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Homework Help: Klein-Gordon operator on a time-ordered product

  1. Nov 4, 2014 #1
    1. The problem statement, all variables and given/known data
    Hey guys,

    So here's the problem I'm faced with. I have to show that

    [itex] (\Box + m^{2})<|T(\phi(x)\phi^{\dagger}(y))|>=-i\delta^{(4)}(x-y) [/itex],

    by acting with the quabla ([itex]\Box[/itex]) operator on the following:


    2. Relevant equations

    3. The attempt at a solution
    So I've split the quabla into its time and spatial derivatives: [itex]\Box = \partial_{0}^{2}-\nabla^{2}[/itex] and I'm applying the time derivative first, using the product rule:

    =\delta(x_{0}-y_{0})\phi(x)\phi^{\dagger}(y) +\theta(x_{0}-y_{0})\dot{\phi}(x)\phi^{\dagger}(y)\\
    +\theta(x_{0}-y_{0})\phi(x)\dot{\phi}^{\dagger}(y) -\delta(x_{0}-y_{0})\phi^{\dagger}(y)\phi(x)
    +\theta(y_{0}-x_{0})\dot{\phi}^{\dagger}(y)\phi(x) +\theta(y_{0}-x_{0})\phi^{\dagger}(y)\dot{\phi}(x)

    However, the two delta terms vanish as the commutator of two fields is 0. so I'm left with

    =\theta(x_{0}-y_{0})\dot{\phi}(x)\phi^{\dagger}(y) +\theta(x_{0}-y_{0})\phi(x)\dot{\phi}^{\dagger}(y)\\
    +\theta(y_{0}-x_{0})\dot{\phi}^{\dagger}(y)\phi(x) +\theta(y_{0}-x_{0})\phi^{\dagger}(y)\dot{\phi}(x)

    At this point I'm meant to be using the equal-time commutation relation: [itex] [\phi(x),\dot{\phi}^{\dagger}(y)] = i\delta^{(3)}(x-y)[/itex] but all my signs are positive...so what do I do?

    Thanks guys...
  2. jcsd
  3. Nov 9, 2014 #2
    Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
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