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We apply the Clebsch-Gordan and Racah coefficients to calculate the double tensors for two equivalent d electrons. We also obtain the commutation relations for these double tensors and choose certain quantum numbers, which produce a subgroup. From the root vectors of the commutation relations, we identify them with Lie algebra B2. Once we have the correct Lie algebra, it is feasible to use the Wigner-Eckart theorem to find matrix elements for transition states among atomic spectra or nuclear shell models.
Comment: 46 pages, no figures. Minor changes in the introduction and correction of some typos. Version accepted for publication in Annales Henri Poincare'
Comment: Added references, some typos corrected, published version
We prove a general theorem about the self-adjointness and domain of Pauli-Fierz type Hamiltonians. Our proof is based on commutator arguments which allow us to treat fields with non-commuting components. As a corollary it follows that the domain of the Hamiltonian of non-relativistic QED with Coulomb interactions is independent of the coupling constant.
Comment: 9 pages, written for proceedings of the AGA program at Newton Institute; To appear in Proc. Symp. Pure Math. (2008)
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