Classical field limit of the electron quantum field

To recover classical electromagnetic fields from the quantum electromagnetic field, we consider coherent states of the form $$\exp \left(\int d\vec{r}\, \vec{A}(\vec{r}) \vec{a}^\dagger(\vec{r}) \right) |0\rangle$$ as described in this blog post. This state corresponds to a classical EM field with the vector potential $\vec{A}(\vec{r})$, containing an indeterminate number of photons.

I'm confused about how this works for spin 1/2 fields such as the electron. For some reason, the classical limit of the electron quantum field that we care about is a particle, not a field, so it looks like this procedure does not apply. Even if we did it, I have no idea what a "classical electron field" would look like, since we never talk about such a thing.

• If I apply the procedure anyway, to get 'electron coherent states', what is the interpretation of these states? They have indeterminate electron number, which is strange.


• What's the resulting classical electron field like?


• How can we recover quantum electron particles from the quantum electron field?

Answer

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.