Conservation of Noether charge for complex scalar field

[spaghetti3451]

Homework Statement

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.

Homework Equations

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

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?

 

[Fightfish]

This:

Homework Statement

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.

and the fact that there is no interesting physics at infinity (i.e. the fields and whatnot vanish)

 

[spaghetti3451]

Got it!

Thanks!