In the context of quantum field theory, what does it mean to "couple" something? Suppose I have the following Lagrangian density
\begin{equation}
\mathcal{L} = - \frac{1}{4} F_{\mu\nu}F^{\mu\nu}
\end{equation} 
The lecture notes I an reading suggest if I want to "couple to matter", I would write a Lagrangian density such as
\begin{equation}
\mathcal{L} = - \frac{1}{4} F_{\mu\nu}F^{\mu\nu} - j^{\mu}A_{\mu}
\end{equation} 
where $j^\mu$ is a function of matter fields. 
My Question:


*

*What does it mean to "couple"?  Why does coupling involve adding additional terms to the Lagrangian density? 

*When he says "couple to matter", what are we coupling? The electromagnetic field to the matter? 
 A: In terms of Feynman diagrams, a "coupling" translates to a vertex factor. The Lagrangian for a free electromagnetic field is
$$\mathcal{L}=-\frac{1}{4}F^2$$
as you well know. Now suppose we have an electron field $\psi$ too. We want this electron field to "interact", or couple, with (to) the photon field. The free Dirac Lagrangian is
$$\mathcal{L}=\bar\psi(i\gamma^\mu\partial_\mu-m)\psi$$
We can construct from the Dirac equation the electron current
$$j^\mu\propto\bar\psi\gamma^\mu\psi$$
The proportionality constant is $e$, the electron charge. In a sense, $e$ describes the strength of the interaction between photons and electrons. The coupling term is $j_\mu A^\mu$ and thus the full Lagrangian is
$$\tag{1}\mathcal{L}=\bar\psi(i\gamma^\mu\partial_\mu-m)\psi-\frac{1}{4}F^2+e\bar\psi\gamma^\mu A_\mu\psi$$
It is also commonly written as
$$\mathcal{L}=\bar\psi(i\gamma^\mu D_\mu-m)\psi-\frac{1}{4}F^2,\quad D_\mu=\partial_\mu-ieA_\mu$$
So what do all these terms mean in terms of QFT? The first term in (1) gives
$$\frac{i}{\gamma^\mu p_\mu-m+i\epsilon}$$
which is the fermion propagator. The second term gives (after a Faddeev-Popov treatment)
$$\frac{i}{k^2+i\epsilon}\left[(1-\xi)\frac{k_\mu k_\nu}{k^2+i\epsilon}-\eta_{\mu\nu}\right]$$
which is the photon propagator. 
The last term is a bit more tricky. It describes an electron interacting with a photon. This is represented by the vertex factor
$$ie\gamma^\mu$$
which describes one of four situations:


*

*An electron emits a photon.

*An electron absorbs a photon. 

*An electron annihilates with a position to form a photon.

*A photon turns into an electron and a positron via pair production. 
