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In the conventional adiabatic regularization the vacuum ultravioletdivergences of a quantum field in curved spacetime are removed by subtractingthe $k$-mode of the stress tensor to the 4th-order. For a scalar field in deSitter space, we find that the 4th-order regularized spectral energy density isnegative. Moreover, the 2nd-order regularization for minimal coupling ($\xi=0$)and the 0th-order regularization for conformal coupling ($\xi=\frac16$) yield apositive and UV-convergent spectral energy density and power spectrum. Theregularized stress tensor in the vacuum is maximally symmetric and can driveinflation, while its $k$-modes representing the primordial fluctuations arenonuniformly distributed. Conventional regularization of a Green's function inposition space is generally plagued by a log IR divergence. Only in themassless case with $\xi=0$ or $\frac16$, we can directly regularize the Green'sfunctions and obtain vanishing results that agree with the adiabaticregularization results. In this case, the regularized power spectrum and stresstensor are both zero, and no trace anomaly exists. To overcome the log IRdivergence problem in the massive cases with $\xi=0$ and $\frac16$, we performFourier transformation of the regularized power spectra and obtain theregularized analytical Green's functions which are IR- and UV-convergent.

Adiabatic regularization and Green's function of a scalar field in de Sitter space: Positive energy spectrum and no trace anomaly