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.
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History
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.
Statement of the theorem
Let
exists, then
Note that when applied to, say, the ground-state, the theorem does not guarantee that the evolved state will be a ground state. In other words, level crossing is not excluded.
Proof
As in the original paper, the theorem is typically proved making use of Dyson's expansion of the evolution operator. Its validity however extends beyond the scope of perturbation theory as has been demonstrated by Molinari. We follow Molinari's method here. Focus on
and the boundary condition that
Focus for the moment on the case
We therefore have that
This result can be combined with the Schrödinger equation and its adjoint
to obtain
The corresponding equation between
The other case we are interested in, namely
We proceed for the negative-times case. Abbreviating the various operators for clarity
Now using the definition of
where