 Lorentzian quasiparticle spectral peak

A normalized retarded amplitude $G^R(\mathbf{k},t)=-i\Theta (t)e^{-i{\Omega }_{\mathbf{k}}t-{\Gamma }_{\mathbf{k}}t}$, with constant ${\Gamma }_{\mathbf{k}}>0$, has Fourier transform $G^R(\mathbf{k},\omega )=(\omega -{\Omega }_{\mathbf{k}}+i{\Gamma }_{\mathbf{k}}{)}^{-1}$. Consequently $A=-\mathrm{Im}{G}^R/\pi$ is the displayed positive Lorentzian line shape, of unit frequency integral. Its maximum is at ${\Omega }_{\mathbf{k}}$ and its half width at half maximum is ${\Gamma }_{\mathbf{k}}$. The energy half-width is $\hslash {\Gamma }_{\mathbf{k}}$. Amplitude decays on time $1/{\Gamma }_{\mathbf{k}}$ whereas probability decays on time $1/(2{\Gamma }_{\mathbf{k}})$. A general interacting electron has quasiparticle residue $Z$ and incoherent spectral weight in addition to this normalized local pole model; frequency-dependent width or self-energy can invalidate a global Lorentzian.