Editing Weyl transformation (section)

==Conformal weight==
A quantity <math>\varphi</math> has [[conformal weight]] <math>k</math> if, under the Weyl transformation, it transforms via 

:<math>
   \varphi \to \varphi e^{k \omega}.
</math>

Thus conformally weighted quantities belong to certain [[density bundle]]s; see also [[conformal dimension]].  Let <math>A_\mu</math> be the [[connection one-form]] associated to the Levi-Civita connection of <math>g</math>.  Introduce a connection that depends also on an initial one-form <math>\partial_\mu\omega</math> via

:<math>
   B_\mu = A_\mu + \partial_\mu \omega.
</math>

Then <math>D_\mu \varphi \equiv \partial_\mu \varphi + k B_\mu \varphi</math> is covariant and has conformal weight <math>k - 1</math>.