Abstract: The recently proposed Wigner function for a particle in an infinite lattice (Hinarejos M, Banuls MC and Perez A 2012 New J. Phys. 14 103009) is extended here to include an internal degree of freedom as spin. This extension is made by introducing a Wigner matrix. The formalism is developed to account for dynamical processes, with or without decoherence. We show explicit solutions for the case of Hamiltonian evolution under a position-dependent potential, and for evolution governed by a master equation under some simple models of decoherence, for which the Wigner matrix formalism is well suited. Discrete processes are also discussed. Finally, we discuss the possibility of introducing a negativity concept for the Wigner function in the case where the spin degree of freedom is included.

Abstract: Gamow-Teller (GT) transitions are the most common weak interaction processes of spin-isospin (sigma tau) type in atomic nuclei. They are of interest not only in nuclear physics but also in astrophysics; they play an important role in supernovae explosions and nucleosynthesis. The direct study of weak decay processes, however, gives relatively limited information about GT transitions and the states excited via GT transitions (GT states); beta decay can only access states at excitation energies lower than the decay Q-value, and neutrino-induced reactions have very small cross-sections. However, one should note that beta decay has a direct access to the absolute GT transition strengths B(GT) from a study of half-lives, Q(beta)-values and branching ratios. They also provide information on GT transitions in nuclei far-from-stability. Studies of M1 gamma transitions provide similar information. In contrast, the complementary charge-exchange (CE) reactions, such as the (p, n) or ((3)He, t) reactions at intermediate beam energies and 0 degrees, can selectively excite GT states up to high excitation energies in the final nucleus. It has been found empirically that there is a close proportionality between the cross-sections at 0 degrees and the transition strengths B(GT) in these CE reactions. Therefore, CE reactions are useful tools to study the relative values of B(GT) strengths up to high excitation energies. In recent ((3)He, t) measurements, one order-of-magnitude improvement in the energy resolution has been achieved. This has made it possible to make one-to-one comparisons of GT transitions studied in CE reactions and beta decays. Thus GT strengths in ((3)He, t) reactions can be normalised by the beta-decay values. In addition, comparisons with closely related M1 transitions studied in gamma decay or electron inelastic scattering [(e, e')1, and furthermore with “spin” M I transitions that can be studied by proton inelastic scattering [(p, p')[ have now been made possible. In these comparisons, the isospin quantum number T and associated symmetry structure in the same mass A nuclei (isobars) play a key role. Isospin symmetry can extend our scope even to the structures of unstable nuclei that are far from reach at present unstable beam factories.