Quantization of electrostatic $\vec E$ field? Can a electrostatic field $\vec E=\vec E(x,y,z)$ (time-independent) or electrostatic potential $\phi=\phi(x,y,z)$ be quantized? If yes, will these quanta be photons again? But we don't have an electromagnetic field here.
 A: In the standard quantization of the free electromagnetic field, the field operators satisfy the (equal time) commutation relations
$$ [E_i(\mathbf{x}, t), B_j(\mathbf{y}, t)] = -i \hbar \epsilon_{ijk} \partial_k \delta^3(\mathbf{x}-\mathbf{y})$$.
Please, see for example the following article by Stewart.
This implies the existence of an uncertainty relation:
$$\Delta E_f \Delta B_g \le \frac{\hbar}{2} \int d^3x \epsilon_{ijk} f^i \partial_k  g^j$$
where $E_f$, $B_g$ are the smeared fields by the vector valued functions $f^i$, $ g^j $ respectively ($E_f = \int d^3x f^i((\mathbf{x}) E_i(\mathbf{x})$). We can assume that these functions are compactly supported in order to ensure the convergence of the integral. 
This means that for almost all the choices of the functions $f^i$, $ g^j $, there an unvanishing uncertainty relation among components of the electric and magnetic fields. Thus a vanishing magnetic field would imply infinite fluctuations of the electric field. 
As a consequence, electrostatics with vanishing magnetic fields would imply infinite uncertainty in the electric field. Therefore, electrostatics cannot be quantized.
A: So, let me try to rephrase your question a little. The "electrostatic" case in, for example, plasma physics refers to the case when $|v| \ll c$, so that the coupling to the vector potential $\vec{A}$ is negligible and we can consider the pure situation of the scalar potential $\phi$. We can TOTALLY write a Lagrangian quantum density for this as
\begin{equation}
\mathcal{L} = \underbrace{i \psi^* \partial_t \psi + \frac{1}{2m}(\nabla \psi^*)\cdot (\nabla \psi)}_{\textrm{Schrodinger equation}} + \underbrace{e \psi^* \psi \phi}_{\textrm{coupling}} + \underbrace{\frac{1}{8 \pi}(\nabla \phi)\cdot (\nabla \phi)}_{\textrm{Maxwell stress tensor}}
\end{equation}
I'm typing this from memory so I'm not sure if I have every single sign and constant right, but you can see that you can certainly have a quantum massless real scalar field $\phi$ that acts like the scalar potential in classical mechanics, but is in a quantum action and, hence, may be canonically quantized.
