Adiabatic and point-splitting regularization of spin-1/2 field in de Sitter space

We study the regularization of a spin-1/2 fieldin the vacuum state in de Sitter space. We find that the 2nd order adiabatic regularization is sufficient to remove all UV divergences for the spectral stress tensor, as well as for the power spectrum. The regularized vacuum stress tensors of the massive field is maximally symmetric with the energy density remaining negative, and behaves as a ``negative" cosmological constant. In the massless limit it reduces smoothly to the zero stress tensor of the massless field, and there is no trace anomaly. We also perform the point-splitting regularization in coordinate space, and obtain the analytical, regularized correlation function and stress tensor, which agree with those from the adiabatic regularization. In contrast, the 4th order regularization is an oversubtraction, and changes the sign of the vacuum energy density. In the massless limit the 4th order regularized auto-correlation becomes singular and the regularized stress tensor does not reduce to the zero stress tensor of the massless field. These difficulties tell that the 4th order regularization is inadequate for the spin-1/2 massive field.