Semiclassical Analysis of the Wigner 9J-Symbol with Small and Large Angular Momenta

ORAL

Abstract

We derive a new asymptotic formula for the Wigner $9j$-symbol, in the limit of one small and eight large angular momenta, using a novel gauge-invariant factorization for the asymptotic solution of a set of coupled wave equations. Our factorization eliminates the geometric phases completely, using gauge-invariant non-canonical coordinates, parallel transports of spinors, and quantum rotation matrices. Our derivation generalizes to higher $3nj$-symbols. We display without proof some new asymptotic formulas for the $12j$-symbol and the $15j$-symbol in the appendices. This work contributes a new asymptotic formula of the Wigner $9j$-symbol to the quantum theory of angular momentum, and serves as an example of a new general method for deriving asymptotic formulas for $3nj$-symbols.

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Authors

  • G.V. Brown

    Department of Physics at CSU, Fresno, Lawrence Livermore National Laboratory, Stanford University, Advanced Light Source, Lawrence Berkeley National Lab, Nanoelectronics Research Institute, Japan Atomic Energy Agency, University of Tokyo, Cornell University, Tokyo Institute of Technology, Suranaree University of Technology, Shandong University, Stanford Synchrotron Radiation Lightsource, UC Davis, CSU Dominguez Hills, Thomas Jefferson National Accelerator Facility, UC Berkeley, ANL, UChicago/ANL, UManitoba, Northwestern U/ANL, LLNL, UCB/LLNL, McGill U, McGill U/ANL, University of Nevada, Reno, Hitachi Global Storage Technology, Advanced Light Source, Stanford Synchrotron Radiation Laboratory, Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Materials Science and Engineering, Stanford University, Radiation Oncology, Stanford University School of Medicine, Missouri University of Science and Technology, International Institute of Physics, University of Missouri, University of Notre Dame du Lac, Department of Physics, California State University, Fresno, California, California State University Long Beach, BNL, IWF Dresden, Department of Physics, University of California, Davis, CA 95616, USA, University of California, Merced, Laboratoire Univers et Particules de Montpellier, IN2P3 (France), SLAC National Accelerator Laboratory/Stanford University/KIPAC, University of Perugia, University of Washington, CEA/Saclay, UNR, UNM, UCSD, RAL, ILE, MIT, LANL, LLE, NRL, SLAC National Accelerator Laboratory, Technische Universitaet Berlin, Max Born Institut, Hiroshima University, Western Michigan University, MPIK

  • G.V. Brown

    Department of Physics at CSU, Fresno, Lawrence Livermore National Laboratory, Stanford University, Advanced Light Source, Lawrence Berkeley National Lab, Nanoelectronics Research Institute, Japan Atomic Energy Agency, University of Tokyo, Cornell University, Tokyo Institute of Technology, Suranaree University of Technology, Shandong University, Stanford Synchrotron Radiation Lightsource, UC Davis, CSU Dominguez Hills, Thomas Jefferson National Accelerator Facility, UC Berkeley, ANL, UChicago/ANL, UManitoba, Northwestern U/ANL, LLNL, UCB/LLNL, McGill U, McGill U/ANL, University of Nevada, Reno, Hitachi Global Storage Technology, Advanced Light Source, Stanford Synchrotron Radiation Laboratory, Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Materials Science and Engineering, Stanford University, Radiation Oncology, Stanford University School of Medicine, Missouri University of Science and Technology, International Institute of Physics, University of Missouri, University of Notre Dame du Lac, Department of Physics, California State University, Fresno, California, California State University Long Beach, BNL, IWF Dresden, Department of Physics, University of California, Davis, CA 95616, USA, University of California, Merced, Laboratoire Univers et Particules de Montpellier, IN2P3 (France), SLAC National Accelerator Laboratory/Stanford University/KIPAC, University of Perugia, University of Washington, CEA/Saclay, UNR, UNM, UCSD, RAL, ILE, MIT, LANL, LLE, NRL, SLAC National Accelerator Laboratory, Technische Universitaet Berlin, Max Born Institut, Hiroshima University, Western Michigan University, MPIK