Quantization of electrostatic $\vec E$ field?

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.