# Expectation value of the time evolution operator

1. Nov 19, 2013

### Catria

This problem pertains to the perturbative expansion of correlation functions in QFT.

1. The problem statement, all variables and given/known data

Show that $\langle0|T\left[exp\left(i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle = \left(\langle0|T\left[exp\left(-i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle\right)^{-1}$

2. Relevant equations

$H_{I}$ satisfies $i \frac{\partial U}{\partial t} = H_{I}U(t)$ and is the Hamiltonian in the interaction picture for the $\varphi_{in}$ fields, which was used in the perturbative expansion. Also, $|0\rangle\langle0| = 1$

3. The attempt at a solution

$1=\langle0|1|0\rangle = \langle0|T\left[exp\left((i-i)\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle$

$1=\langle0|T\left[exp\left(i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle\langle0|T\left[exp\left(-i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle$

From there it follows that $\langle0|T\left[exp\left(i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle = \left(\langle0|T\left[exp\left(-i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle\right)^{-1}$ simply by dividing...

I can't see what's wrong with my solution... but I have the feeling that there is something wrong.

Last edited: Nov 19, 2013