Using Noether's Theorem find a continuity equation for KG

[It's me]

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.

 

[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.

 

[It's me]

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

 

[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.