 The propagators in real and imaginary time can both be related to the spectral density (or spectral weight), given by
where |⟩ refers to a (many-body) eigenstate of the grand-canonical Hamiltonian , with eigenvalue .The imaginary-time propagator is then given by
and the retarded propagator by
where the limit as formula_55 is implied.The advanced propagator is given by the same expression, but with formula_56 in the denominator.