I would like to know if for a linear system, the Green function and the linear response function finally represent the same thing ?
Indeed, if I have an "ouput" A of a system that is caused by an excitation F, I have by definition of the linear response function :
$$ A(t)=\int \chi(t,t')F(t')dt' $$
Because of the linearity of the system, the ouput $A(t)$ is solution of a linear differential equation with a source term.
The green function is how the system react to a dirac.
We can proove that :
$$A(t)=\int dt' G(t,t') F(t') dt'$$
So can we say that susceptibility and green functions are exactly the same thing ?
Then why would it be two different notions ?