Editing Weyl transformation (section)

{{short description|Local rescaling of a metric tensor}}
{{see also|Wigner–Weyl transform}}

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

<math display=block>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 (mathematics)|invariant]] under this transformation is called [[conformally invariant]], or is said to possess '''Weyl invariance''' or '''Weyl symmetry'''.  The Weyl symmetry is an important [[symmetry]] in [[conformal field theory]].  It is, for example, a symmetry of the [[Polyakov action]]. When quantum mechanical effects break the conformal invariance of a theory, it is said to exhibit a [[conformal anomaly]] or '''Weyl anomaly'''.

The ordinary [[Levi-Civita connection]] and associated [[spin connection]]s are not invariant under Weyl transformations.  [[Weyl connection]]s are a class of affine connections that is invariant, although no Weyl connection is individual invariant under Weyl transformations.