Weyl transformation

See also Weyl quantization, for another definition of the Weyl transform.

In theoretical physics, the Weyl transformation is a local rescaling of the metric tensor:

$g_{ab}\rightarrow g_{ab} \exp(2\omega(x))$

The invariance of a theory or an expression under this transformation is called the Weyl symmetry. It is an important symmetry in conformal field theory - for example a symmetry of the Polyakov action.

Note that the ordinary affine connection (and also spin connection) are no longer covariant under Weyl transformations. To restore covariance, we need to introduce a Weyl connection.

Let's say we have a scalar field with a conformal dimension of Δ. (The metric has a conformal dimension of -2). Then, under a Weyl transformation,

$\phi \to \phi e^{-\Delta \omega}$

and unless Δ=0, the partial derivative is no longer Weyl covariant. We need to introduce a Weyl connection which goes as

$A_\mu \to A_\mu + \partial_\mu \omega$

Then, $D_\mu \phi \equiv \partial_\mu \phi + \Delta A_\mu \phi$ is now covariant and has a conformal dimension of $Δ + 1$ .

For more, see Weyl curvature.

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