# Weyl transformation

*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:

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 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 connections are not invariant under Weyl transformations. Weyl connections are a class of affine connections that is invariant, although no Weyl connection is individual invariant under Weyl transformations.

## Conformal weight[edit]

A quantity has conformal weight if, under the Weyl transformation, it transforms via

Thus conformally weighted quantities belong to certain density bundles; see also conformal dimension. Let be the connection one-form associated to the Levi-Civita connection of . Introduce a connection that depends also on an initial one-form via

Then is covariant and has conformal weight .

## Formulas[edit]

For the transformation

We can derive the following formulas

Note that the Weyl tensor is invariant under a Weyl rescaling.

## References[edit]

- Weyl, Hermann (1993) [1921].
*Raum, Zeit, Materie*[*Space, Time, Matter*]. Lectures on General Relativity (in German). Berlin: Springer. ISBN 3-540-56978-2.