- #1
spaghetti3451
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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}^{*})##.
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