## 2 Super-Operator Integral Equation for Scattering of Quantum Light by a Dielectric Body
### Observable Values  
We have considered quantum light scattering from the point of view of open quantum systems (for example, applied to the modeling of lossy beam splitter in [44]). The concept of composite quantum systems is used in it as a main framework. The composite quantum system is considered as a set of two (or more) subsystems, whose statistical properties completely define the behavior of the total system. For example, the two subsystems may be two atoms in molecules, two modes in the EM field, or two elements in an electrical circuit, etc. In our case, the fields in two geometrical regions (the area inside the dielectric body and the whole outside space, near-field and far-field zones of antennas, etc.) play the role of subsystems. As a starting point, the Hilbert space of a composite system defined as a tensor product \(\mathbb{S}=\mathbb{S}^{(1)}\otimes\mathbb{S}^{(2)}\) of Hilbert spaces of its subsystems \(\mathbb{S}^{(1)}\) and \(\mathbb{S}^{(2)}\) (superscripts 1 and 2 indicate the areas inside and outside the dielectric body, respectively) should be introduced. The incident field is produced by the set of external sources distributed over the whole space. Such a set may be considered as a composite quantum system and characterized by the density matrix, which is the operator in the state space \(\mathbb{S}\). We will decompose the total source into two subsystems, in which the first one consists of sources outside the body and the second consists of inside ones. We will assume that the sources of different subsystems are uncorrelated. Therefore, the total density matrix takes the form of a tensor product of the density matrices of the partial regions: \(\rho=\rho^{(1)}\otimes\rho^{(2)}\). Due to the noise component, the operators \(\hat{U}_{k}(\mathbf{x})\), \(\hat{U}_{k}^{+}(\mathbf{x})\) for the field inside the body act on the total space \(\mathbb{S}\). For the calculation of observable values, the field operators are conveniently present in the form of the tensor products. Operator (9) will have the form  
\[\hat{U}_{k}(\mathbf{x})=\left(\hat{U}_{k0}(\mathbf{x})\otimes\hat{I}^{(2)} \right)-k^{2}(\varepsilon-1)\int\limits_{\mathbb{S}}\Gamma\left(\mathbf{x}, \mathbf{x}^{\prime};k\right)\left(\hat{U}_{k0}\left(\mathbf{x}^{\prime}\right) \otimes\hat{I}^{(2)}\right)d\mathbf{x}^{\prime}+\left(\hat{I}^{(1)}\otimes \hat{F}_{k}(\mathbf{x})\right), \tag{13}\]  
similar to the negative frequency field \(\hat{U}_{k}^{+}(\mathbf{x})\) (operators \(\hat{I}^{(1,2)}\) denote the identity super-operators in the subspaces \(\mathbb{S}^{(1,2)}\)). For an arbitrary operator \(\hat{O}\) acting on the total space \(\mathbb{S}\), the observable value is \(\left\langle\hat{O}\right\rangle=Tr\big{(}\hat{O}\rho\big{)}\). Assuming that the fluctuating part is in vacuum state, we obtain for the observable field:  
\[\left\langle\hat{U}_{k}(\mathbf{x})\right\rangle=\left\langle\hat{U}_{k0}( \mathbf{x})\right\rangle-k^{2}(\varepsilon-1)\int\limits_{\mathbb{S}}\Gamma \Big{(}\mathbf{x},\mathbf{x}^{\prime};k\Big{)}\Big{\langle}\hat{U}_{k0}\Big{(} \mathbf{x}^{\prime}\Big{)}\Big{\rangle}d\mathbf{x}^{\prime} \tag{14}\]  
(with no support of fluctuations). For the intensity \(\hat{O}(\mathbf{x})=\hat{U}_{k}^{+}(\mathbf{x})\hat{U}_{k}(\mathbf{x})\), we have  
\[\left\langle\hat{O}(\mathbf{x})\right\rangle=\left\langle\hat{U}_{k}^{+}( \mathbf{x})\hat{U}_{k}(\mathbf{x})\right\rangle=\left\langle\hat{U}_{k0}^{+}( \mathbf{x})\overline{\mathbf{R}}^{+}\left|\overline{\mathbf{R}}\hat{U}_{k0}( \mathbf{x})\right\rangle+\left\langle\hat{F}_{k}^{+}(\mathbf{x})\hat{F}_{k}( \mathbf{x})\right\rangle_{0}, \tag{15}\]  
where \(\overline{\mathbf{R}}(\cdot)=(\cdot)+v\int_{\mathbb{S}}\Gamma\big{(}\mathbf{x },\mathbf{x}^{\prime};k\big{)}(\cdot)d\mathbf{x}^{\prime}\) is the resolvent operator, and \(\left\langle\ldots\right\rangle_{0}\) denotes averaging over the vacuum state \(|0\rangle\langle 0|\).