Time-reversal properties in the coupling of quantum angular momenta

After a synoptic panorama about some still unsolved foundational problems involving time-reversal, we show that the double time-reversal superselection rule of Nonrelativistic Quantum Mechanics is redundant. We then analyze which, among the symmetries of the Clebsch-Gordan coefficients, may be inferred through time-reversal considerations. Finally, we show how Coupling Theory allows to improve our comprehension of the fact that, in presence of hidden symmetries, Wigner's Theorem concerning Kramers degeneration cannot be applied, by analyzing a set of Z uncoupled massive particles, both in the case in which they are bosons of spin zero and in the case in which they are fermions of spin 1/2, subjected to a Keplerian force's field and considering the coupling of the 2Z quantum angular momenta resulting by the hidden SO(4) symmetry owed to the fact that, apart from the rotational SO(3) symmetry responsible of the conservation of the angular momentum, the Laplace-Runge-Lenz vector is also conserved.