Infinite time limit in two-point correlation function

Question

I am reading the derivation of the two-point correlation function in Peskin and Schroeder (section 4.2). I don't understand the infinite time limit that is taken between eq. (4.26) and (4.27).

They write

$$e^{-i H T} |0\rangle=e^{-iE_0 T}|\Omega\rangle+\sum_{n\neq 0}e^{-i E_n T}|n\rangle \langle n|0\rangle\tag{4.27}$$

where $|\Omega\rangle$ is the ground state in the interaction theory, $E_0<E_n$ are the eigenvalues of the Hamiltonian $H|n\rangle=E_n|n\rangle$.

Peskin and Schröder are now considering the limit $T\to \infty (1-i\epsilon)$. It is clear that in this limit only the $|\Omega\rangle$ term remains (then one can relate $|\Omega\rangle$ and $|0\rangle$). However I am wondering why it is allowed to take this limit? The time variable should always be real and this looks like cheating!?

Answer 1

It is a trick, no more, no less. It is not meant to be rigorous -- this is another one of those formal manipulations that you find all over the place in introductory textbooks.

The conclusion, the result you need to remember, is that propagators carry a $+i\epsilon$ in the denominator. P&S derive this using the $t\to(1-i\epsilon)\infty$ trick. There are other derivations that you may find more convincing. You could even take this prescription as a definition. Whatever argument you use, the end-result is the same: propagators carry a $+i\epsilon$ in the denominator.

For example, you could reach the same conclusion by sending $t\to\infty$ along the real axis, and using the Riemann–Lebesgue lemma to kill oscillatory terms. Making this precise, though, requires being careful about analyticity, so you'd have to work harder. Specifically, you need to make sure that you really understand the structure of poles so you know from which side your functions are boundary-values.

Another alternative is to derive the prescription using appropriate boundary conditions on the path integral. See e.g. M. Schwartz QFT textbook, §14.4. Long story short, projecting onto the ground state can be achieved by inserting a damping term $$ \exp\biggl[ -\frac12\epsilon\phi^2\biggr] $$ which, of course, is equivalent to adding a small imaginary mass term, i.e., a $+i\epsilon$ in the denominator.

Answer 2

This comes from the physics lore that quantum field theory (QFT) in Minkowski spacetime can be analytically continued (Wick rotated $\beta=iT$) to statistical physics (SP) in Euclidean spacetime.

The upshot is that QFT in Minkowski spacetime should be regularized with the Feynman $i\epsilon$ prescription.

In SP the ground state dominates at low temperature $\beta\to\infty$, which corresponds to the infinite time limit $T\to\infty$ in QFT.