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

Abstract

We perform the point-splitting regularizationon the vacuum stress tensor of a coupling scalar field in de Sitter spaceunder the guidance from the adiabatically regularized Green's function.For the massive scalar field with the minimal coupling ξ = 0,the 2nd order point-splitting regularizationyields a finite vacuum stress tensorwith a positive, constant energy density,which can be identified as the cosmological constantthat drives de Sitter inflation.For the coupling ξ ≠ 0, we find that,even if the regularized Green's function is continuous, UV and IR convergent,the point-splitting regularization does notautomatically lead to an appropriate stress tensor.The coupling ξ R causes log divergent terms,as well as higher-order finite termswhich depend upon the path of the coincidence limit.After removing these unwanted terms by extra treatments,the 2nd-order regularizationfor small couplings ξ ∈ (0,1/7.04),and respectively the 0th-order regularizationfor the conformal coupling ξ=1/6,yield a finite, constant vacuum stress tensor, in analogy to the case ξ=0.For the massless field with ξ = 0 or ξ = 1/6,the point-splitting regularizationyields a vanishing vacuum stress tensor,and there is no conformal trace anomaly for ξ = 1/6.If the 4th-order regularization were taken,the regularized energy density for general ξ would be negative,which is inconsistent with the de Sitter inflation,and the regularized Green's function would be singular at the zero mass,which is unphysical.In all these cases,the stress tensor from the point-splitting regularization is equal to that from the adiabatic one. We also discuss the issue ofthe adequate order of regularization.