Feynman Rules for mixed Bilinears

Question

I stumbled over an odd problem of two point functions I coundn't not figure out:

Essentially, how to derive Feynman rules for bilinears with different quantum numbers?

Let me give an example: Suppose the action of a complex field $\phi_n$, with some function g(n), takes the form:

$S=\int dt \sum_n ( \bar{\phi}_n ( \partial_t + \omega n ) \phi_n + g(n) ( \bar{\phi}_{n+1} \phi_n + \bar{\phi}_n \phi_{n+1}))$

Is it possible to work out diagrammatic rules, without going into an appropriate function basis, which combindes the two terms?

Answer 1

I found the solution:

The action can be written in matrix from:

$S= \int dt \sum_{n,m} \phi_n D_{n,m}^{-1} \phi_m$

Where $D_{n,m}$ is the propagator, which can be seen in the functional integral by adding external currents coupling to the field and calculating the Gaussian integral.
Therefore whats left to find the propagator is to solve the equation:

$((\partial_t + \omega n) \delta_{n,m} + g(n) ( \delta_{n,m+1} + \delta_{n+1,m} )) D_{n,m}= \delta_{n,m}$