# Using Noether's Theorem find a continuity equation for KG

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1. Sep 11, 2016

### It's me

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. Sep 12, 2016

### Twigg

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. Sep 12, 2016

### It's me

If I show that, does Noether's theorem immediately guarantee such a continuity equation?

4. Sep 12, 2016

### Twigg

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.