# The retarded green's function in the extended nambu basis

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

• Since that $i G^{Ret}(xt,x't') = \langle GS \vert \text{T} \psi_m(xt) \psi^\dagger_{m'}(x't') \vert GS \rangle$ so $G^{Ret}_{ij}(\omega)$ is the propagator in the $\omega$ space. – Algebrato Jul 6 '17 at 22:45
• What do you mean what is $\tilde{G}^R_{ij}$? You ask about the physical meaning? Or do you want a mathematical equation in terms of specific operators? – gingras.ol Jul 9 '17 at 3:41
• @gingras.ol Well, a mathematical equation would do. – Arnab Barman Ray Jul 9 '17 at 11:37

As a general recipe, the retarded Green's function is defined as $$G^{R}_{ij}(t,t') = -i \theta(t,t') \langle \{\psi_{i}, \psi_{j}^{\dagger} \} \rangle,$$ where $\theta$ is the step-function, and $\{ ,\}$ denotes anticommutator. $\psi_{k}$ based on your suggested language is k-th elemnt of $\Psi$. For instance, $\psi_{1}= \Psi_{\uparrow}$. The full retarded Green's function would be $$G^{R}(t,t')= -i \theta(t,t') \begin{pmatrix} \langle \{ \Psi_{\uparrow}(t) , \Psi^{\dagger}_{\uparrow}(t') \} \rangle& \langle \{ \Psi_{\uparrow}(t), \Psi^{\dagger}_{\downarrow}(t') \} \rangle& \langle \{ \Psi_{\uparrow}(t), \Psi_{\downarrow}(t') \} \rangle & -\langle \{ \Psi_{\uparrow}(t), \Psi_{\uparrow}(t') \} \rangle\\ \langle \{ \Psi_{\downarrow}(t) , \Psi_{\uparrow}^{\dagger}(t') \} \rangle & \langle \{ \Psi_{\downarrow}(t) , \Psi_{\downarrow}^{\dagger}(t') \}\rangle & \langle \{ \Psi_{\downarrow}(t) , \Psi_{\downarrow}(t') \} \rangle & -\langle \{ \Psi_{\downarrow}(t) ,\Psi_{\uparrow}(t') \} \rangle \\ \langle \{ \Psi^{\dagger}_{\downarrow}(t) , \Psi_{\uparrow}^{\dagger}(t') \} \rangle & \langle \{ \Psi^{\dagger}_{\downarrow}(t) , \Psi_{\downarrow}^{\dagger}(t') \} \rangle& \langle \{ \Psi^{\dagger}_{\downarrow}(t) , \Psi_{\downarrow}(t') \}\rangle & -\langle \{ \Psi^{\dagger}_{\downarrow}(t) ,\Psi_{\uparrow}(t') \} \rangle\\ -\langle \{ \Psi^{\dagger}_{\uparrow}(t) , \Psi_{\uparrow}^{\dagger}(t') \} \rangle& -\langle \{ \Psi^{\dagger}_{\uparrow}(t) , \Psi_{\downarrow}^{\dagger}(t') \} \rangle& -\langle \{ \Psi^{\dagger}_{\uparrow}(t) , \Psi_{\downarrow}(t') \} \rangle& \langle \{ \Psi^{\dagger}_{\uparrow}(t) ,\Psi_{\uparrow}(t') \} \rangle \end{pmatrix}$$
The suggested generic definition of the retarded Green's function in your question is just the Fourier transform of the above matrix in the basis where $\overline{H}$is defined.