Weyl_transformation Knowpia

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:

g_{ab}\rightarrow e^{-2\omega (x)}g_{ab}

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 $\varphi$ has conformal weight $k$ if, under the Weyl transformation, it transforms via

\varphi \to \varphi e^{k\omega }.

Thus conformally weighted quantities belong to certain density bundles; see also conformal dimension. Let $A_{\mu }$ be the connection one-form associated to the Levi-Civita connection of $g$ . Introduce a connection that depends also on an initial one-form $\partial _{\mu }\omega$ via

B_{\mu }=A_{\mu }+\partial _{\mu }\omega .

Then $D_{\mu }\varphi \equiv \partial _{\mu }\varphi +kB_{\mu }\varphi$ is covariant and has conformal weight $k-1$ .

Formulas edit

For the transformation

g_{ab}=f(\phi (x)){\bar {g}}_{ab}

We can derive the following formulas

{\begin{aligned}g^{ab}&={\frac {1}{f(\phi (x))}}{\bar {g}}^{ab}\\{\sqrt {-g}}&={\sqrt {-{\bar {g}}}}f^{D/2}\\\Gamma _{ab}^{c}&={\bar {\Gamma }}_{ab}^{c}+{\frac {f'}{2f}}\left(\delta _{b}^{c}\partial _{a}\phi +\delta _{a}^{c}\partial _{b}\phi -{\bar {g}}_{ab}\partial ^{c}\phi \right)\equiv {\bar {\Gamma }}_{ab}^{c}+\gamma _{ab}^{c}\\R_{ab}&={\bar {R}}_{ab}+{\frac {f''f-f^{\prime 2}}{2f^{2}}}\left((2-D)\partial _{a}\phi \partial _{b}\phi -{\bar {g}}_{ab}\partial ^{c}\phi \partial _{c}\phi \right)+{\frac {f'}{2f}}\left((2-D){\bar {\nabla }}_{a}\partial _{b}\phi -{\bar {g}}_{ab}{\bar {\Box }}\phi \right)+{\frac {1}{4}}{\frac {f^{\prime 2}}{f^{2}}}(D-2)\left(\partial _{a}\phi \partial _{b}\phi -{\bar {g}}_{ab}\partial _{c}\phi \partial ^{c}\phi \right)\\R&={\frac {1}{f}}{\bar {R}}+{\frac {1-D}{f}}\left({\frac {f''f-f^{\prime 2}}{f^{2}}}\partial ^{c}\phi \partial _{c}\phi +{\frac {f'}{f}}{\bar {\Box }}\phi \right)+{\frac {1}{4f}}{\frac {f^{\prime 2}}{f^{2}}}(D-2)(1-D)\partial _{c}\phi \partial ^{c}\phi \end{aligned}}

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.

Summary

Conformal weight edit

Formulas edit

References edit