  Consequences of Schr\"{o}dinger's antipodal identification on quantum field
theory in de Sitter space are investigated. The elliptic $\mathbb{Z}_2$
identification provides observers with complete information. We show that a
suitable confinement on dimension of the elliptic de Sitter space guarantees
the existence of globally defined spinors and orientable $dS/\mathbb{Z}_2$
manifold. In Beltrami coordinates, we give exact solutions of scalar and spinor
fields. The CPT invariance of quantum field theory on the elliptic de Sitter
space is presented explicitly.
