Gell-Mann Low formula vs time independent perturbation

Question

Consider a nonperturbed Hamiltonain $H_0$ and an eigenstate $|\Phi\rangle$ satisfying $$H_0|\Phi\rangle=E_0|\Phi\rangle.$$

Now consider the perturbed Hamiltonian $H=H_0+\lambda H_1$ and let $H_\epsilon=H_0+\lambda e^{-\epsilon|t|}H_1$ to be adiabatic switching of interaction. Gell-Mann and Low's theorem says the state $$|\Psi\rangle=\lim_{\epsilon\rightarrow 0}\frac{U_\epsilon(0,-\infty)|\Phi\rangle}{\langle\Phi|U_\epsilon(0,-\infty)|\Phi\rangle}$$ is an eigenstate of $H_0+H_1$ with the energy shift $$\Delta=E-E_0=\lim_{\epsilon\rightarrow 0}\text{i}\epsilon\lambda\frac{\partial}{\partial\lambda}\ln\langle\Phi|U_\epsilon(0,-\infty)|\Phi\rangle.$$

My question: what is the relation (difference) between this formula and time independent perturbation theory? Why do we need Gell-Mann and Low's formula when doing scattering theory in QFT but not in nonrelativistic quantum mechanics?

Answer 1

The main reason modern textbooks introduce the Gell-Mann-Low formula is that it leads to a very simple proof of the Feynman rules.

The argument roughly goes like this: first, we write $$ U(t_1,t_2)=\mathrm T e^{i\int_{t_1}^{t_2}\mathcal H_\text{int}(\tau)\mathrm d\tau} $$ where $\mathrm T$ denotes the time-ordering symbol and $\mathcal H_\text{int}$ is the interaction Hamiltonian. Then, using the GML formula, we can write $$ \color{red}{\langle \Omega|\mathrm T \phi_1\cdots\phi_n|\Omega\rangle\propto\langle 0|\mathrm T e^{i \int \mathcal H_\text{int}}\phi'_1\cdots\phi'_n|0\rangle} $$ where $\Omega$ is the full vacuum state and $|0\rangle$ is the perturbative vacuum. Also, $\phi$ is the Heisenberg picture field and $\phi'$ is the interaction picture field. Basically, $|\Omega\rangle\propto U(0,-\infty)|0\rangle$, and similarly with $\phi\sim U\phi'U^\dagger$. All $U$'s combine to give a single $U(+\infty,-\infty)$, which gives the formula in red.

Expanding the exponential in power series, this formula becomes $$ \langle \Omega|\mathrm T \phi_1\cdots\phi_n|\Omega\rangle\propto \sum_{p=0}^\infty\frac{i^p}{p!}\int \langle 0|\mathrm T \mathcal H_\text{int}^p\phi'_1\cdots\phi'_n|0\rangle $$

This basically proves the Feynman rules, since the right-hand-side is a formula that involves free fields only, i.e., its value is fixed by Wick's theorem. The formula tells us that the full n-point function is given by a sum over free correlation functions, with suitable insertions of $\mathcal H_\text{int}$ -- which is nothing but what we usually call the "vertices" of a Feynman diagram. You will have to check a textbook if you want more than a two-paragraph answer, but hopefully this sketch is enough to illustrate the idea.

A standard reference is Peskin&Schroder, sections 4.2-4.4.

I didn't really assume relativistic invariance so this argument is also valid in non-relativistic systems. You can also use Feynman diagrams when doing perturbation theory in standard QM. This is not usually done for historical reasons. Old-school perturbation theory, in its algebraic formulation, was properly understood before Feynman introduced his graphical method. But there is nothing intrinsically relativistic about Feynman diagrams and path integrals.

The formula for $\Delta$ is not usually discussed in textbooks because it has no immediate applications at the introductory level. It is hard to compute $\Delta$, in practice, for non-trivial systems. If I recall correctly, Coleman uses this formula in his textbook to compute the energy pumped into a system by an external current. This may be the only example where the solution can actually be calculated exactly.