To study excited states in meny-fermion systems, the limit of the two-particle (tp) propagator is used
G p h ( α , β − 1 ; γ , δ − 1 ; t − t ′ ) = lim t β → t + lim t γ → t ′ + G I I ( α t , δ t ′ , β t β , γ t γ ) = − i ℏ ⟨ ψ o N | T [ a β H + ( t ) a α H ( t ) a γ H + ( t ′ ) a δ H ( t ′ ) ] | ψ o N ⟩ {\displaystyle G_{ph}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};t-t')=\lim _{t_{\beta }\rightarrow t^{+}}\lim _{t_{\gamma }\rightarrow t'^{+}}G_{II}(\alpha t,\delta t',\beta t_{\beta },\gamma t_{\gamma })=-{\frac {i}{\hbar }}\langle \psi _{o}^{N}|T[a_{\beta }^{H+}(t)a_{\alpha }^{H}(t)a_{\gamma }^{H+}(t')a_{\delta }^{H}(t')]|\psi _{o}^{N}\rangle } (Eq. 1)
![{\displaystyle G_{ph}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};t-t')=\lim _{t_{\beta }\rightarrow t^{+}}\lim _{t_{\gamma }\rightarrow t'^{+}}G_{II}(\alpha t,\delta t',\beta t_{\beta },\gamma t_{\gamma })=-{\frac {i}{\hbar }}\langle \psi _{o}^{N}|T[a_{\beta }^{H+}(t)a_{\alpha }^{H}(t)a_{\gamma }^{H+}(t')a_{\delta }^{H}(t')]|\psi _{o}^{N}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/49be2b225bbbca1de8a59e6a990c9d7e75fb2085)
(Eq. 1)