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

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  • $\begingroup$ 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. $\endgroup$ – Algebrato Jul 6 '17 at 22:45
  • $\begingroup$ 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? $\endgroup$ – gingras.ol Jul 9 '17 at 3:41
  • $\begingroup$ @gingras.ol Well, a mathematical equation would do. $\endgroup$ – Arnab Barman Ray Jul 9 '17 at 11:37
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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.

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  • $\begingroup$ Thanks. I don't fully understand the relation between Green the "function" and Green the "operator". Could you give me a source so that I can get this cleared up? $\endgroup$ – Arnab Barman Ray Jul 11 '17 at 17:08
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    $\begingroup$ Please remember that the Green's function is a particular two-point "function" of "operators". Most likely a review on real-time Green's functions can be illustrative. $\endgroup$ – Shasa Jul 12 '17 at 8:23
  • $\begingroup$ Could you please suggest something specific? I have read the usual treatments in Doniac and Walecka, but neither of them seems to treat operator-wise aspects of the Green's function. $\endgroup$ – Arnab Barman Ray Jul 12 '17 at 18:14
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    $\begingroup$ Reading "Nonequilibrium Many-Body Theory of quantum systems" by Stefanucci and van Leeuwen is most likely helpful. You can also try to learn the "Keldysh Green's function" formalism. $\endgroup$ – Shasa Jul 12 '17 at 18:24

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