\vec{k},\vec{p'})\ \cdot\
\underline{A}(\omega , \vec{q},\vec{p'})$$ also satisfy the (\[c32\]) where $ \underline{A}$ is a tensor with orthogonality condition $$\label{c34}
\int d^3p\ \underline{A}(\omega , \vec{p},\vec{p'})\ \cdot\
\underline{A}^\dag(\omega , \vec{p},\vec{p''})=I\delta^3(
\vec{p'}-\vec{p''})$$ Although for a given $\underline{\chi}^e$ various choices of the tensor $\underline{f}$ satisfying ( \[c32\]) affect the space-time dependence of electromagnetic field operators, but all of these choices are equivalent and the commutation relations between electromagnetic field operators remain unchanged [@22]. That is, the commutation relations between electromagnetic operators and the physical observables finally are dependent only on the given susceptibility tensor and not on the coupling functions $\underline{f} , \underline{g}$.\
In (\[c31\]), $\underline{\vec{P}}_N$ is the spatial Fourier transform of the noise electric polarization density which is expressed in terms of the the dynamical variables of the “$E$ field” at $t=0$ $$\label{c35}
\underline{\vec{P}}_N(\vec{k},t)=\int_0^\infty d\omega\int d^3p \
\underline{f}( \omega ,
\vec{k},\vec{p})\cdot\left(\underline{\vec{Q}}_\omega(\vec{p},0)\frac{\sin
\omega t }{\omega}+ \underline{\vec{X}}_\omega(\vec{p},0)\cos
\omega t\right)$$ In a similar fashion the constitutive equation relating the magnetic polarization density of the medium to the magnetic field $\underline{B}(\vec{q},t)=
\imath\vec{q}\times\underline{A}(\vec{q},t)$ can be obtained straightforwardly using the Heisenberg equations for the conjugate dynamical variables $ \underline{Y}_{\omega\lambda}$ and $\underline{\Pi}_{\omega\lambda}$ as the form $$\label{c36}
\underline{\vec{M}}(\vec{k},t)=\underline{\vec{M}}_N(\vec{k},t)+\frac{1}{\mu_0}\int_0^{|t|}
dt' \int d^3q\ \underline{\chi}^m(\vec{k},\vec{q},
|t|-t')\cdot\underline{\vec{B}}(\vec{q},\pm t')$$ where the $\underline{\chi}^m$ is the spatial Fourier 