Quantizing the electrostatic field operator

[Konte]

Hello everybody,

It is known that electric field operator is shown as

$\hat{E}(r,t)=-i\sum_{k,\lambda}\sqrt{\frac{\hbar\omega_k}{2\epsilon V}}\left(a(t)^\dagger_{k,\lambda} e^{-ik.r} - a(t)_{k,\lambda} e^{ik.r} \right) \hat{e}_{k,\lambda}$

But if I need to represent an electrostatic field in a quantized form, how to proceed?

Thank you everybody.

 

[Konte]

Please.

 

[RockyMarciano]

Apparently you have mixed in your expression above the electric field in the quantized and not quantized forms. AFAIK the electric field operator(electrostatic field is not a meaningful distinction here) is: $E(r)=i\sum_{k,\lambda}\sqrt{\frac{\hbar\omega}{2\epsilon V}}\left(ξ(\lambda)a^{(\lambda)}(k) e^{ik.r} - ξ(\lambda)a^{\dagger(\lambda)}(k) e^{-ik.r} \right)$

 

[Konte]

Apparently you have mixed in your expression above the electric field in the quantized and not quantized forms. AFAIK the electric field operator(electrostatic field is not a meaningful distinction here) is: $E(r)=i\sum_{k,\lambda}\sqrt{\frac{\hbar\omega}{2\epsilon V}}\left(ξ(\lambda)a^{(\lambda)}(k) e^{ik.r} - ξ(\lambda)a^{\dagger(\lambda)}(k) e^{-ik.r} \right)$

Thank you for your response.
All of electromagnetic field quantization still really dark for me. I don't even know what question I have to ask. So, static field are not define in quantum field formulation?
(excuse for my english)

Thanks.
Konte