Symmetries of the nuclear Hamiltonian (exact or almost exact) Translational invariance Galilean invariance (or Lorentz invariance) Rotational invariance Time reversal Parity (space reflection) Charge independence and isobaric symmetry Baryon and lepton number symmetry Permutation between the two nucleons (imposed by the exclusion principle) Continuous transformations (appear to be universally valid) Dynamical symmetries apply in certain cases, provide useful coupling schemes Chiral symmetry (broken by a quark condensate; valid for massless quarks) SU(4) symmetry (Wigner supermultiplet) SU(2) symmetry (seniority) SU(3) symmetry (Elliott model)

Symmetries in quantum mechanics (see "Symmetry in Physics", J.P. Elliott and P.G. Dawber, The Macmillan Press, London) Wave equation for the Hamiltonian operator: Group of transformations G whose elements G commute with H: We say that H is invariant under G or totally symmetric with respect to the elements of G What are the properties of ? Reminder: representation of the group (the basis, a set of operators acting on the basis functions and corresponding to a symmetry group) dimension of the representation matrix representation of the group basis If all matrices D can be put into a block-diagonal form, the representation is reducible

Hence is also an eigenfunction of H with eigenvalue Ek . If Ek is nondegenerate If Ek is n-fold degenerate, there are n partner functions one-dimensional irrep of G We can thus label the wave function fully as n-dimensional irrep of G , except for accidental degeneracy We can thus label the wave function fully as

Classification of eigenstates with respect to symmetry group Wave functions for different energy levels Ek transform as basis functions of irreducible representations of the group G If we know the properties of G, we can classify the wave functions Further, the same group-theoretical structure will tell us about the spectroscopy of the system Point groups: geometric symmetries that keep at least one point fixed Lie groups: continuous transformation groups D1: (dihedral) reflection group (2 element group: identity and single reflection) Cn: cyclic n-fold rotation (C1 is a trivial group containing identity operation) SO(3): group of rotations in 3D (isomorphic with SU(2)) Poincare group (Translations, Lorentz transformations) What are generators and irreducible representations of SO(3)?

Scalars, Vectors, Tensors… Orthogonal transformations U preserve lengths of vectors and angles between them map orthonormal bases to orthonormal bases Orthogonal transformations in two- or three-dimensional Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection (also known as improper rotations). det(U)=1 – usual (stiff) rotations (scalars, vectors,…) det(U)=-1 – improper rotations (pseudo-scalars, axial vectors, …) Improper rotation operation S4 in CH4

Translational Invariance Total momentum (nucleons, mesons, photons, leptons, etc.) Transformation generator Time displacement

Rotations in 3D (space isotropy) (+ cycl.) Total angular momentum Transformation generator SU(2) group! Rotational states of the system labeled by the total angular momentum quantum numbers JM see examples at http://www.nndc.bnl.gov

Galilean (Lorentz) Invariance In atomic nucleus v2/c2<0.1, i.e., kinematics is nonrelativistic Such a separation can be done for Galilean-invariant interactions Depends only on relative coordinates and velocities!

Relativistic generalization no new conservation laws and quantum numbers! Relativistic generalization Center-of-mass coordinate cannot be introduced in a relativistically covariant manner All powers of c.m. momentum are present Unitary transformation contains gradient terms and spin-dependent pieces!

Space Reflection (Parity) Parity is violated by weak interaction. The simplest of a parity-violated interaction requires a pseudoscalar field. If one assumes rotational invariance, the field looks like (*) Why not take instead? The interaction (*) produces a very small parity mixing J= Parity-violating matrix elements are of the order of 0.1 eV. This leads to the mixing amplitude of the order of 10-7 J=+

Experimental test of parity violation (Lee and Yang, 1956; Wu et al., 1957) Parity violation in a beta decay of polarized 60Co: the emission of beta particles is greater in the direction opposite to that of the nuclear spin. pseudoscalar

Time Reversal T cannot be represented by an unitary operator. Unitary operations preserve algebraic relations between operators, while T changes the sign of commutation relations. In order to save the commutation relations, on has to introduce: antiunitary takes complex conjugate of all c numbers unitary

Time Reversal symmetry and nuclear reactions normal and inverse kinematics!

C - interchanges particles & antiparticles (charge conjugation) Other Symmetries C - interchanges particles & antiparticles (charge conjugation) CP - violated in K0 decay (1964 Cronin & Fitch experiment) CPT - follows from relativistic invariance The CPT theorem appeared for the first time in the work of Julian Schwinger in 1951 to prove the connection between spin and statistics. In 1954, Lüders and Pauli derived more explicit proofs. At about the same time, and independently, this theorem was also proved by John Stewart Bell. These proofs are based on the principle of Lorentz invariance and the principle of locality in the interaction of quantum fields. Since CP is violated, T has to be violated as well!

Isospin Symmetry Introduced 1932 by Heisenberg wave functions Protons and neutrons have almost identical mass: Dm/m = 1.4x10-3 Low energy np scattering and pp scattering below E=5 MeV, after correcting for Coulomb effects, is equal within a few percent in the 1S scattering channel. Energy spectra of “mirror” nuclei, (N,Z) and (Z,N), are almost identical. up and down quarks are very similar in mass, and have the same strong interactions. Particles made of the same numbers of up and down quarks have similar masses and are grouped together. wave functions Pauli isospin matrices SU(2) commutations Using spin and isospin algebra, and Pauli principle, find two-nucleon wave functions. Assume that the spatial part of the wave functions corresponds to an s-wave.

total isospin Tz component conserved! (charge conservation) charge independence T is conserved! The concept of isospin symmetry can be broadened to an even larger symmetry group, now called flavor symmetry. Once the kaons and their property of strangeness became better understood, it started to become clear that these, too, seemed to be a part of an enlarged symmetry that contained isospin as a subgroup. The larger symmetry was named the Eightfold Way by Gell-Mann, and was recognized to correspond to the adjoint representation of SU(3). While isospin symmetry is broken slightly, SU(3) symmetry is badly broken, due to the much higher mass of the strange quark compared to the up and down.

Group of permutations Exchange operators - exchanges particles i and j is hermitian and unitary: eigenvalues of are (identical particles cannot be distinguished) For identical particles, measurements performed on quantum states and have to yield identical results A principle, supported by experiment

is a basis of one-dimensional representation of the permutation group This principle implies that all many-body wave functions are eigenstates of is a basis of one-dimensional representation of the permutation group There are only two one-dimensional representations of the permutation group: for all i,j - fully symmetric representation for all i,j - fully antisymmetric representation Consequently, systems of identical particles form two separate classes: bosons (integer spins) fermions (half-integer spins) For spin-statistics theorem, see W. Pauli, Phys. Rev. 58, 716-722(1940)