From the commutation relations, ⟨ ⟨ α α ′ ∣ ∣ ψ ψ k ψ ψ k † † ∣ ∣ α α ′ ⟩ ⟩ = ⟨ ⟨ α α ′ ∣ ∣ ( 1 + ζ ζ ψ ψ k † † ψ ψ k ) ∣ ∣ α α ′ ⟩ ⟩ , with possible factors of the volume again. The sum, which involves the thermal average of the number operator, then gives simply [ 1 + ζ ζ n ( ξ ξ k ) ] Z , leaving ρ ρ 0 ( k , ω ω ) = 2 π π δ δ ( ξ ξ k − − ω ω ) .