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Conservation of Noether charge for complex scalar field

  1. Sep 18, 2016 #1
    1. The problem statement, all variables and given/known data

    Prove that the Noether charge ##Q=\frac{i}{2}\int\ d^{3}x\ (\phi^{*}\pi^{*}-\phi\pi)## for a complex scalar field (governed by the Klein-Gordon action) is a constant in time.

    2. Relevant equations

    ##\pi=\dot{\phi}^{*}##

    3. The attempt at a solution

    ##\frac{dQ}{dt}=\frac{i}{2}\int\ d^{3}x\ \frac{d}{dt}(\phi^{*}\pi^{*}-\phi\pi)##

    ##=\frac{i}{2}\int\ d^{3}x\ (\dot{\phi}^{*}\pi^{*}+\phi^{*}\dot{\pi}^{*}-\dot{\phi}\pi-\phi\dot{\pi})##

    ##=\frac{i}{2}\int\ d^{3}x\ (\pi\pi^{*}+\phi^{*}\ddot{\phi}-\pi^{*}\pi-\phi\ddot{\phi}^{*})##

    ##=\frac{i}{2}\int\ d^{3}x\ (\phi^{*}\ddot{\phi}-\phi\ddot{\phi}^{*})##.

    What do I do next?
     
  2. jcsd
  3. Sep 19, 2016 #2
    This:
    and the fact that there is no interesting physics at infinity (i.e. the fields and whatnot vanish)
     
  4. Sep 19, 2016 #3
    Got it!

    Thanks!
     
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