I've recently been trying to compute the Green's function for a non-interacting system of fermions. Since this is a site for mathematicians, for context, let me provide the following definition:

Definition:Anoninteracting system of fermionsis a quantum dynamical system along with the following data:

- A single-particle Hilbert space $\mathfrak h$, for which the full Hilbert space of the dynamical system is the exterior algebra $\Lambda(\mathfrak h):=\oplus_{n\geq 0}\,\Lambda^n(\mathfrak h)$.
- A non-interacting Hamiltonian $H$, which, for some basis $f_1,\cdots f_{\dim \mathfrak h}$ of the single particle Hilbert space reads as $H=\sum_{ij}A_{ij}c^\dagger(f_i)c(f_j).$

I've been puzzled by how physicists go about computing the Green's function of a non-interacting Hamiltonian $H$. To see what I mean, here is a theorem:

Theorem (Classification of Non-interacting Systems):For $(\Lambda(\mathfrak h), A_{ij})$ a general non-interacting system of fermions, the time-evolution of any $k$-particle state factors in the following manner: \begin{align*} e^{itH}(g_1\wedge \cdots \wedge g_k)=e^{it\mathcal H}g_1\wedge \cdots \wedge e^{it\mathcal H}g_k \end{align*} Where $\mathcal H= \sum_{ij}A_{ij}\,\left|f_i\right>\left<f^j\right|$ is a single-particle Hamiltonian, and the raised index indicates dualization. In other words, a "non-interacting system of identical fermions" always factors into a set of identical, non-interacting, single-particle systems, where the single-particle dynamics has the replacements \begin{align*} c^*(f_i)\,c(f_j)\mapsto \left|f_i\right>\left<f_j\right|.\\ \end{align*}

Enough background. When physicists say, "The Green's function of this non-interacting Hamiltonian $H$", I would think that they mean
$$G:=\frac{1}{\frac{i}{\hbar}H-\partial_t},~~~~~~ H=\sum_{ij}A_{ij}c^\dagger(f_i)c(f_j). $$
However, they really mean the Green's function of the associated single-particle Hamiltonian:
$$G':=\frac{1}{\frac{i}{\hbar}\mathcal H-\partial_t},~~~~~~\mathcal H= \sum_{ij}A_{ij}\,\left|f_i\right>\left<f^j\right|.$$
However, this does not generalize straightforwardly to interacting systems, and therefore, I am actually curious: ** is there a nice formula for $G$ in terms of $G'$?** Naively, using the direct-sum decomposition $\Lambda(\mathfrak h)=\oplus_{k\geq 0}\,\Lambda^k(\mathfrak h)$, we get
$$\frac{1}{\frac{i}{\hbar}H-\partial_t}=\bigoplus_{k\geq 0} \frac{1}{\frac{i}{\hbar}H_k-\partial_t}$$
So this reduces to computing the Green's function of the $k$-particle Hamiltonian: $(\frac{i}{\hbar}H_k-\partial_t)^{-1}$. However, this is as far as I can get on my own.