# Many-body Green Functions equation

In many-body physics the concept of Green Functions is essential especially when you deal with things like superconductivity that are strictly linked to the presence of off-diagonal long-range order in the two-body green function. I've been studying and using it myself but there is a question that remains for me unexplained and is the following.

For the one-body green function (also called two-points green function) we know that it is the solution of the associated equation: $$\Big( i\frac{d}{dt} - \hat{H} \Big) G(x) = \delta(\vec{x})$$ where $$G(x) = -i \langle T \psi(x) \psi^\dagger (0) \rangle$$ and $$x$$ contains all the operator coordinates while $$\vec{x}$$ is only the spacial coordinates component.

But as I already said for some other phenomena other green functions are much more important as the two-body GF $$G(x_1,x_2,x_3,x_4) = -i\langle T \psi(x_1) \psi(x_2) \psi^\dagger (x_3) \psi^\dagger (x_4) \rangle.$$

What is not clear to me is if also this last definition is the green function, mathematically speaking, of some equation or the name has been just adopted due to the previous case and that is only a component of the two-body density matrix.

The defining equation of the one-body Green-function that you wrote down is in fact just the real-time-version and zero-temperature-limit of Green-functions that one encounters in many-particle-physics. It might be motivated by this differential-equation, but it is defined differently. In the book from Negele you find generalizations of that one-body-Green-function to $$n$$-body-Green-functions (resembling the real-time-version of the second Green-function that you are interested in, in your question); In the book from Alexander Altland and in the book from Elk and Gasser you find generalizations of many-body-correlation-functions which are also called Green-functions.