Suppose I would like to compute (time ordered) vacuum expectation values for a quantum field theory by using the path integral approach. Using the Lagrangian for the theory, we define a generating functional $Z[f]$ where $f$ is the source term. The vacuum expectation values are obtained by taking functional derivatives of $Z[f]$.
I have read that $f$ can be interpreted as either a purely mathematical tool or as a physical field with sources/sinks that creates/annihilate particles. In either case, the functional derivatives will depend on our choice of $f$, so how do we choose it so that it is "physically relevant"?
