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
a
b
→
e
−
2
ω
(
x
)
g
a
b
{\displaystyle g_{ab}\rightarrow e^{-2\omega (x)}g_{ab}}
which produces another metric in the same
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 A quantity
φ
{\displaystyle \varphi }
conformal weight
k
{\displaystyle k}
if, under the Weyl transformation, it transforms via
φ
→
φ
e
k
ω
.
{\displaystyle \varphi \to \varphi e^{k\omega }.}
Thus conformally weighted quantities belong to certain
. Let
A
μ
{\displaystyle A_{\mu }}
be the
connection one-form
associated to the Levi-Civita connection of
g
{\displaystyle g}
. Introduce a connection that depends also on an initial one-form
∂
μ
ω
{\displaystyle \partial _{\mu }\omega }
via
B
μ
=
A
μ
+
∂
μ
ω
.
{\displaystyle B_{\mu }=A_{\mu }+\partial _{\mu }\omega .}
Then
D
μ
φ
≡
∂
μ
φ
+
k
B
μ
φ
{\displaystyle D_{\mu }\varphi \equiv \partial _{\mu }\varphi +kB_{\mu }\varphi }
k
−
1
{\displaystyle k-1}
Formulas For the transformation
g
a
b
=
f
(
ϕ
(
x
)
)
g
¯
a
b
{\displaystyle g_{ab}=f(\phi (x)){\bar {g}}_{ab}}
We can derive the following formulas
g
a
b
=
1
f
(
ϕ
(
x
)
)
g
¯
a
b
−
g
=
−
g
¯
f
D
/
2
Γ
a
b
c
=
Γ
¯
a
b
c
+
f
′
2
f
(
δ
b
c
∂
a
ϕ
+
δ
a
c
∂
b
ϕ
−
g
¯
a
b
∂
c
ϕ
)
≡
Γ
¯
a
b
c
+
γ
a
b
c
R
a
b
=
R
¯
a
b
+
f
″
f
−
f
′
2
2
f
2
(
(
2
−
D
)
∂
a
ϕ
∂
b
ϕ
−
g
¯
a
b
∂
c
ϕ
∂
c
ϕ
)
+
f
′
2
f
(
(
2
−
D
)
∇
¯
a
∂
b
ϕ
−
g
¯
a
b
◻
¯
ϕ
)
+
1
4
f
′
2
f
2
(
D
−
2
)
(
∂
a
ϕ
∂
b
ϕ
−
g
¯
a
b
∂
c
ϕ
∂
c
ϕ
)
R
=
1
f
R
¯
+
1
−
D
f
(
f
″
f
−
f
′
2
f
2
∂
c
ϕ
∂
c
ϕ
+
f
′
f
◻
¯
ϕ
)
+
1
4
f
f
′
2
f
2
(
D
−
2
)
(
1
−
D
)
∂
c
ϕ
∂
c
ϕ
{\displaystyle {\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 Weyl, Hermann (1993) [1921]. Raum, Zeit, Materie [Space, Time, Matter ]. Lectures on General Relativity (in German). Berlin: Springer. .
