# Relation between adiabatic approximation and imaginary time

Tags:
1. May 7, 2015

### taishizhiqiu

Regarding interacting green's function, I found two different description:

1. usually in QFT:
$<\Omega|T\{ABC\}|\Omega>=\lim\limits_{T \to \infty(1-i\epsilon)}\frac{<0|T\{A_IB_I U(-T,T)\}|0>}{<0|T\{U(-T,T)\}|0>}$

2. usually in quantum many body systems:
$<\Omega|T\{ABC\}|\Omega>=\frac{<0|T\{A_IB_I S\}|0>}{<0|T\{S\}|0>}$
where interaction is switched off at $T=\pm\infty$ (adiabatic approximation)

Is there any connection between the two descriptions?

2. May 11, 2015

### thierrykauf

yes but it will take weeks of study to get there. Basically the difficulty is that ordinary QFT assumes zero temperature (otherwise there is no longer a definition for asymptotic states. QFT at finite temperarure is basically the interaction of many particles so the proper formalism is Statistical Mechanics, more precisely the expectation over states which can be reinterpretated into a scattering process in time (even though there is no "time" in the expectation value) by the substitution
Temperature = imaginary time

3. May 11, 2015

### taishizhiqiu

So can you recommend some materials for me?

4. May 11, 2015

### thierrykauf

5. May 12, 2015

### thierrykauf

Thermal field theory is a world of its own. Quantum field theory rests on the assumption that you can define an "in" and an "out" state, that is if you go far enough from the interacting region you can speak of an initial unperturbed state and a final state that has been perturbed by the interaction but no longer is. That's because at zero temperature, which is what the vacuum is, you can speak of an unperturbed initial state, but at finite temperature, there are particles everywhere, so the notion of <in|out> with interaction only in between is no longer as simple. Here is a good source for you.
https://workspace.imperial.ac.uk/th...issertations/2011/Yuhao Yang Dissertation.pdf

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook