External radiation in quantum particle systems In describing system of quantum particles external radiation is often assumed to be classical. Is there any text book that give a proof why can we assume that?
 A: It is possible to prove that one obtains the dynamics generated by the interaction of quantized particles with a classical electromagnetic field starting from a completely quantized model (for both particles and fields), in a suitable limit called quasi-classical limit.
The idea is to consider a quantum state (density matrix) $\rho_\epsilon$, and to evolve it for example with the Pauli-Fierz Hamiltonian:
$$H_\epsilon=\frac{1}{2}\bigl(\hat{p}+\hat{A}_{\epsilon}(x)\bigr)^2+ \frac{1}{\epsilon}\int \mathrm{d}k \lvert k\rvert a_{\epsilon}^\dagger(k)a_{\epsilon}(k)\; ,$$
where $\hat{A}_\epsilon$ is the quantum electromagnetic vector potential (smeared by the particle's charge distribution, that is assumed to be regular enough), and the second term is the radiation field's free Hamiltonian in second quantization.
The parameter $\epsilon$ is the quasi-classical parameter, that one would like to use to obtain the classical EM field in the limit $\epsilon\to 0$. Physically, it represents the regime in which the systems has a very large number of photons, and thus the EM energy is large compared to the particle's energy scale. Mathematically, it enters as Planck's constant on the canonical commutation relations of the field only:
$$[a_\epsilon(k),a^\dagger_\epsilon(k')]=\epsilon\delta(k-k')\; .$$
Of course, in general also the quantum state depends on $\epsilon$, and therefore this is why I denoted it by $\rho_\epsilon$.
Now, consider the evolved state $\rho_\epsilon(t)=e^{-itH_\epsilon}\rho_\epsilon e^{itH_\epsilon}$. This state encodes the evolution of the fully quantum system of the particle and EM field (I took for simplicity a spinless, non-relativistic, particle but one could modify the Hamiltonian accordingly for other cases). Since we would like to derive an effective evolution for the particle only (under the action of an external classical EM field), we should focus on the subsystem consisting of the particle alone. This is done mathematically taking the partial trace. Let me denote by $\gamma_\epsilon(t)$ the partial trace, w.r.t. the radiation degrees of freedom, of $\rho_\epsilon(t)$.
Now, the last step in order to obtain the effective quasi-classical evolution is to take the limit $\epsilon \to 0$ of $\gamma_{\epsilon}(t)$. Modulo mathematical technicalities (the topology on which one should take the limit, $\gamma_{\epsilon}(t)$ may be "rapidly oscillating" and thus a convergent subsequence shall be extracted), one obtains a limit quantum state
$$\gamma_0(t)\; ,$$
acting on the particle alone.
The hope is that the following identity is true:
$$\gamma_0(t)=U(t,0)\gamma_0 U^{\dagger}(t,0)\; ,$$
with $U(t,0)$ is the evolution generated by
$$\mathcal{H}_0(t)=\frac{1}{2}\bigl(\hat{p}+A(t,x)\bigr)^2\; ,$$
where $A(t,x)$ is now the classical EM vector potential, evolved in time by the free Maxwell equations, starting from a suitable initial condition $A(x)$ that depends on the microscopic initial configuration $\rho_\epsilon$.
The latter, however, is true only for very special initial configurations $\rho_\epsilon$, and in general the situation is more complicated (but nonetheless explicit). 
So, to conclude, it is possible to prove that in suitable situations the dynamics of quantum particles subjected to an external classical EM field is the correct effective description of a fully quantized microscopic system. Concerning eventual textbook references to the thing I discussed above, I am afraid that there are none available, and in fact the above is an ongoing research topic in mathematical physics right now.
