Finally, the spectral density simplifies to give ρ ρ α α β β = 2 π π δ δ ( ξ ξ α α − − ω ω ) δ δ α α β β , so that the thermal Green function is G α α β β ( ω ω n ) = δ δ α α β β − − i ω ω n + ξ ξ β β and the retarded Green function is G α α β β ( ω ω ) = δ δ α α β β − − ( ω ω + i η η ) + ξ ξ β β . Note that the noninteracting Green function is diagonal, but this will not be true in the interacting case.