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A geometrical theorem on the asymptotic space-time properties of conservation laws in a classical field theory
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Nuclear Rotation and the Random-Phase Approximation. EUGENE R. MARSHALEK. Department of Physics, University of Notre Dame, Notre Dame, Indiana; JOSEPHWENESER.Department of Physics, Brookhaven National Laboratory, Upton, New York. The paper begins with a review of the extraction of the rotational energy and corresponding collective coordinate within the formalism of the random-phase approximation (RPA) for a two-dimensional deformed “nucleus.” An intrinsic difficulty is observed: the random-phase approximation to the angular-momentum operator appears to have a continuous rather than discrete spectrum. Correspondingly, the RPA Hamiltonian has a continuum of unnormalizable eigenstates. One of the main results of the paper is a diagnosis of the origin of this difficulty and a method to deal with it. It then becomes possible to calculate matrix elements of static and transition operators, and illustrative examples are given. Similar difficulties are also encountered in mechanical small-oscillation theory; simple examples are used to illustrate and motivate the procedures applied to the nuclear RPA problem. In the latter connection, the recent general formulation of rotational motion due to Villars is used as a convenient formal framework. Next, the relation between the RPA and the self-consistent cranking (constrained Hartree-Fock) model is simply exhibited, and subsequently it is shown that Kelson’s “external consistency criterion” is fulfilled up to terms that are of the order of corrections to the RPA itself. A Geometrical Theorem on the Asymptotic Space-Time Properties of Conservation Laws in a Classical Field Theory. ROBERT G. CAWLEY. U. S. Naval Ordnance Laboratory, Silver Spring, Maryland. A result, having to do with the asymptotic space-time properties of conservation laws and which we call the cone bands theorem, is proved for a classical field theory model. With the aid of the theorem it is possible to define a tensor reduction procedure on conserved and asymptotically conserved quantities. The reduction procedure applied to such objects permits identitication of components of radiation by means of conical hypersurfaces, or in analogy to particle world lines, radiation world cones, whose axes lie along the four velocities U, of the radiation. It is shown that for the classical theory of a massive real scalar field (retarded solution), produced by a collection of point sources experiencing arbitrary accelerations in a finite region of space-time, these conditions are met by the stress-energy tensor Tuy and by a set of asymptotically conserved amplitude vectors Ar)(x). A dimensionless quantity d3N(U) arises which measures the amount of radiation lying along the direction U, and which is identified with the radiated particle number, or particle “number-probability.” Because the cone bands property possesses a physical interpretation which is rooted in measurement procedures, it is noted that a necessary requirement on a classical field theory, for a particle interpretation to be meaningful in the asymptotic region is that all asymptotically conserved quantities which are regarded as being carried along by the particles possess the cone bands property. Relationship of the present work to quantum field theory is discussed.
ABSTRACTS
OF
PAPERS
TO
APPEAR
IN
FUTURE
ISSUES
Nuclear Rotation and the Random-Phase Approximation. EUGENE R. MARSHALEK. Department of Physics, University of Notre Dame, Notre Dame, Indiana; JOSEPHWENESER.Department of Physics, Brookhaven National Laboratory, Upton, New York. The paper begins with a review of the extraction of the rotational energy and corresponding collective coordinate within the formalism of the random-phase approximation (RPA) for a two-dimensional deformed “nucleus.” An intrinsic difficulty is observed: the random-phase approximation to the angular-momentum operator appears to have a continuous rather than discrete spectrum. Correspondingly, the RPA Hamiltonian has a continuum of unnormalizable eigenstates. One of the main results of the paper is a diagnosis of the origin of this difficulty and a method to deal with it. It then becomes possible to calculate matrix elements of static and transition operators, and illustrative examples are given. Similar difficulties are also encountered in mechanical small-oscillation theory; simple examples are used to illustrate and motivate the procedures applied to the nuclear RPA problem. In the latter connection, the recent general formulation of rotational motion due to Villars is used as a convenient formal framework. Next, the relation between the RPA and the self-consistent cranking (constrained Hartree-Fock) model is simply exhibited, and subsequently it is shown that Kelson’s “external consistency criterion” is fulfilled up to terms that are of the order of corrections to the RPA itself. A Geometrical Theorem on the Asymptotic Space-Time Properties of Conservation Laws in a Classical Field Theory. ROBERT G. CAWLEY. U. S. Naval Ordnance Laboratory, Silver Spring, Maryland. A result, having to do with the asymptotic space-time properties of conservation laws and which we call the cone bands theorem, is proved for a classical field theory model. With the aid of the theorem it is possible to define a tensor reduction procedure on conserved and asymptotically conserved quantities. The reduction procedure applied to such objects permits identitication of components of radiation by means of conical hypersurfaces, or in analogy to particle world lines, radiation world cones, whose axes lie along the four velocities U, of the radiation. It is shown that for the classical theory of a massive real scalar field (retarded solution), produced by a collection of point sources experiencing arbitrary accelerations in a finite region of space-time, these conditions are met by the stress-energy tensor Tuy and by a set of asymptotically conserved amplitude vectors Ar)(x). A dimensionless quantity d3N(U) arises which measures the amount of radiation lying along the direction U, and which is identified with the radiated particle number, or particle “number-probability.” Because the cone bands property possesses a physical interpretation which is rooted in measurement procedures, it is noted that a necessary requirement on a classical field theory, for a particle interpretation to be meaningful in the asymptotic region is that all asymptotically conserved quantities which are regarded as being carried along by the particles possess the cone bands property. Relationship of the present work to quantum field theory is discussed.