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phyz2

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- TL;DR Summary
- Confused bout KG Lagrangian in curved space

Hello!

I'm starting to study curved QFT and am slightly confused about the invariance of the Klein Gordon Lagrangian under a linear diffeomorphism.

This is $$L=\sqrt{-g}\left(g^{\mu\nu}\partial_\mu \phi \partial_\nu \phi-\frac{m^2}{2}\phi^2\right),$$

I don't see how ##g^{\mu\nu}\to g^{\mu\nu}+\partial^{(\mu}\xi^{\nu)}## should leave this invariant.. I get and extra piece which is ##\sqrt{-g} \partial^{(\mu}\xi^{\nu)} \partial_\mu \phi \partial_\nu \phi## and I don't see how this should be zero, even at the linear level.

Thanks for any help!

I'm starting to study curved QFT and am slightly confused about the invariance of the Klein Gordon Lagrangian under a linear diffeomorphism.

This is $$L=\sqrt{-g}\left(g^{\mu\nu}\partial_\mu \phi \partial_\nu \phi-\frac{m^2}{2}\phi^2\right),$$

I don't see how ##g^{\mu\nu}\to g^{\mu\nu}+\partial^{(\mu}\xi^{\nu)}## should leave this invariant.. I get and extra piece which is ##\sqrt{-g} \partial^{(\mu}\xi^{\nu)} \partial_\mu \phi \partial_\nu \phi## and I don't see how this should be zero, even at the linear level.

Thanks for any help!