Partial Dynamical Symmetries

Two examples of partial dynamical symmetries are presented. 1) The j=9/2 shell of identical particles e.g. neutrons affords the first shell where one can have seniority mixing via a two-body interaction.It was however noted by Escuderos and Zamick that even with a seniority violating interaction certain states remain pure. For 4 neutrons in the 9/2 shell for total ang. momentum I=4 there is one pure seniority v=4 state. This does not mix with the single seniority V=2 state or with the other v=4 state. A proof is presented showing that this special state does not mix with V=2. A similar scenario plays out for I=6. 2) In 44Ti if we sent the two=body T=0 matrix elements to zero, keeping only T=1, then in the single j shell model we get degeneracies of certain states e.g. a 3+,7+,9+, and 10+ state are all degenerate. The ``symmetry'' is partial because we do not get degeneracies for I=0,2,4,6,8. The explanation is that the symmetry only occurs if in 44T (2protons and 2 neutrons) the total angular momenta are ones that cannot occur for 4 identical particles i.e. 44Ca. Where the partial dynamical symmetry applies Jp and Jn are good ``dual'' quantum numbers for all the T=0 states.