In quantum scattering theory, Green's Function is defined as [1]
$$G_0(z)=(z-H_0)^{-1},$$
$$G(z)=(z-H)^{-1},$$
where $H_0$ and $H=H_0+V$ are separately non-interacting and interacting Hamiltonian. $V$ is interaction.
One can then use the identity
$$\tag{1}V=G_0^{-1}-G^{-1},$$
to obtain Lippmann-Schwinger equation
$$\tag{2}G=G_0 + G_0 V G. $$
However, on the other hand, in quantum field theory(QFT), Green's function is defined as correlation function. For 2-point Green's function, we have Dyson equation
$$\tag{3} G=G_0+ G_0 \Sigma G, $$
where $\Sigma$ is here defined as self-energy. Equivalently
$$\tag{4} \Sigma:=G_0^{-1}-G^{-1}.$$
My questions are
Are the two Green's functions the same? What's the relation between the two formalisms? And the relation between Lippmann-Schwinger equation and Dyson equation? If they are actually the same thing, then does it mean $V=\Sigma$(this sounds very stupid)?
Are the possible differences relating to the discrepancy between S-matrix theory and QFT?
[1]: John R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions.