We then rewrite ρ ρ α α β β ( ω ω ) = 1 Z ∑ ∑ m , n 2 π π δ δ ( ξ ξ α α − − ω ω ) δ δ ξ ξ α α , ξ ξ β β ⟨ ⟨ m ∣ ∣ ψ ψ α α ∣ ∣ n ⟩ ⟩ ⟨ ⟨ n ∣ ∣ ψ ψ β β † † ∣ ∣ m ⟩ ⟩ e − − β β E m ( 1 − − ζ ζ e − − β β ξ ξ α α ) , therefore ρ ρ α α β β ( ω ω ) = 1 Z ∑ ∑ m 2 π π δ δ ( ξ ξ α α − − ω ω ) δ δ ξ ξ α α , ξ ξ β β ⟨ ⟨ m ∣ ∣ ψ ψ α α ψ ψ β β † † e − − β β ( H − − μ μ N ) ∣ ∣ m ⟩ ⟩ ( 1 − − ζ ζ e − − β β ξ ξ α α ) , use ⟨ ⟨ m ∣ ∣ ψ ψ α α ψ ψ β β † † ∣ ∣ m ⟩ ⟩ = δ δ α α , β β ⟨ ⟨ m ∣ ∣ ζ ζ ψ ψ α α † † ψ ψ α α + 1 ∣ ∣ m ⟩ ⟩ and the fact that the thermal average of the number operator gives the Bose–Einstein or Fermi–Dirac distribution function.