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.
Answer
There are very important differences between this two approaches, that can be summarized by noting that the Lippmann-Schwinger is the (formal) solution of a one-body problem (scattering of a particle by an external potential) whereas the Dyson equation gives the solution of a many-body problem. I focus here on the non-relativistic many-body case (it is also the case of the scattering problem).
It is only in the case where the system is in an external potential and non-interacting (or empty,assuming conservation of the number of particle) that the two approaches are equivalent (or more precisely the Dyson equation gives back the Lippmann-Schwinger equation).
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