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Under a Weyl transformation $g_{\mu\nu}(x)\rightarrow e^{2\omega(x)}g_{\mu\nu}(x)$

So a scalar $X_\mu X^{\mu}$ should transform as following: $$X_\mu X^{\mu}=g_{\mu\nu}X^\nu X^{\mu}\rightarrow e^{2\omega(x)}g_{\mu\nu}X^\nu X^{\mu}$$ $$=e^{2\omega(x)}X_\mu X^{\mu}$$ where $X^{\mu}$ is a $(1,0)$ tensor on spacetime whose coordinate is given by $x^\alpha$

The above calculation indicates the above-constructed scalar doesn't remain invariant under a Weyl transformation. Is it correct?

aitfel
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    It all depends on the conformal weight attached to the tensor $X^{\mu}$ https://en.wikipedia.org/wiki/Weyl_transformation – KP99 Sep 26 '21 at 08:05
  • Is your transformation global? If so and if the conformal weight of a quantity is the number of length dimensions in it, then there is rescaling symmetry and everything is invariant https://physics.stackexchange.com/questions/620794/cosmology-an-expansion-of-all-length-scales – John Hunter Sep 26 '21 at 08:47

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