  In the conventional adiabatic regularization the vacuum ultraviolet
divergences of a quantum field in curved spacetime are removed by subtracting
the $k$-mode of the stress tensor to the 4th-order. For a scalar field in de
Sitter space, we find that the 4th-order regularized spectral energy density is
negative. Moreover, the 2nd-order regularization for minimal coupling ($\xi=0$)
and the 0th-order regularization for conformal coupling ($\xi=\frac16$) yield a
positive and UV-convergent spectral energy density and power spectrum. The
regularized stress tensor in the vacuum is maximally symmetric and can drive
inflation, while its $k$-modes representing the primordial fluctuations are
nonuniformly distributed. Conventional regularization of a Green's function in
position space is generally plagued by a log IR divergence. Only in the
massless case with $\xi=0$ or $\frac16$, we can directly regularize the Green's
functions and obtain vanishing results that agree with the adiabatic
regularization results. In this case, the regularized power spectrum and stress
tensor are both zero, and no trace anomaly exists. To overcome the log IR
divergence problem in the massive cases with $\xi=0$ and $\frac16$, we perform
Fourier transformation of the regularized power spectra and obtain the
regularized analytical Green's functions which are IR- and UV-convergent.
