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Using Noether's Theorem find a continuity equation for KG

  1. Sep 11, 2016 #1
    1. The problem statement, all variables and given/known data

    Consider the Klein-Gordon equation ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##. Using Noether's theorem, find a continuity equation of the form ##\partial_\mu j^{\mu}=0##.

    2. Relevant equations

    ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##

    3. The attempt at a solution

    I really haven't been able to solve this problem because I don't understand why Noether's Theorem would be useful in this case. Any help would be greatly appreciated.
     
  2. jcsd
  3. Sep 12, 2016 #2

    Twigg

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    Gold Member

    The Klein-Gordon equation, like any quantum wave equation, is invariant under a complex phase shift of the wave function. You can show that this is a 1-parameter continuous symmetry.
     
  4. Sep 12, 2016 #3
    If I show that, does Noether's theorem immediately guarantee such a continuity equation?
     
  5. Sep 12, 2016 #4

    Twigg

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    Gold Member

    The conserved current you're looking for is the one predicted by Noether's theorem given that the Lagrangian is invariant under a phase shift of the wavefunction.
     
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