# Why are Green functions in many body physics Green functions?

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?

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

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.
• @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. Jan 14, 2023 at 12:23