Since | α α ⟩ ⟩ and | α α ′ ⟩ ⟩ are eigenstates of H − − μ μ N , the Heisenberg operators can be rewritten in terms of Schrödinger operators, giving G ( x , τ τ | 0 , 0 ) = 1 Z ∑ ∑ α α , α α ′ e − − β β E α α ′ e τ τ ( E α α ′ − − E α α ) ⟨ ⟨ α α ′ ∣ ∣ ψ ψ ( x ) ∣ ∣ α α ⟩ ⟩ ⟨ ⟨ α α ∣ ∣ ψ ψ † † ( 0 ) ∣ ∣ α α ′ ⟩ ⟩ . Performing the Fourier transform then gives G ( k , ω ω n ) = 1 Z ∑ ∑ α α , α α ′ e − − β β E α α ′ 1 − − ζ ζ e β β ( E α α ′ − − E α α ) − − i ω ω n + E α α − − E α α ′ ∫ ∫ k ′ d k ′ ⟨ ⟨ α α ∣ ∣ ψ ψ ( k ) ∣ ∣ α α ′ ⟩ ⟩ ⟨ ⟨ α α ′ ∣ ∣ ψ ψ † † ( k ′ ) ∣ ∣ α α ⟩ ⟩ .