Lets say we have a non-interacting translation invariant Hamiltonian $H$ in a matrix form:

$$H=\Psi^{\dagger}\bar{H}\Psi$$

where $\Psi^{\dagger}=(\Psi_{\uparrow}^{\dagger}\quad \Psi_{\downarrow}^{\dagger} \quad-\Psi_{\downarrow}\quad\Psi_{\uparrow})$ is the extended nambu spinor.

In such a basis what is the meaning of a particular element of the matrix representation of the retarded green function:

$$\bar{G}^{R}(\omega)=\frac{1}{\omega-\bar{H}+i\delta}$$

what is $\bar{G}^{R}_{ij}$?