# The relation between full Green's function and S-matrix

I'm learning Green's function in condensed matter. The full Green's function is defined as

$$G(k_2,t_2;k_1,t_1) = \langle\Omega |T a_{k_1}(t_1)a_{k_2}^{\dagger}(t_2) |\Omega \rangle$$

The $$\Omega$$ is the interaction ground state and the operators are in the Heisenberg picture.

We want $$k_1,k_2$$ to be the quantum number of free Hamiltonian(like free electron gas). In that case, we should impose the condition that when $$a_{k_1}(t_1),a_{k_2}^{\dagger}(t_2)$$ are identical to the free electron creation or annihilation operators. In literatures (e.g. Mahan), they pick $$a_{k_1}(-\infty )=a_{k_1}^{free},a_{k_2}^{\dagger}(-\infty)= a_{k_2}^{\dagger,free}$$.

It occurs to me that, when I learn scattering theory, the S-matrix has similar structure. For potential scattering,

$$S_{k_2,k_1}=\langle{k_2}^{out}|k_1^{in}\rangle=\langle k_2^{free}|U(\infty,-\infty)|k_1^{free}\rangle=\langle\Omega_0|a_{k_2}U(\infty,-\infty)a^\dagger_{k_1}|\Omega_0\rangle$$

In this case, the $$\Omega_0$$ is the ground state of non-interacting Hamiltonian. They are even more likely to each other when you pick an interacting picture.

My question is, what is the relationship of these quantities? The book of QFT discusses free Green's functions and use them to expand the S matrix, but I really need the discussion of full Green's function and S-matrix.

My naive observation tell me $$G(k_2,\infty;k_1,-\infty)=S_{k_2,k_1}$$, but the convention I mentioned above denied it...

• And I also wonder what the physical meaning of $a\dagger_k(t_1)$? will it create an eigen state of full Hamiltonian when acting on $\Omega$? Commented May 25, 2022 at 11:39
• Consider to use \langle and \rangle instead of $<$ and $>$. Commented May 25, 2022 at 12:58