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I'm studying from the book "Fundamentals of many body physics" by Nolting, and I've encountered the retarded, advanced and casual green functions. For example, the retarded green function of operators $A(t)$ and $B(t)$ is $$G_{AB}^{ret}(t,t')=-i\theta(t-t')\left\langle [A(t),B(t')] \right\rangle,$$ where the square brackets represent a commutator, the angle brackets represent a grancanonical average and $\theta$ is the Heaviside's theta function.

I am curious as why are they are called "Green functions"; I know that Green's functions of an operator $D$ are the functions that give an impulse (a delta function) as an output $$Df(x,y)=\delta(x-y).$$

Do the retarded, advanced and casual Green's functions have this property? If yes, with respect to which operator? Are these two concepts related somehow?

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  • $\begingroup$ Have a look here. Hope this helps for the start. $\endgroup$ Jan 12, 2023 at 15:25

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Indeed, the retarded and advance Green's functions satisfy this property. $$ \partial_t G_{AB}^{ret}(t,t')=-i\delta(t-t')\langle[A(t),B(t)]\rangle -i\theta(t-t')\langle[\partial_t A(t),B(t')]\rangle $$ The last term can be transformed using the equation of motion for operator $A(t)$, leading to the form similar to the mathematical definition of Green's function.

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  • $\begingroup$ But it's not that similar, there is a term without any delta, which is a higher order Green's function $\endgroup$
    – Rhino
    Jan 14, 2023 at 9:28
  • $\begingroup$ @Rhino whether it is a higher order green's function depends on the Hamiltonian - sometimes the equations close. It can also be viewed as a part of complex operator $D$. Kadanoff&Baym in their book use an approach like this. $\endgroup$
    – Roger V.
    Jan 14, 2023 at 12:23

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