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

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 ($ξ=0$) and the 0th-order regularization for conformal coupling ($ξ=\frac16$) 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 $ξ=0$ 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 $ξ=0$ 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.