## 2 Quantum-Field Theory for Photonic Transport
### Bethe-Salpeter Equation for Photonic Transport in Samples of Disordered Active Mie Resonators
\(\Psi\) is characterized by the wavelength \(\lambda\), whereas the change of the light intensity \(\Phi\) is characterized by the transport mean-free path \(\xi\), and the correlation length in random lasers respectively, which is one of the central characteristics that we derive in due course of the paper. In \((x,y)\)-plane the standard Fourier transform is used, in the limited \(z\)-direction we Fourier-transform the relative coordinate but the center-of-motion coordinate \(Z\) remains in real space. The separation of scales allows for the physical incorporation of loss at the samples boundaries at the level of the transport theory. The break of the T-symmetry, the break of the time reversal invariance of the multiple scattering processes, is incorporated in a macrocanonical sense [42] on the level of the integration of the real space coordinate \(Z\). We apply these arguments to the equation of motion for the intensity correlation, known as the _Bethe-Salpeter equation_, see Figure 1c,  
\[\Phi_{ee}=G^{R}G^{A}[1+\int\frac{d^{3}k}{(2\pi)^{3}}\gamma\Phi_{ee}] \tag{6}\]  
and we obtain the Boltzmann or kinetic equation for transport Equation (8). We introduce here the following abbreviations, consistent with previous work, \(\Delta\Sigma=\Sigma^{A}(\omega_{-})-\Sigma^{R}(\omega_{+})\), \(\square\Sigma=\Sigma^{A}(\omega_{-})+\Sigma^{R}(\omega_{+})\), and equivalent expressions for \(\Delta G_{\vec{k}_{-}}^{\omega_{-}}\) and \(\square G_{\vec{k}_{-}}^{\omega_{-}}\). The term \(\gamma_{\vec{k}\,\vec{k}^{\prime\prime}}^{\omega}(Z,Z^{\prime\prime},\vec{Q} _{||},\Omega)\) represents the irreducible vertex, which is physically interpreted here as the coherent light matter interaction in disordered granular non-linear systems. The Ward identity is derived in the generalized form for the scattering of photons in _non-conserving_ media. Absorption or gain yield an additional contribution, and a form of the Ward-Takahashi identity for photons in complex matter [40; 43; 44] is derived. Effectively the additional contribution is not negligible and thus presents in the theoretical results of the transport characteristics of the self-consistent framework. It also renormalizes the energy transport velocity [31; 34; 40]. The solution to Equation (6), is derived as the energy density response \(\Phi_{ee}\) in the form of a diffusion pole, Equation (7)  
\[\Phi_{ee}=\frac{N_{\omega}}{\Omega+iDQ_{||}^{2}+iD\xi^{-2}}. \tag{7}\]  
\[\left[2\text{Re }(\epsilon_{b})\,\omega\Omega\right. - \left.2\vec{k}_{||}\cdot\vec{Q}_{||}+2 ik_{z}\partial_{Z}\ +\ \Delta\Sigma\ -\ \Delta\epsilon_{b}\omega^{2}\right]\Phi_{\vec{k}\,\vec{k}^{\prime}}^{ \omega}(Z,Z^{\prime},\vec{Q}_{||},\Omega)\] \[= \Delta G\delta(\vec{k}-\vec{k}^{\prime}\ )\ +\ \sum  
The full diffusion coefficient \(D(\Omega=0;Z)\) is derived as  
\[D(\Omega)=D_{0}^{tot}-\tau^{2}D(\Omega)M(\Omega) \tag{11}\]  
where the bare diffusion \(D_{0}=\frac{2\
u_{E}c_{p}}{\omega\Delta\Omega}\int\frac{d^{3}k}{(2\pi)^{3}}( \vec{k}\cdot\vec{Q})^{2}\Delta G\) is complemented by the contributions originating from the active medium as the scatterers \(D_{s}=\frac{1}{8}r_{e}A_{e}\tau^{2}\tilde{D}_{0}\) or the background \(D_{b}=\frac{1}{4}(\omega\tau)^{2}\Delta e_{b}\tilde{D}_{0}\). Thus the diffusion coefficient without memory effects reads \(D_{0}^{tot}=D_{0}+D_{s}+D_{b}\). The term \(-\tau^{2}D(\Omega)M(\Omega)\) contains the memory kernel \(M(\Omega)\), Equation (9), including the [PERSON] contribution and consequentially all interferences. \(\tilde{D}_{0}\) equals \(D_{0}\) where the imaginary part \(\Delta G\) is replaced by the real part \(\square G\). The renormalized density of states \(N_{\omega}\) is derived as follows  
\[N_{\omega}=\frac{\omega^{2}\Delta G_{0}(\vec{Q},\Omega)}{c_{p}^{2}g_{\omega}^{ (1)}[1+\Delta(\omega)]}. \tag{12}\]  
We further use the notation and the abbreviations of [34].