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 $Z_{in/out}$ is divergent at late times. By using dimensional regularization, we extract the finite part of $log Z_{in/out}$ for $d$ even and the logarithmically divergent part of $log Z_{in/out}$ 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.
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