Lesser Green's function

[rbphysics]

TL;DR

Physical interpretation of Lesser green function G<(t,t') for both t>t' and t<t'.

The lesser Green's function is defined as:
$$G^{<}(t,t')=i\langle C_{\nu}^{\dagger}(t')C_{\nu}(t)\rangle=i\bra{n}C_{\nu}^{\dagger}(t')C_{\nu}(t)\ket{n}$$ where $\ket{n}$ is the many particle ground state.
$$G^{<}(t,t')=i\bra{n}e^{iHt'}C_{\nu}^{\dagger}(0)e^{-iHt'}e^{iHt}C_{\nu}(0)e^{-iHt}\ket{n}$$
First consider the case t <t'
Define,
$$\ket{\alpha}=e^{-iH(t'-t)}C_{\nu}(0)e^{-iHt}\ket{n}$$
$$\ket{\beta}=C_{\nu}(0)e^{-iHt'}\ket{n}$$
$$G^{<}(t,t')=i\bra{\beta}\ket{\alpha}$$
$\ket{\alpha}$ is the state of the system at time t' given that a hole was created at single particle state $\nu$ at earlier time t. And $\ket{\beta}$ is the state of the system at time t' given that a hole is created at state $\nu$ at that instant. It gives me good physical interpretation that lesser function for t<t' is proportional to probability amplitude for the hole created at time t in state $\nu$ would remain in state nu at time t' also.
Now consider t>t'
Define,
$$\ket{\alpha}=C_{\nu}(0)e^{-iHt}\ket{n}\\$$
$$\ket{\beta}=e^{-iH(t-t')}C_{\nu}(0)e^{-iHt'}\ket{n}\\$$
$$G^{<}(t,t')=i\bra{\beta}\ket{\alpha}$$
$\ket{\alpha}$ is the state of the system at time t with hole created at that instant and $\ket{\beta}$ is the state of the system at time t with hole created at earlier time t' . What does physical interpretation now it have? What are forward and backward propagation often described in time.

 

[PeterDonis]

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