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Peskin and Schroeder - Derivation of equation (2.45)

  1. Oct 20, 2015 #1
    I'm having trouble deriving equation (2.45) on page 25. In particular, in the derivation of

    ##i\frac{\partial}{\partial t}\pi({\bf{x}},t) = -i(-\nabla^{2}+m^{2}) \phi({\bf{x}},t)##,

    I need to show that

    ##\frac{1}{2}\pi({\bf{x}},t) \phi({\bf{x'}},t)(-\nabla^{2}+m^{2}) \phi({\bf{x'}},t) - \frac{1}{2} \phi({\bf{x'}},t)(-\nabla^{2}+m^{2})\phi({\bf{x'}},t)\pi({\bf{x}},t) = -i \delta^{(3)}({\bf{x}}-{\bf{x'}})(-\nabla^{2}+m^{2})\phi({\bf{x'}},t)##.

    Now, this is what I've done so far:

    ##\frac{1}{2}\pi({\bf{x}},t) \phi({\bf{x'}},t)(-\nabla^{2}+m^{2}) \phi({\bf{x'}},t) - \frac{1}{2} \phi({\bf{x'}},t)(-\nabla^{2}+m^{2})\phi({\bf{x'}},t)\pi({\bf{x}},t)##

    ##=\frac{1}{2}\pi({\bf{x}},t) \phi({\bf{x'}},t)(-\nabla^{2}+m^{2}) \phi({\bf{x'}},t)-\frac{1}{2}\phi({\bf{x'}},t)\pi({\bf{x}},t) (-\nabla^{2}+m^{2}) \phi({\bf{x'}},t) +\frac{1}{2}\phi({\bf{x'}},t)\pi({\bf{x}},t) (-\nabla^{2}+m^{2}) \phi({\bf{x'}},t)- \frac{1}{2} \phi({\bf{x'}},t)(-\nabla^{2}+m^{2})\phi({\bf{x'}},t)\pi({\bf{x}},t)##

    ##=\frac{1}{2}[\pi({{\bf{x}},t}), \phi({{\bf{x'}},t})](-\nabla^{2}+m^{2}) \phi({\bf{x'}},t) + \frac{1}{2}\phi({\bf{x'}},t)[\pi({\bf{x}},t), (-\nabla^{2}+m^{2}) \phi({\bf{x'}},t)]##

    ##=\frac{1}{2}(-i)\delta^{(3)}({\bf{x'}}-{\bf{x}})(-\nabla^{2}+m^{2}) \phi({\bf{x'}},t) + ??##.

    How do I evaluate the second commutator?
     
  2. jcsd
  3. Oct 23, 2015 #2
  4. Oct 23, 2015 #3

    bapowell

    User Avatar
    Science Advisor

    Did you try breaking the second commutator up as [itex][\pi,m^2\phi] - [\pi,\nabla^2\phi][/itex] and convincing yourself that both the [itex]m^2[/itex] and [itex]\nabla^2[/itex] can be pulled out of their respective commutators?
     
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