The propagators in real and imaginary time can both be related to the spectral density (or spectral weight), given by ρ ρ ( k , ω ω ) = 1 Z ∑ ∑ α α , α α ′ 2 π π δ δ ( E α α − − E α α ′ − − ω ω ) | ⟨ ⟨ α α ∣ ∣ ψ ψ k † † ∣ ∣ α α ′ ⟩ ⟩ | 2 ( e − − β β E α α ′ − − ζ ζ e − − β β E α α ) , where | α ⟩ refers to a (many-body) eigenstate of the grand-canonical Hamiltonian H − μN, with eigenvalue E α.