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## Main Question or Discussion Point

Wald Appendix D talks on why [tex]g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi [/tex] is not conformally invariant when n is not equal to 2.

I want to prove that the Klein Gordon Action (V=0) is not conformally invariant.

However the term that I have in the action is just

[tex] g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi[/tex] which IS clearly invariant in any dimension since [tex]g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi = \nabla^{\nu}\phi\nabla_{\nu}\phi[/tex] where the RHS is just an inner product of two vectors which indeed remains invariant.

PS: [tex]\phi\rightarrow\Omega^s\phi[/tex] for s=0;

I want to prove that the Klein Gordon Action (V=0) is not conformally invariant.

However the term that I have in the action is just

[tex] g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi[/tex] which IS clearly invariant in any dimension since [tex]g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi = \nabla^{\nu}\phi\nabla_{\nu}\phi[/tex] where the RHS is just an inner product of two vectors which indeed remains invariant.

PS: [tex]\phi\rightarrow\Omega^s\phi[/tex] for s=0;