Electron phonon coupling field integral I am stuck on the problem "electron-phonon coupling" part a in Altland and Simons page 187. There is a related question on PSE, but that concerns part b.
Borrowing from the notation in the link above:
We want to formulate the coherent state action of the electron-phonon system. The action is given by
$$
S[\bar{\phi},\phi,\bar{\psi},\psi] = S_{ph}[\bar{\phi},\phi] + S_{el}[\bar{\psi},\psi] + S_{el-ph}[\bar{\phi},\phi,\bar{\psi},\psi],
$$
where $S_{el}$ is the electronic non interacting action, which we don't need here, $S_{ph}$ is the free phonons action, and $S_{el-ph}$ is the action of the interaction:
$$
S_{ph} \; [\bar{\phi},\phi] = \sum_{q,j} \bar{\phi}_{qj}( - i\omega_n + \omega_q ) \phi_{qj}
$$
$$
S_{el-ph} \; \; [\bar{\phi},\phi,\bar{\psi},\psi] = \gamma \sum_{qj} \frac{i \vec{q}\cdot \vec{e}_j}{\sqrt{2m\omega_q}} \sum_{k,\sigma} \bar{\psi}_{k+q,\sigma} \psi_{k\sigma} (\phi_{qj} + \bar{\phi}_{-qj}).
$$
Here $\omega_q$ is the phonon dispersion relation, supposed to be dependent on $q$ only and such that $\omega_{-q}=\omega_q$, $j$ labels the phononic branches, $\sigma$ labels the electronic spin, $i\omega_n$ are Matsubara bosonic frequencies, $m$ and $\gamma$ are constants.
We are supposed to get the formula for $S_{el-ph} \; \; [\bar{\phi},\phi,\bar{\psi},\psi]$ from
$$
H_{el-ph} \, = \gamma \sum_{k, q, j} \, \frac{i q_j}{(2 m \omega_q)^{1/2}} \; n_q \; (a_{q, j} + a^\dagger_{-q, j}).
$$
I was able to get the formula for $S_{ph}\; [\bar{\phi},\phi]$, as the Hamiltonian mirrors the Hamiltonian given in equation (4.26) on page 167. I'm stuck getting the formula for $S_{el-ph} \; \; [\bar{\phi},\phi,\bar{\psi},\psi]$ because
i. it seems we have both the Grassman field $\psi$ and the complex field $\phi$, and as opposed to the Hamiltonian discussed previously on page 167, we now have creation/annihilation operators ($a_{q, j} + a^\dagger_{-q, j}$,) appearing by themselves and not in pairs.
ii. I would have expected to have a term like $- i\omega_n$, like there is in $S_{ph} \; [\bar{\phi},\phi]$ and equation 4.27 on page 167, but it is not present in  the desired expression for $S_{el-ph} \; \; [\bar{\phi},\phi,\bar{\psi},\psi]$.
 A: First, there is a typo in the book: in the first equation on page 188, you shouldn't have the summation over $\mathbf{k}$, but whatever.
Let's start with the Hamiltonian:
$$
H = H_e + H_p + H_{ep} = H_e + \sum_{\mathbf{q}} \omega_{\mathbf{q}}a_{\mathbf{q}}^\dagger a_{\mathbf{q}} + \sum_\mathbf{q}\Gamma_\mathbf{q}\hat{n}_\mathbf{q}(a_\mathbf{q}+a^\dagger_{-\mathbf{q}})\,.
$$
To make the notation shorter, I suppressed the $j$ index and lumped the entire coupling term into $\Gamma_\mathbf{q}$.
We assume that $H_e$ is normal-ordered and write out the remaining part explicitly to show the fermionic operators:
$$
H = H_e + \sum_{\mathbf{q}} \omega_{\mathbf{q}}a_{\mathbf{q}}^\dagger a_{\mathbf{q}} + \sum_{\mathbf{qk}}\Gamma_\mathbf{q}c_{\mathbf{q}+\mathbf{k}}^\dagger c_\mathbf{k}(a_\mathbf{q}+a^\dagger_{-\mathbf{q}})\,.
$$
Using the fact that a product of two fermionic operators behaves like a bosonic one, we write
$$
H = H_e + \sum_{\mathbf{q}} \omega_{\mathbf{q}}a_{\mathbf{q}}^\dagger a_{\mathbf{q}} 
+ \sum_{\mathbf{qk}}\Gamma_\mathbf{q}c_{\mathbf{q}+\mathbf{k}}^\dagger c_\mathbf{k}a_\mathbf{q}
+ \sum_{\mathbf{qk}}\Gamma_\mathbf{q}a^\dagger_{-\mathbf{q}} c_{\mathbf{q}+\mathbf{k}}^\dagger c_\mathbf{k}\,,
$$
which is now normal-ordered.
Next, we do the usual dance:
$$
\sum_n\langle n|e^{-\beta H}|n\rangle = \lim_{N\rightarrow\infty} \sum_n\langle n|e^{-\delta H}e^{-\delta H}\dots e^{-\delta H}|n\rangle\,,
$$
where $\delta = \beta / N$.
Now, between each pair of $e^{-\delta H}$ terms, we insert a coherent resolvent sums for the two operator species. Let's focus on one of the $e^{-\delta H}$ terms:
$$
1 \, e^{-\delta H} \,1
\approx
1\left[1-\delta H\right]1 
\\
= \int d(\phi_l,\bar\phi_l)d(\bar\psi_l,\psi_l)
e^{-\sum_{\; \mathbf{q}} \, \bar\phi_{\; l,\mathbf{q} \;} \, \phi_{\; l,\mathbf{q} \; }}
\; e^{-\sum_\mathbf{q} \, \bar\psi_{\; l,\mathbf{q \; }\; } \, \psi_{\; 
 l,\mathbf{q} \; }}\; |\phi_l\rangle\otimes|\psi_l\rangle
\langle\phi_l|\otimes\langle\psi_l|
\\
\left[1-\delta H\right]
\\
\int d(\phi_{l+1 \; },\bar\phi_{l+1 \; })d(\bar\psi_{l+1 \; },\psi_{l+1 \; })
\; e^{-\sum_\mathbf{\; q} \, \bar\phi_{\, l+1,\mathbf{q \; }} \; \;  \phi_{\, l+1,\mathbf{q \; }}}
\, \; \; e^{-\sum_\mathbf{\; q \; } \, \bar\psi_{\; 
 l+1,\mathbf{q} \; } \; \; \psi_{ \; 
 l+1,\mathbf{q \; }}}\; \; |\phi_{l+1}\rangle\otimes|\psi_{l+1}\rangle
\langle\phi_{l+1}|\otimes\langle\psi_{l+1}|
\\
= \int d(\dots)
e^{-\sum_\mathbf{\; q \; } \; \bar\phi_{\; l,\mathbf{q\; }}\; \phi_{\; l,\mathbf{q\; }} \; - \; \sum_\mathbf{q} \; \bar\phi_{l+1,\mathbf{q}} \; \; \;  \phi_{l+1,\mathbf{q}}} \; \; \; \; 
e^{- \sum_\mathbf{q} \; \bar\psi_{\, l,\mathbf{q}} \; \psi_{\, l,\mathbf{q}} \; \; - \;  \sum_\mathbf{q} \; \bar\psi_{\, l+1,\mathbf{q}} \; \; \; \psi_{\, l+1,\mathbf{q}}}
\\
|\phi_l\rangle\otimes|\psi_l\rangle
\langle\phi_l|\otimes\langle\psi_l|
\left[1-\delta H\right]
|\phi_{l+1}\rangle\otimes|\psi_{l+1}\rangle
\langle\phi_{l+1}|\otimes\langle\psi_{l+1}|\,.
$$
Here, we put all the integrals together to keep the expression shorter. Now, we look at the bracket term inside:
$$
\langle\phi_l|\otimes\langle\psi_l|
\left[1-\delta H\right]
|\phi_{l+1}\rangle\otimes|\psi_{l+1}\rangle = 
\langle\phi_l|\otimes\langle\psi_l|
|\phi_{l+1}\rangle\otimes|\psi_{l+1}\rangle-\delta
\langle\phi_l|\otimes\langle\psi_l|
H
|\phi_{l+1}\rangle\otimes|\psi_{l+1}\rangle
\\
=\langle\phi_l|\phi_{l+1}\rangle\langle\psi_l|\psi_{l+1}\rangle-\delta
\langle\phi_l|\otimes\langle\psi_l|
H
|\phi_{l+1}\rangle\otimes|\psi_{l+1}\rangle
\\
=\langle\phi_l|\phi_{l+1}\rangle\langle\psi_l|\psi_{l+1}\rangle-\delta
\langle\phi_l|\otimes\langle\psi_l|
(H_e + H_p + H_{ep})
|\phi_{l+1}\rangle\otimes|\psi_{l+1}\rangle
\\
=\langle\phi_l|\phi_{l+1}\rangle\langle\psi_l|\psi_{l+1}\rangle
-
\delta
\langle\phi_l|\otimes\langle\psi_l|
H_e 
|\phi_{l+1}\rangle\otimes|\psi_{l+1}\rangle
-
\delta
\langle\phi_l|\otimes\langle\psi_l|
 H_p 
|\phi_{l+1}\rangle\otimes|\psi_{l+1}\rangle
-
\delta
\langle\phi_l|\otimes\langle\psi_l|
 H_{ep}
|\phi_{l+1}\rangle\otimes|\psi_{l+1}\rangle
\\
=\langle \phi_l| \phi_{l+1} \rangle \langle \psi_l |\psi_{l+1} \rangle
- \delta \langle \phi_l |\phi_{l+1}\rangle \langle\psi_l|H_e |\psi_{l+1}\rangle
-
\delta
\langle\phi_l|H_p|\phi_{l+1}\rangle\langle\psi_l|\psi_{l+1}\rangle
-
\delta
\langle\phi_l|\otimes\langle\psi_l|
 H_{ep}
|\phi_{l+1}\rangle\otimes|\psi_{l+1}\rangle\,.
$$
The first 3 terms should be familiar from other examples where the types of the operators are not mixed. For the last term, the bosonic operators act on $\phi$ states and the fermionic act on $\psi$. This means that $a$ gets replaced by $\psi$ and $c$ by $\phi$, as usual. After that, we have
$$
\langle \phi_l| \phi_{l+1} \rangle \langle \psi_l |\psi_{l+1} \rangle
- S_e\delta \langle \phi_l |\phi_{l+1}\rangle \langle\psi_l |\psi_{l+1}\rangle
-
S_p\delta
\langle\phi_l|\phi_{l+1}\rangle\langle\psi_l|\psi_{l+1}\rangle
-
S_{ep}\delta
\langle\phi_l|\otimes\langle\psi_l|
|\phi_{l+1}\rangle\otimes|\psi_{l+1}\rangle
\\
= e^{-\delta (S_e + S_p + S_{ep})}\langle \phi_l| \phi_{l+1} \rangle \langle \psi_l |\psi_{l+1} \rangle\,.
$$
The rest follows the usual procedure: reinsert the term into the product and collect all the multiples to form a "time" integral. From there, you can go to the Matsubara form, like the one in the book.
