  Spin dependence of quasiparticle mass has been observed recently in CeCoIn5
and other systems. It emerges from strong electronic correlations in a
magnetically polarized state and was predicted earlier. Additionally, the
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)phase has also been discovered in
CeCoIn5 and therefore, the question arises as to what extent these two basic
phenomena are interconnected, as it appears in theory. Here we show that the
appearance of the spin-split masses essentially extends the regime of
temperature and applied magnetic field, in which FFLO state is stable, and
thus, it is claimed to be very important for the phase detectability.
Furthermore, in the situation when the value of the spin z-component \sigma
differentiates masses of the particles, the fundamental question is to what
extent the two mutually bound particles are indistinguishable quantum
mechanically? By considering first the Cooper-pair state we show explicitly
that the antisymmetry of the spin-pair wave function in the ground state may be
broken when the magnetic field is applied.
