1. Symmetries in Nuclei, Tokyo, 2008 Symmetries in Nuclei Symmetry and its mathematical description The role of symmetry in physics Symmetries of the nuclear shell model Symmetries of the interacting boson model

2. Symmetries in Nuclei, Tokyo, 2008 Symmetry in physics Geometrical symmetries Example: C 3 symmetry of O 3, nucleus 12 C, nucleon (3q). Permutational symmetries Example: S A symmetry of an A-body hamiltonian:

3. Symmetries in Nuclei, Tokyo, 2008 Symmetry in physics Space-time (kinematical) symmetries Rotational invariance, SO(3): Lorentz invariance, SO(3,1): Parity: Time reversal: Euclidian invariance, E(3): Poincaré invariance, E(3,1): Dilatation symmetry:

4. Symmetries in Nuclei, Tokyo, 2008 Symmetry in physics Quantum-mechanical symmetries, U(n) Internal (global gauge) symmetries. Examples: p  n, isospin SU(2) u  d  s, flavour SU(3) (Local) gauge symmetries. Example: Maxwell equations, U(1). Dynamical symmetries. Example: the Coulomb problem, SO(4).

5. Symmetries in Nuclei, Tokyo, 2008 Symmetry in quantum mechanics Assume a hamiltonian H which commutes with operators g i that form a Lie algebra G:  H has symmetry G or is invariant under G. Lie algebra: a set of (infinitesimal) operators that close under commutation.

6. Symmetries in Nuclei, Tokyo, 2008 Consequences of symmetry Degeneracy structure: If  is an eigenstate of H with energy E, so is g i  : Degeneracy structure and labels of eigenstates of H are determined by algebra G: Casimir operators of G commute with all g i :

7. Symmetries in Nuclei, Tokyo, 2008 Hydrogen atom Hamiltonian of the hydrogen atom is Standard wave quantum-mechanics gives Degeneracy in m originates from rotational symmetry. What is the origin of l-degeneracy? E. Schrödinger, Ann. Phys. 79 (1926) 361

8. Symmetries in Nuclei, Tokyo, 2008 Energy spectrum

9. Symmetries in Nuclei, Tokyo, 2008 Classical Kepler problem Conserved quantities: Energy (a=semi-major axis): Angular momentum (  =eccentricity): Runge-Lentz vector: Newtonian potential gives rise to closed orbits with constant direction of major axis.

10. Symmetries in Nuclei, Tokyo, 2008 Classical Kepler problem

11. Symmetries in Nuclei, Tokyo, 2008 Quantization of operators From p  -ih  : Some useful commutators & relations:

12. Symmetries in Nuclei, Tokyo, 2008 Conservation of angular momentum The angular momentum operators L commute with the hydrogen hamiltonian: L operators generate SO(3) algebra: H has SO(3) symmetry  m-degeneracy.

13. Symmetries in Nuclei, Tokyo, 2008 Conserved Runge-Lentz vector The Runge-Lentz vector R commutes with H: R does not commute with the kinetic and potential parts of H separately: Hydrogen atom has a dynamical symmetry.

14. Symmetries in Nuclei, Tokyo, 2008 SO(4) symmetry L and R (almost) close under commutation: H is time-independent and commutes with L and R  choose a subspace with given E. L and R'  (-M/2E) 1/2 R form an algebra SO(4) corresponding to rotations in four dimensions.

15. Symmetries in Nuclei, Tokyo, 2008 Energy spectrum Isomorphism of SO(4) and SO(3)  SO(3): Since L and A' are orthogonal: The quadratic Casimir operator of SO(4) and H are related: W. Pauli, Z. Phys. 36 (1926) 336

16. Symmetries in Nuclei, Tokyo, 2008 Dynamical symmetry Two algebras G 1  G 2 and a hamiltonian  H has symmetry G 2 but not G 1 ! Eigenstates are independent of parameters  m and n in H. Dynamical symmetry breaking “splits but does not admix eigenstates”.

17. Symmetries in Nuclei, Tokyo, 2008 Empirical observations: About equal masses of n(eutron) and p(roton). n and p have spin 1/2. Equal (to about 1%) nn, np, pp strong forces. This suggests an isospin SU(2) symmetry of the nuclear hamiltonian: Isospin symmetry in nuclei W. Heisenberg, Z. Phys. 77 (1932) 1 E.P. Wigner, Phys. Rev. 51 (1937) 106

18. Symmetries in Nuclei, Tokyo, 2008 Isospin SU(2) symmetry Isospin operators form an SU(2) algebra: Assume the nuclear hamiltonian satisfies  H nucl has SU(2) symmetry with degenerate states belonging to isobaric multiplets:

19. Symmetries in Nuclei, Tokyo, 2008 Isospin symmetry breaking: A=49 Empirical evidence from isobaric multiplets. Example: T=1/2 doublet of A=49 nuclei. O’Leary et al., Phys. Rev. Lett. 79 (1997) 4349

20. Symmetries in Nuclei, Tokyo, 2008 Isospin symmetry breaking: A=51

21. Symmetries in Nuclei, Tokyo, 2008 Isospin SU(2) dynamical symmetry Coulomb interaction can be approximated as  H Coul has SU(2) dynamical symmetry and SO(2) symmetry. M T -degeneracy is lifted according to Summary of labelling:

22. Symmetries in Nuclei, Tokyo, 2008 Isobaric multiplet mass equation Isobaric multiplet mass equation: Example: T=3/2 multiplet for A=13 nuclei. E.P. Wigner, Proc. Welch Found. Conf. (1958) 88

23. Symmetries in Nuclei, Tokyo, 2008 Isospin selection rules Internal E1 transition operator is isovector: Selection rule for N=Z (M T =0) nuclei: No E1 transitions are allowed between states with the same isospin. L.E.H. Trainor, Phys. Rev. 85 (1952) 962 L.A. Radicati, Phys. Rev. 87 (1952) 521

24. Symmetries in Nuclei, Tokyo, 2008 E1 transitions and isospin mixing B(E1;5 -  4 + ) in 64 Ge from: lifetime of 5 - level;  (E1/M2) mixing ratio of 5 -  4 + transition; relative intensities of transitions from 5 -. Estimate of minimum isospin mixing: E.Farnea et al., Phys. Lett. B 551 (2003) 56

25. Symmetries in Nuclei, Tokyo, 2008 Pairing SU(2) dynamical symmetry The pairing hamiltonian, …has a quasi-spin SU(2) algebraic structure: H has SU(2)  SO(2) dynamical symmetry: Eigensolutions of pairing hamiltonian: A. Kerman, Ann. Phys. (NY) 12 (1961) 300

26. Symmetries in Nuclei, Tokyo, 2008 Interpretation of pairing solution Quasi-spin labels S and M S are related to nucleon number n and seniority  : Energy eigenvalues in terms of n, j and  : Eigenstates have an S-pair character: Seniority  is the number of nucleons not in S pairs (pairs coupled to J=0). G. Racah, Phys. Rev. 63 (1943) 367

27. Symmetries in Nuclei, Tokyo, 2008 Quantal many-body systems Take a generic many-body hamiltonian: Rewrite H as (bosons: q=0; fermions: q=1) Operators u ij generate U(n) for bosons and for fermions (q=0,1):

28. Symmetries in Nuclei, Tokyo, 2008 Quantal many-body systems With each chain of nested algebras …is associated a particular class of many-body hamiltonian Since H is a sum of commuting operators …it can be solved analytically!