- #1
Dixanadu
- 254
- 2
Homework Statement
Hey guys, so I gota prove that the charge
[itex]Q=\int d^{3}xJ^{0}(\vec{x},t)[/itex]
is constant in time, that [itex]\dot{Q}=0[/itex]
Homework Equations
[itex]J^{\mu}=i[\phi^{\dagger}(\partial^{\mu}\phi)-(\partial^{\mu}\phi^{\dagger})\phi][/itex]
The Attempt at a Solution
So first what I did was find [itex]J^{0}=i[\phi^{\dagger}\dot{\phi}-\dot{\phi^{\dagger}}\phi][/itex]
Then plug this into Q and differentiate it w.r.t. time, which gives us:
[itex]\dot{Q}=i\int d^{3}x(\phi^{\dagger}\ddot{\phi}-\ddot{\phi^{\dagger}}\phi)[/itex]
And erm, provided I've done it all right (which I probably haven't lol!) i don't know how to show that this is 0?
Thanks in advance guys