Classical field limit of the electron quantum field

The classical limit of bosonic quantum mechanical systems with both finite and infinite degrees of freedom is pretty well understood from a mathematical standpoint (with complete rigour, and for quite general quantum states; see the references in the end).

With fermions on the other hand, the situation is more involved. The point is essentially that hinted by @Virgo 's answer, i.e. that by exclusion principle it is quite tricky to treat the limit of a large number of fermionic particles.

As far as I know, the question may be tackled in two separate steps. 1) A first step is to pass from the Fock space and quantum field dynamics to a sort of classical field fermion dynamics (you can see that as a mean field approximation $N\to\infty$). While for bosons the result of such a procedure is an effective dynamics on the one-particle space, for fermions you still have a dynamics involving all the different components with different number of particles. However, it may be possible (e.g. in the case of non-self-interacting-fermions) that the dynamics can be actually restricted to each $n$-particle sector. 2) Now suppose that you have a mean field dynamics that restricts to each sector. On the one particle sector, you would have let's say a Dirac equation for one electron in an external potential. Now you can there use the standard tools of (bosonic/quantum mechanic) semiclassical analysis to obtain the classical particle dynamics, taking also into account spin, in the limit $\hbar\to 0$.

The step (2) has been done e.g. here, for relativistic fermions. In the case of non-relativstic self-interacting fermions, this two-step procedure has been done with complete rigour recently, obtaining Hartree-Fock for the mean-field $N\to\infty$ approximation, and Vlasov equation then in the $\hbar\to 0$ limit (because for self-interacting fermions you can't just restrict to a fixed-particle sector because the dynamics involves other sectors, so you have to consider all particles at once, getting therefore a Vlasov equation). The references are for the step (1) this and for the step (2) this.

I remark that also for non-relativstic bosons it is possible to do both a mean filed and classical limit, and it is easier than with fermions. It has been done e.g. here. However for relativstic bosonic fields and quantum mechanical systems it is possible to simply do the classical limit $\hbar\to 0$ to obtain either classical field or classical particle dynamics. The latter has been done rigorously for essentially any "conceivable" system, using the results originating in the Weyl-Hörmander pseudodifferential calculus and the so-called Wigner measures. The former is much more complicated, but some (few) results have been recently obtained, essentially here, here, and here.

That formula is for bosons. In the large particle limit the boson quantum field can be approximated by the classical field. Electrons are fermions and by the Pauli exclusion principle you can have only one particle per state. So you cannot go to the large particle limit in the same way as you can for a boson field which can have many particles in each state. For this reason it is not possible to get a classical field limit for the electron field. For non-relativistic electrons, quantum field theory reduces to multi-particle quantum mechanics. This limit does not apply to photons as they are always relativistic.

Fermionic coherent states with a truly classical behavior are constructed in the survey

Zhang, W. M., & Gilmore, R. (1990). Coherent states: theory and some applications. Reviews of Modern Physics, 62(4), 867. http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.62.867

A second kind of fermionic coherent states is obtained by treating the coefficients in your expression as Grassmann variables. This gives a generalized classical limit in which one has classical anticommuting variables - the same kind as one has in fermionic path integrals.