Gell-Mann and Low theorem

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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.

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 be an eigenstate of with energy and let the 'interacting' Hamiltonian be , where is a coupling constant and the interaction term. We define a Hamiltonian which effectively interpolates between and in the limit and . Let denote the evolution operator in the interaction picture. The Gell-Mann and Low theorem asserts that if the limit as of

exists, then are eigenstates of .

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 let . From Schrödinger's equation for the time-evolution operator

and the boundary condition that we can formally write

Focus for the moment on the case . Through a change of variables we can write

We therefore have that

This result can be combined with the Schrödinger equation and its adjoint

to obtain

The corresponding equation between is the same. It can be obtained by pre-multiplying both sides with , post-multiplying with and making use of

The other case we are interested in, namely can be treated in an analogous fashion and yields an additional minus sign in front of the commutator (we are not concerned here with the case where have mixed signs). In summary, we obtain

We proceed for the negative-times case. Abbreviating the various operators for clarity

Now using the definition of we differentiate and eliminate derivatives using the above expression, finding

where . We can now let as by assumption the left hand side is finite. We then clearly see that is an eigenstate of and the proof is complete.

References

1. M. Gell-Mann and F. Low: "Bound States in Quantum Field Theory", Phys. Rev. 84, 350 (1951)

2. K. Hepp: Lecture Notes in Physics (Springer-Verlag, New York, 1969), Vol. 2.

3. G. Nenciu and G. Rasche: "Adiabatic theorem and Gell-Mann-Low formula", Helv. Phys. Acta 62, 372 (1989).

4. L.G. Molinari: "Another proof of Gell-Mann and Low's theorem", J. Math. Phys. 48, 052113 (2007)

5. A.L. Fetter and J.D. Walecka: "Quantum Theory of Many-Particle Systems", McGraw–Hill (1971)