Gell-Mann and Low theorem

The Gell-Mann and Low theorem is a theorem in quantum field theory that allows one to relate the ground (or vacuum) state of an interacting system to the ground state of the corresponding non-interacting theory. It was proved in 1951 by Murray Gell-Mann and Francis E. Low. The theorem is useful because, among other things, by relating the ground state of the interacting theory to its non-interacting ground state, it allows one to express Green's functions (which are defined as expectation values of Heisenberg-picture fields in the interacting vacuum) as expectation values of interaction picture fields in the non-interacting vacuum. While typically applied to the ground state, the Gell-Mann and Low theorem applies to any eigenstate of the Hamiltonian. Its proof relies on the concept of starting with a non-interacting Hamiltonian and adiabatically switching on the interactions.

The theorem was proved first by Gell-Mann and Low in 1951, making use of the Dyson series. In 1969 Klaus Hepp provided an alternative derivation for the case where the original Hamiltonian describes free particles and the interaction is norm bounded. In 1989 Nenciu and Rasche proved it using the adiabatic theorem. A proof that does not rely on the Dyson expansion was given in 2007 by Molinari.

Let ${\displaystyle |\Psi _{0}\rangle }$ be an eigenstate of ${\displaystyle H_{0}}$ with energy ${\displaystyle E_{0}}$ and let the 'interacting' Hamiltonian be ${\displaystyle H=H_{0}+gV}$, where ${\displaystyle g}$ is a coupling constant and ${\displaystyle V}$ the interaction term. We define a Hamiltonian ${\displaystyle H_{\epsilon }=H_{0}+e^{-\epsilon |t|}gV}$ which effectively interpolates between ${\displaystyle H}$ and ${\displaystyle H_{0}}$ in the limit ${\displaystyle \epsilon \rightarrow 0^{+}}$ and ${\displaystyle |t|\rightarrow \infty }$. Let ${\displaystyle U_{\epsilon I}}$ denote the evolution operator in the interaction picture. The Gell-Mann and Low theorem asserts that if the limit as ${\displaystyle \epsilon \rightarrow 0^{+}}$ of

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