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:''See also [[Wigner–Weyl transform]], for another definition of the Weyl transform.''

In [[theoretical physics]], the '''Weyl transformation''', named after [[Hermann Weyl]], is a local rescaling of the [[metric tensor]]:

:<math>g_{ab}\rightarrow e^{-2\omega(x)}g_{ab}</math>

which produces another metric in the same [[conformal class]].  A theory or an expression invariant under this transformation is called [[conformally invariant]], or is said to possess '''Weyl symmetry'''.  The Weyl symmetry is an important [[symmetry]] in [[conformal field theory]].  It is, for example, a symmetry of the [[Polyakov action]].

The ordinary [[Levi-Civita connection]] and associated [[spin connection]]s are not invariant under Weyl transformations. An appropriately invariant notion is the '''Weyl connection''', which is one way of specifying the structure of a [[conformal connection]].

A quantity φ has [[conformal weight]] ''k'' 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 ''A''<sub>μ</sub> be the [[connection one-form]] associated to the Levi-Civita connection of ''g''.  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>.
==Literature==
* Hermann  Weyl, ''Raum, Zeit, Materie'' (Space, Time, Matter), Lectures on General Relativity, in German. Berlin, Springer 1921, with later reprints in 1993. ISBN 3-540-56978-2
[[Category:Theoretical physics]]
[[Category:Differential geometry]]
[[Category:Symmetry]]
[[Category:Scaling symmetries]]


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