Gell-Mann & Low theorem in QFT and in many-body physics at $T = 0$

Question

I couldn't help but noticed that the main physical arguments are the same as they can be found in practically all the books on many body published in English. The finite temperature case is very well understood, at least in the thermodynamic equilibrium case, but the $T = 0$ case is still very problematic. Indeed, starting with Abrikosov et al., and going through Pines, Nozieres, Pitaevskii, Fetter, etc., one starts with the adiabatic switching-on of interparticle interaction and with the celebrated Gell-Mann & Low theorem (GMLT). This theorem is correct, but it doesn't guarantee that one obtains the interacting ground-state (g.s.) state vector in this way. Indeed, the theorem ONLY shows that starting from the free g.s. one obtains an eigenvector of the full (i.e., with interaction) Hamiltonian, but it doesn't show that the corresponding energy eigenvalue is the minimum one, so that this interacting eigenvector to REALLY represent the interacting g.s. Can one prove that the eigenvalue is a minimum, or one just has to POSTULATE this? Any ideas and references to the literature would be greatly appreciated.

Concerning systems with symmetry breaking, again at $T = 0$, such as BEC and BCS one introduces the corresponding condensates by hand. In the BEC case it's not that bad since if one follows the GMLT construction starting with the free g.s., there is a condensate to begin with for the free g.s. (and this can be rigorously proven from a math point of view) for a free system. However, one still has to show that free g.s. is REALLY mapped onto the interacting g.s. as I've already asked. But in the BCS case there is no condensate for the free system. One really needs an interaction to form a condensate. One way is, of course, to introduce symmetry breaking terms into the free system but in the end, after the thermodynamic limit is taken, one has to take the limit of their strength (coupling constant) to zero. How can one justify the GMLT construction in this case? How can one prove that the energy is minimum?

I would be very grateful if you could give me your insight and possible answers to my questions above.

Answer 1

1. The statement of Gell-Mann and Low theorem:

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• Let $\left|\Psi_0\right>$ be an eigenstate of the non-interacting Hamiltonian $H_0$ with energy $E_0$ and let the interacting Hamiltonian be $$H = H_0 + g V,$$ where $g$ is a coupling constant and $V$ the interaction term.We define a Hamiltonian $$H_{\epsilon} = H_0 +e^{-\epsilon\left|t\right|}g V,$$ where $\epsilon$ is a positive parameter,and then $H_\epsilon$ will effectively interpolates between $H$ and $H_0$ .

• For the limiting cases we have \begin{align} t \rightarrow \pm\infty \Rightarrow & H_\epsilon = H_0 ,\\ \epsilon \rightarrow 0^+ \Rightarrow & H_\epsilon = H . \end{align}

• Let $U_{\epsilon I}$ denote the evolution operator in the interaction picture.The Gell-Mann and Low's theorem asserts that if the limit of

\begin{equation} \left|\Psi_{\epsilon}^{(\pm)}\right> = \dfrac{U_{\epsilon I}(0,\pm\infty)\left|\Psi_0\right>}{\left<\Psi_0\right|U_{\epsilon I}(0,\pm\infty)\left|\Psi_0\right>} \end{equation}

exists as $\epsilon\rightarrow 0^+$,then $\left|\Psi_{\epsilon}^{(\pm)}\right>$ are eigenstates of $H$ .

• Note that when applied to,say,the ground-state,the theorem does not guarantee that the evolved state will still be a ground state.In other words,level crossing in not excluded.

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2.Here is a nice proof by Molinario: ${J. Math. Phys. 48, 052113 (2007)}$ .You will find this in the references for Gell Mann and Low's theorem on Wikipedia,which will tell you more details.

Hope this helps.

Answer 2

this puzzled me a lot for a long time as well. My current understanding is there is no intrinsic difference between GMLT in QFT and AT in QM. In fact, the idea of adiabatic switching should work even better in QFT. As the adiabaticy of scattering is almost always guaranteed, whereas no QM experiments guarantees real adaiabaticy. So one has to come up with different criterions to check if the theorem apply.

The great thing that Gell Mann and Low did is, by inserting the correct normalization factor, contributions of vacuum bubbles are canceled so that the expression for the reulatant state after adiabatic time evolution does not contain a divergent phase.

But other than that, it's the same with the QM version of AT. Indeed, Gell Mann and Low hadn't proved that the interacting vacuum has to be arrived from a free vacuum. But Kato did. See T. Kato, J. Phys. Soc. Jpn \textbf{5}, 435, (1950). He has his great insight to define dynamical transformation and adiabatic transformation. And he managed to prove the two distinct types of transformations coincide as one takes the adiabatic limit ( a sufficiently slowly varying H ). With his formilism, the interacting vacuum could be obtained by adiabatically evolving the free vacuum.