Is it possible to Vectorialize Quantum Field Theories? If I take the rules for classical electrodynamics in the covariant formulation (the closest to QFT), I have a tensor that describes the field, $F_{\mu\nu}$. Now we know that we can take some of the components of that tensor and find two vectors, $\mathbf{E}$ and $\mathbf{B}$,  that obey Maxwell equations and have an identity on their own.
In my understanding the "photon field" has been reduced to something much more material, that I can visualize. I can see easily, for example, a wave propagating in a particular direction. I am aware that I tacitly assumed that the "photon field" is described by the classical theory, but I think that the concept is clear nonetheless.
Now, is it possible to do something like that to the "electron field", the "positron field", the "gluon field" and so on? If you take all the forces together and you have a giant field with many components (as in my understanding is the Standard Model), is it possible to find some combinations of those components that behave like vectors?
 A: If you start from QED, the classical EM fields, in principle, could be interpreted as expectation values, say $$F^{\Psi}_{\mu\nu}(x) := \langle \Psi| \hat{F}_{\mu\nu}(x) |\Psi\rangle\:.$$  The point is that if the state $\Psi$ contains a finite number of photons (in particular one photon) you immediately obtain $$\langle \Psi| \hat{F}_{\mu\nu}(x) |\Psi\rangle=0\:.$$ This is a trivial consequence of the fact that field operators are a linear combination of $a_k$ and $a^*_k$. To obtain $F^{\Psi}_{\mu\nu}(x) \neq 0$, in order to give a quantum interpretation of things like classical EM waves, you should take $\Psi$ as a coherent state, i.e., an eigenstate of the annihilation operators $a_k$. These states contain "an infinite number of photons in the same one-particle state" (actually the number of photons is not defined) and have properties that can be considered classical. For instance they verify Maxwell's equations since fields operators do (notice that everything works because  these equations are linear, with non-Abelian gauge fields the situation would be much more complicated). 
It is worth stressing that, however, what I said above concerns only radiative fields. The "quantum" interpretation of static macroscopic electric and magnetic fields is much more complicated.
Unfortunately similar states do not exist for fermion fields in view of the antisymmetry property of the states (you cannot have more than electron in a given state). So there is nothing like a macroscopic Dirac field (similar to macroscopic EM field).  
