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:

:$g\_\{ab\}\; ightarrow\; g\_\{ab\}\; exp(2omega(x))$

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 covariantunder 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 conformaldimension of -2). Then, under a Weyl transformation,

:$phi\; o\; phi\; e^\{-Delta\; omega\}$

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

:$A\_mu\; o\; A\_mu\; +\; partial\_mu\; omega$

Then, $D\_mu\; phi\; equiv\; partial\_mu\; phi\; +\; Delta\; A\_mu\; phi$ is now covariant and has a conformal dimension of $Delta\; +\; 1$.

For more, see Weyl curvature.