Coherent state vs. classical world

Question

I am very curious as to how the classical field theory emerges from quantum field theory. In free quantum field theories, coherent states have classical properties, namely that the observables' uncertainties could be made relatively small: $\Delta \phi << \langle \phi \rangle$. So this is one notion in which you could associate classical behavior to a QFT state. But if a QFT is interacting then a state that starts out coherent loses coherence in the future. But we clearly observe classical electrodynamics which is an interacting theory. So, how does classical behavior emerge from interacting QFTs?

Answer 1

Suppose free quantum system. It can be well described in terms of creation-annihilation operators $\hat{c}_{\mathbf k}^{\dagger},\hat{c}_{\mathbf k}$. The classical description is good approximation of this quantum system if the number of particles $N$ in this system is macroscopical, i.e. for the system treated and the state $|N\rangle$ we have $$ \hat{c}_{\mathbf k}^{\dagger}|N\rangle = \sqrt{N+1}|N+1\rangle \approx \sqrt{N}|N+1\rangle + \mathcal{O}(N^{-\frac{1}{2}}), $$ $$ \hat{c}_{\mathbf k}|N\rangle = \sqrt{N}|N-1\rangle \approx \sqrt{N}|N+1\rangle \Rightarrow $$ $$ [\hat{c}_{\mathbf k},\hat{c}_{\mathbf k}^{\dagger}]|N\rangle = \mathcal{O}(N^{-\frac{1}{2}}) \to 0 \ \ \text{ as } \ \ N\to \infty $$ Then the creation-annihilation operators can be treated as ordinary c-numbers. The coherent state is just a state on which the equality $[\hat{c}_{\mathbf k},\hat{c}_{\mathbf k}^{\dagger}] = 0$ is exact. Parametrizing it by the number of particles $N$, $|N\rangle_{\text{coh}}$, we have $$ \tag 1 |N\rangle_{\text{coh}} = e^{-\frac{N}{2}+\sqrt{N}\hat{c}_{\mathbf k}^{\dagger}}|0\rangle $$ In this case $$ \hat{c}_{\mathbf{k}}|N\rangle_{\text{coh}} = \hat{c}^{\dagger}_{\mathbf k}|N\rangle_{\text{coh}} = \sqrt{N}|N\rangle_{\text{coh}}, $$ the quantum system can be equivalently described in terms of classical variables after considering the VEV on the coherent state of all the quantum operators. Say, for the field operator $$ \hat{\phi}(x) = \sum_{\mathbf k}\frac{1}{\sqrt{2\omega_{\mathbf k}V}}(e^{ikx}\hat{c}_{\mathbf k}+e^{-ikx}\hat{c}^{\dagger}_{\mathbf k}), $$ where $V$ is the volume of the system and $\omega_{\mathbf k} = \sqrt{k^{2}+m^{2}}$, the VEV on the coherent state $(1)$ for, say, $\mathbf k = 0$ is $$ \langle N|_{\text{coh}}\hat{\phi}(x)|N\rangle_{\text{coh}} = \sqrt{\frac{N}{mV}}\cos(mt) $$ Let's now turn on quantum interactions, assuming for simplicity that they're sufficiently small, i.e. parametrized by dimensionless couplings $\alpha << 1$. Then the interactions can be viewed as a perturbation around the coherent vacuum. Their destruction of the coherent state (and therefore the classical description) can therefore be described as decays and rescatterings of the particles from the coherent state, because of which the particles leave the condensate. Deriving the corresponding reaction rate $\Gamma$ (using Fermi golden rule for this), one can then estimate the time $t \simeq N\Gamma^{-1}$ for which the quantum interaction will lead to decay of the condensate. If this time is sufficiently large (often this is the case) in compare to time scales relevant for description of classical processes, then approximately the classical description holds.