with the results obtained by various other techniques. Consequences of Analyticity and Unitarity for Partial-Wave Amplitudes. A. P. BALACHANDRAN AND FRANK VON HIPPEL, The Enrico Fermi Institute for Nuclear Studies, The University of Chicago, Chicago, Illinois. Asymptotic and integral relationships between the functions associated with the unphysical and physical spectral functions of partial-wave amplitudes with 1 2 1 for the scattering of spinless particles are developed. The results are used to show that: (a) The function associated with the unphysical singularities must have a closely limited asymptotic behavior if the dispersion relation is to be consistent. (b) The unphysical singularities determine precisely the leading asymptotic term in the function associated with the physical cut, and in the absence of unphysical singularities, there can be no scattering. (c) In many cases, the unphysical singularities determine the asymptotic limits of the transmission
OF PHYSICS:
a8,
Abstracts
560-502
(1964)
of Papers
to Appear
in Future
Issues
Electrostatic Oscillations in Inhomogeneous Cold Plasmas. E. M. BARSTON, Microwave Laboratory, W. W. Hansen Laboratories of Physics, Stanford University, Stanford, California. A theoretical investigation of small-amplitude electrostatic oscillations in cold inhomogeneous plasmas is presented. It is demonstrated that for those systems possessing continuous nonconstant unperturbed electron-ion densities, a Fourier or normal-mode analysis in time leads in general to singular modes and a continuous frequency spectrum. Wellbehaved solutions to the linearized partial differential equations describing the system are obtained by taking a Fourier integral superposition (inverse Fourier transform) of the singular modes over their continuous spectrum. The analysis leads, in particular, to the following results : 1. A finite plasma with an everywhere continuous and nowhere constant plasma frequency CO,(X) admits of only singular modes and an entirely continuous spectrum. No dispersion relation exists. 2. If, for a collision-free plasma, wpp(x) is continuous in a region R, that region contributes to the spectrum precisely those values of w such that I.O~ = wpz(x) for some x in R. 3. A finite plasma with an everywhere continuous op2(x) will resonate in the presence of an applied field, oscillating with the frequency W, precisely at those points x of the plasma where up’(x) = w2. 4. The introduction of jump discontinuities into op2(x) is necessary for the existence of well-behaved modes and a discrete spectrum (dispersion relation), if ~,~is nowhere constant. If up%(x) jumps discontinuously from a value W? to a value 02~ > W? upon crossing a surface S, and if (~‘2, ~22) and the range of W,“(X) have no points in common, then a well-behaved mode will exist with a frequency w such that ml2 < w2 < wz2. Unitarity
Corrections and Dispersion Modijications of One-Particle Exchange Theories. J. MORAVCSIK, Lawrence Radiation Laboratory, University of California, Livermore, California. Amplitudes for elementary particle reactions calculated from a one-particle exchange theory can be modified to satisfy unitarity and to conform to the constraints of a dispersion relation. Unitarity corrections are discussed for uncoupled and coupled phases, and expressions are given for the pure unitarity corrections to phases. Dispersion modifications are discussed, and it is concluded that the schemes so far proposed for such modifications are either inconsistent with the unsubtracted dispersion relations or are ambiguous in their definition of the pole contributions. The conclusion is reached that it is not possible to determine experimentally from a dispersion relation the parameters of a one-particle exchange model. A prescription is given for obtaining the modest amount of information that can in fact be gleaned from the use of such dispersion relations. The specific and quantitative summary of conclusions is given. MICHAEL
Some Aspects of Short-Range Institute for Theoretical Short-range correlations
Correlations in Nuclei. J. DA PROVIDENCIA AND Physics, University of Copenhagen, Coperhagen, are introduced into the nuclear wave function 500
C. M. SHAKIN, Denmark. by a unitary
ABSTRACTS
501
operator whose form was originally suggested by Villars. The modification of electricdipole transition rates as calculated in the shell model is discussed and is found to be small for the correlation we have assumed. Further, the effective interaction to be used for shellmodel calculations is considered in the context of this theory, and specific suggestions as to the nature of this interaction are put forth. Finally, it is shown how a modified HartreeFock procedure can be constructed for a finite system having short-range correlations. On Quasi-Free PoEyloeal Fields and Fields oj Infinite Spin. R. F. STREATER, Department of Physics, Imperial College, London. A method is given for constructing polylocal fields (depending on many space-time variables) similar to the generalized free field in one variable. The fields satisfy all the axioms given in a previous paper. A special case of the polylocal field may be said to correspond to a local field of infinite spin. The one-particle state of such a theory is infinitely degenerate unless the “one-particle unitarity condition” holds. In certain cases the existence of localized von Neumann algebras (Haag-Araki algebras) can be proved. The “spacelike” asymptotic condition in the form suggested by Haag and Ruelle holds, and so we may expect the interpretation of the theory in terms of asymptotic states. Octuplet Transformation Properties of Weak Interactions. S. COLEMAN, S. L. GLASHOM-, Harvard University, Cambridge, Massachusetts, AND B. W. LEE, University of Pennsylvania, Philadelphia, Pennsylvania. The transformation properties of the weak interaction Lagrangian under the group SU(3) are discussed. Selection rules (statements about the representation of the symmetry group according to which the Lagrangian transforms) and conservation laws (statements about members of the symmetry group which leave the Lagrangian invariant) are considered for leptonic and nonleptonic weak interactions. The In&ite Number of Particles Limit of a Solvable Model. G. T. SCHAPPERT, AFCRL. The energy spectrum of a simple many-body problem, which is solvable for any finite number of particles, is solved in the infinite particle number limit. The model consists of N fermions, in two N-fold degenerate levels, interacting through a monopole-monopole force of strength V. The Hamiltonian can be written in terms of ordinary angular momentum operators, and for a given N, the finite matrix representation can in principle be diagonalized. In this paper, the limit N + a, but NV finite is investigated from the point of view of diagonalizing the resulting infinite matrix. The complete energy spectrum is obtained, and the first exited state agrees with the results obtained by various other techniques. Consequences of Analyticity and Unitarity for Partial-Wave Amplitudes. A. P. BALACHANDRAN AND FRANK VON HIPPEL, The Enrico Fermi Institute for Nuclear Studies, The University of Chicago, Chicago, Illinois. Asymptotic and integral relationships between the functions associated with the unphysical and physical spectral functions of partial-wave amplitudes with 1 2 1 for the scattering of spinless particles are developed. The results are used to show that: (a) The function associated with the unphysical singularities must have a closely limited asymptotic behavior if the dispersion relation is to be consistent. (b) The unphysical singularities determine precisely the leading asymptotic term in the function associated with the physical cut, and in the absence of unphysical singularities, there can be no scattering. (c) In many cases, the unphysical singularities determine the asymptotic limits of the transmission