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We perform the point-splitting regularization on the vacuum stress tensor ofa coupling scalar field in de Sitter space under the guidance from theadiabatically regularized Green's function. For the massive scalar field withthe minimal coupling $\xi=0$, the 2nd order point-splitting regularizationyields a finite vacuum stress tensor with a positive, constant energy density,which can be identified as the cosmological constant that drives de Sitterinflation. For the coupling $\xi\ne 0$, we find that, even if the regularizedGreen's function is continuous, UV and IR convergent, the point-splittingregularization does not automatically lead to an appropriate stress tensor. Thecoupling $\xi R$ causes log divergent terms, as well as higher-order finiteterms which depend upon the path of the coincidence limit. After removing theseunwanted terms by extra treatments, the 2nd-order regularization for smallcouplings $\xi \in(0,\frac{1}{7.04})$, and respectively the 0th-orderregularization for the conformal coupling $\xi=\frac16$, yield a finite,constant vacuum stress tensor, in analogy to the case $\xi=0$. For the masslessfield with $\xi=0$ or $\xi=\frac16$, the point-splitting regularization yieldsa vanishing vacuum stress tensor, and there is no conformal trace anomaly for$\xi=\frac16$. If the 4th-order regularization were taken, the regularizedenergy density for general $\xi$ would be negative, which is inconsistent withthe de Sitter inflation, and the regularized Green's function would be singularat the zero mass, which is unphysical. In all these cases, the stress tensorfrom the point-splitting regularization is equal to that from the adiabaticone.

Point-splitting regularization of the stress tensor of a coupling scalar field in de Sitter space