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Expectation value of the time evolution operator

  1. Nov 19, 2013 #1
    This problem pertains to the perturbative expansion of correlation functions in QFT.

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

    Show that [itex]\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}[/itex]

    2. Relevant equations

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

    3. The attempt at a solution

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

    [itex]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[/itex]

    From there it follows that [itex]\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}[/itex] 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
  2. jcsd
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