- Weyl transformation
:"See also
Weyl quantization , for another definition of the Weyl transform."In
theoretical physics , the Weyl transformation is a local rescaling of themetric 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 importantsymmetry inconformal field theory - for example a symmetry of thePolyakov action .Note that the ordinary
affine connection (and alsospin connection ) are no longer covariantunder Weyl transformations. To restore covariance, we need to introduce aWeyl 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 .
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