 The sum, which involves the thermal average of the number operator, then gives simply formula_99, leaving

The imaginary-time propagator is thus
and the retarded propagator is

As →∞, the spectral density becomes
where = 0 corresponds to the ground state.Note that only the first (second) term contributes when is positive (negative).We can use 'field operators' as above, or creation and annihilation operators associated with other single-particle states, perhaps eigenstates of the (noninteracting) kinetic energy.