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Using Noether's Theorem find a continuity equation for KG

  • #1
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Homework Statement



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##.

Homework Equations



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

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.
 

Answers and Replies

  • #2
Twigg
Gold Member
196
35
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.
 
  • #3
35
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If I show that, does Noether's theorem immediately guarantee such a continuity equation?
 
  • #4
Twigg
Gold Member
196
35
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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