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Abstract In this paper, we compute the effective action of both a scalar field and a Dirac spinor field in the global de Sitter space of any dimension d using the in-/out-state formalism. We show that there is particle production in even dimensions for both scalar field and spinor field. The in-out vacuum amplitude Z in / out $$ {\mathcal{Z}}_{\mathrm{in}/\mathrm{out}} $$ is divergent at late times. By using dimensional regularization, we extract the finite part of log Z in / out $$ {\mathcal{Z}}_{\mathrm{in}/\mathrm{out}} $$ for d even and the logarithmically divergent part of log Z in / out $$ {\mathcal{Z}}_{\mathrm{in}/\mathrm{out}} $$ for d odd. We also find that the regularized in-out vacuum amplitude equals the ratio of determinants associated with different quantizations in AdS d upon the identification of certain parameters in the two theories.