After reading through a number of questions on SE including What are field quanta? and What are quantum fields mathematically?, I am still struggling with what specific operators are associated to quantum fields like the electron or electromagnetic field.
From my understanding, a quantum field in QFT is an operator-valued distribution. This means that it associates an operator to each distribution in spacetime, and in particular to single points in spacetime given by Dirac delta distributions. My understanding is also that operators are associated to observables, for instance that of position/momentum/energy.
Now take the electron field in particular (or the electromagnetic field if easier). What are the operators given by the electron field? What is the associated observable? Is it the creation operator, or the annihilation operator, or both, or the particle number operator, or the energy operator?
My current thought process is that it should be something like the creation operator, after reading that electrons are quanta of the electron field and photons are quanta of the electromagnetic field... Since then it would make sense to call 'electrons' the first application of the electron field's corresponding operator (the creation operator) on the vacuum state $\left|0\right>$.