Discrete symmetries of Dirac's theory in the de Sitter manifold

The discrete symmetries of Dirac's free field on the de Sitter manifold are studied, taking into account that this has two portions that can play the role of physical space-times, namely an expanding and a collapsing universe. The proper discrete isometries which preserve the portion have a physical meaning, in contrast to the improper ones, which change the portion being thus merely of mathematical interest. The discrete symmetries generated by the proper isometries and charge conjugation are studied in physical frames on the expanding portion, showing that all discrete transformations reversing the cosmic time are local, depending on a local boost matrix. The discrete group of Dirac's theory in the de Sitter expanding universe is obtained, showing that this is of the order 16, having a multiplication table similar to that of Dirac's theory in special relativity. Moreover, all the discrete de Sitter isometries, including the improper ones, are studied in conformal frames for obtaining a global image of the de Sitter isometries, despite the fact that these cannot be gathered in a larger discrete group with physical meaning.