Wilson line and external source Let's consider free Maxwell theory:
$$
L = -\frac{1}{4g^2} F^{\mu\nu}F_{\mu\nu}
$$
As I understand, one can describe external particles with help of Wilson lines:
$$
W(q,l) = e^{iq\int_l dx^\mu A_\mu}
$$
I don't understand, how to work with such operators? Could I calculate interaction between particles? Could I see how such operators deform free equations of motion?
 A: Let’s say you want to calculate a path integral with some local insertions and Wilson line operators.
Step 1. Taylor-expand the exponentials in the Wilson line operators.
Step 2. For each order in $q$, calculate the path integral as you normally would using Wick’s theorem. You need to use a gauge-fixed propagator for the photon.
Step 3. For each order in $q$, take the integrals over Wilson loops in the result of step 2. This is nontrivial unless simplifying assumptions are made about the shape of Wilson loops. You will encounter situations where two end points of the photon propagator are the same point on the Wilson line — regularize by taking each integral over the same Wilson line to be an integral over a slightly disposed line.
Step 4. Sum up the resulting series. Since there are no interactions, the series is combinatorial and easily summable.
Step 5. Renormalize the operator by multiplying it by an infinite constant (it is an exponential, so infinite summands become infinite factors after exponentiation), and take the limit where displaced lines exactly match the original one.
This algorithm actually works — for example, Witten used it for Abelian Chern-Simons theory to demonstrate that the result is a topological invariant of the Wilson loops called the linking number.
The corresponding operators can be defined from such path integrals by a procedure analogous to Wightman reconstruction, only with integral operators. There can be obstructions, however, which have to do with infinities that appear for intersecting lines. Some sort of smearing is needed to make the Wilson lines into well-defined operators (remember that a quantum field is an operator valued distribution?). It is unclear to me how to achieve that at the moment, I don't know if this problem is solved in Constructive QFT literature.
