  Starting with a new theory of symmetries generated by isometries in field
theories with spin, one finds the generators of the spinor representation in
backgrounds with a given symmetry. In this manner one obtains a collection of
conserved operators from which one can chose the complete sets of commuting
operators defining quantum modes. In this framework, the quantum modes of the
free Dirac field on de Sitter or anti-de Sitter spacetimes can be completely
derived in static or moving charts. One presents the discrete quantum modes, in
the central static charts of the anti-de Sitter spacetime, whose eigenspinors
can be normalized. The consequence is that the second quantization can be done
in this case in canonical manner. For the free Dirac field on de Sitter
manifolds this can not be done in static charts being forced to consider the
moving ones. The quantum modes of the free Dirac field in these charts are used
for writing down the quantum Dirac field and its one-particle operators.
