Research article Topical Sections

Differential geometry of collective models

  • Received: 01 January 2019 Accepted: 13 February 2019 Published: 05 March 2019
  • MSC : 53Z05, 70F25, 81T13

  • The classical astrophysical theory of Riemann ellipsoids and the quantum nuclear theory of Bohr and Mottelson share a common mathematical foundation in terms of the differential geometry of a principal bundle ${\cal P}$ and its associated vector bundle E, respectively. The bundle ${\cal P} = GL_+(3, R)$ is the connected component of the general linear group, the structure group G = SO(3) is the vorticity group, and the base manifold is the space of positive-definite real $3\times 3$ symmetricmatrices, identified geometrically with the space of inertia ellipsoids. The bundle is a Riemannian manifold whose metric is inherited from three-dimensional Euclidean space. A nonholonomic constraint force, like irrotational flow, determines a connection on the bundle. Wave functions of the Bohr-Mottelson model are sections of the associated vector bundle E = ${\cal P}\times_\rho$V, where $\rho$ denotes an irreducible representation of the vorticity group on the vector space Ⅴ. Using the de Rham Laplacian $\triangle = \star d_\nabla \star d_\nabla$ for the kinetic energy introduces a "magnetic" term due to the connection between base manifold rotational and fiber vortex degrees of freedom. A class of Ehresmann connections creates a new model of nuclear rotation that predicts moments of inertia in agreement with experiment.

    Citation: George Rosensteel. Differential geometry of collective models[J]. AIMS Mathematics, 2019, 4(2): 215-230. doi: 10.3934/math.2019.2.215

    Related Papers:

  • The classical astrophysical theory of Riemann ellipsoids and the quantum nuclear theory of Bohr and Mottelson share a common mathematical foundation in terms of the differential geometry of a principal bundle ${\cal P}$ and its associated vector bundle E, respectively. The bundle ${\cal P} = GL_+(3, R)$ is the connected component of the general linear group, the structure group G = SO(3) is the vorticity group, and the base manifold is the space of positive-definite real $3\times 3$ symmetricmatrices, identified geometrically with the space of inertia ellipsoids. The bundle is a Riemannian manifold whose metric is inherited from three-dimensional Euclidean space. A nonholonomic constraint force, like irrotational flow, determines a connection on the bundle. Wave functions of the Bohr-Mottelson model are sections of the associated vector bundle E = ${\cal P}\times_\rho$V, where $\rho$ denotes an irreducible representation of the vorticity group on the vector space Ⅴ. Using the de Rham Laplacian $\triangle = \star d_\nabla \star d_\nabla$ for the kinetic energy introduces a "magnetic" term due to the connection between base manifold rotational and fiber vortex degrees of freedom. A class of Ehresmann connections creates a new model of nuclear rotation that predicts moments of inertia in agreement with experiment.


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