Dirac fermions in de Sitter and anti-de Sitter backgrounds

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. The discrete quantum modes are presented 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 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 the its one-particle operators.