Retarded Green function ($G_R$)

For a bosonic operator $A$, the retarded Green function is $G_R(t)=-i\theta (t)⟨[A(t),A(0)]⟩$. Causality makes its Fourier transform analytic in the upper half frequency plane.

The retarded Green function for $d^2/d{t}^2+1$ is $G_R(t)=\Theta (t)\sin t$, with Fourier transform $[1-(\omega +i{0}^{+}{)}^2{]}^{-1}$ in the $e^{-i\omega t}$ convention. Both poles lie below the real axis. Convolution with a force gives the solution with no incoming homogeneous oscillation. A source that is nonzero at negative times can have a nonzero response before a chosen zero of time without violating causality.

For $g(t)=\alpha e^{-\alpha |t|}/2$ with $\alpha >0$, convolution givesThe coordinate and velocity are continuous at zero. At late times the undamped oscillation has amplitude ${\alpha }^2/(1+{\alpha }^2)$; causality fixes the homogeneous solution only after the no-incoming-oscillation condition is specified.

The Lehmann representation expands a thermal correlation function in energy eigenstates. Its poles occur at transition frequencies, and its residues are thermal occupation differences times operator matrix elements.

The spectral function records the weighted energy differences connected by an operator. With the convention $G_R=-i\theta ⟨[A,A]⟩$, it is proportional to $-\mathrm{Im}{G}_R$.

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.

The electron single-particle spectral function $A(\mathbf{k},ϵ)=-\mathrm{Im}{G}^R(\mathbf{k},ϵ)/\pi$, in a convention normalized to unit energy integral per orbital and spin, describes addition and removal energies. For the fermionic retarded Green function, $G^R(\mathbf{k},t)=-i\Theta (t)⟨\{c_{\mathbf{k}}(t),c_{\mathbf{k}}^{†}(0)\}⟩$, energy is referenced to the chemical potential. The occupied removal spectrum is $f(ϵ)A(\mathbf{k},ϵ)$, with Fermi-Dirac distribution $f$. In the noninteracting limit, $A$ is a delta peak at the band energy; interactions can shift, broaden, and redistribute its weight.

The fluctuation-dissipation theorem relates an equilibrium correlation function to the dissipative part of the corresponding linear response. If $\rho =-2\mathrm{Im}{G}_R$, one common convention is $S(\omega )=\rho (\omega )/(1-e^{-\omega /T})$, which becomes $S(\omega )\simeq T\rho (\omega )/\omega$ in the classical limit $|\omega |\ll T$.

A Matsubara correlator is obtained from the retarded spectral density by a Cauchy transform evaluated at an imaginary frequency.

A Drude response has a single relaxation pole, $G_R(\omega )={\sigma }_0/(1-i\omega {\tau }_0)$. Causality places the pole in the lower half plane when ${\tau }_0>0$.