In non-relativistic QM, the position of a particle is an observable. In QFT, fields are the observables. However, particles must have some sort of position, otherwise we wouldn't see pictures like the ones below. What is the (linear, Hermitian) operator that describes what we are observing?
I've tried considering making a "local" version of $N = \int dk\, a^+ a^- = \int dx \, \phi^+(x) \phi^-(x)$
by replacing this with $N_R =\int dx \, f(x) \phi^+ \phi^-$ where $f$ is some function that is concentrated in the localized region $R$. However, this is not Hermitian!