# What's the relation or difference between Lippmann-Schwinger equation and Dyson equation?

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.

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@Mathieu: you are right, you can use the LS equation for more than one body, but it conserves the number of particles (the ground state of the system is trivial: it is the vacuum). Stated otherwise, it is a reformulation of the Schrodinger equation. Of course, in principle, if you can solve the N-body ($N\to\infty$) SL equation, you could describe a many-body state (say, a supraconductor), but that's definitely not doable in practice. And yes, as I said in the answer, Dyson equation is more general and give the SL equation in the appropriate limit. – Adam Mar 26 '14 at 13:34