Editing Weyl transformation

{{short description|Local rescaling of a metric tensor}}
{{see also|Wigner–Weyl transform}}

In [[theoretical physics]], the '''Weyl transformation''', named after German mathematician [[Hermann Weyl]], is a local rescaling of the [[metric tensor]]:

<math display=block>g_{ab} \rightarrow e^{-2\omega(x)} g_{ab}</math>

which produces another metric in the same [[conformal class]].  A theory or an expression [[Invariant (mathematics)|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 connection]]s are not invariant under Weyl transformations.  [[Weyl connection]]s are a class of affine connections that is invariant, although no Weyl connection is individual invariant under Weyl transformations.

==Conformal weight==
A quantity <math>\varphi</math> has [[conformal weight]] <math>k</math> if, under the Weyl transformation, it transforms via 

:<math>
   \varphi \to \varphi e^{k \omega}.
</math>

Thus conformally weighted quantities belong to certain [[density bundle]]s; see also [[conformal dimension]].  Let <math>A_\mu</math> be the [[connection one-form]] associated to the Levi-Civita connection of <math>g</math>.  Introduce a connection that depends also on an initial one-form <math>\partial_\mu\omega</math> via

:<math>
   B_\mu = A_\mu + \partial_\mu \omega.
</math>

Then <math>D_\mu \varphi \equiv \partial_\mu \varphi + k B_\mu \varphi</math> is covariant and has conformal weight <math>k - 1</math>.

==Formulas==
For the transformation
:<math> 
    g_{ab} = f(\phi(x)) \bar{g}_{ab} 
</math>    
We can derive the following formulas
:<math> 
\begin{align}
    g^{ab} &= \frac{1}{f(\phi(x))} \bar{g}^{ab}\\
    \sqrt{-g} &= \sqrt{-\bar{g}} f^{D/2} \\
    \Gamma^c_{ab} &= \bar{\Gamma}^c_{ab} + \frac{f'}{2f} \left(\delta^c_b \partial_a \phi + \delta^c_a \partial_b \phi - \bar{g}_{ab} \partial^c \phi \right) \equiv \bar{\Gamma}^c_{ab} + \gamma^c_{ab} \\
     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{align}
</math>
Note that the Weyl tensor is invariant under a Weyl rescaling.

==References==
*{{cite book |first=Hermann |last=Weyl |title=Raum, Zeit, Materie |trans-title=Space, Time, Matter |series=Lectures on General Relativity |language=German |location=Berlin |publisher=Springer |orig-date=1921 |year=1993 |isbn=3-540-56978-2 }}

[[Category:Conformal geometry]]
[[Category:Differential geometry]]
[[Category:Scaling symmetries]]
[[Category:Symmetry]]
[[Category:Theoretical physics]]


{{relativity-stub}}
{{quantum-stub}}
{{theoretical-physics-stub}}
{{differential-geometry-stub}}