136
6 Appendix
L ξ , L ξ
≡ L (L ξ ξ ) ,
(6.2.13)
and obey a Jacobi identity
L ξ ,
L ξ , L ξ
+
L ξ ,
L ξ , L ξ
+
L ξ ,
L ξ , L ξ
≡ 0 .
(6.2.14)
These properties imply that the pull-back transformation is nothing but the exponential map of the generator
1
ϕ ∗ (T ) = e
−L ξ T .
(6.2.15)
In this form we obtain a generalization of the Taylor expansion in the generallycovariant context. For instance, for constant ξ
μ in some coordinate system we retrieve
the translations of field theory in flat space-time, where the generators are L ξ = ξ
μ
∂ μ
f (x − ξ) = e
−ξ
μ ∂ μ f (x) ,
ξ
μ
= constant .
(6.2.16)
In general ξ
μ is an arbitrary vector field and, depending on its shape, it can generate
translations, rotations, etc., so (6.2.6) corresponds to translating T along the integral
lines of ξ
μ .
In conclusion, on the one hand we have coordinate transformations, while on the
other hand we have pull-back transformations of tensor fields. In the former case, the
coordinate-independent expression of the tensor T is invariant, by definition, i.e. we
do not change the configuration, only the way it is parametrized (6.2.5), hence the
name “passive” diffeomorphism. In the latter case, the tensor field T is genuinely
transformed (6.2.7), hence the name “active” diffeomorphism. The latter are therefore
the “true” transformations since they do modify the physical configuration and can
be defined without requiring a coordinate system.
Nevertheless, these two transformations are different manifestations of the same
symmetry, since they are derived from the same relation (6.2.5) through two different interpretations. In particular, if an equation is invariant under passive diffeomorphisms, it is also invariant under the active ones, and vice-versa. Therefore, for questions of covariance, the distinction is irrelevant. In contrast, one of the cases where
the distinction between the two transformations is relevant are the transformation
properties of integrals over M. So let us consider, as an example, a four-dimensional
submanifold U ⊂ M and the integral of some scalar density of weight one L over U
S U :=
U
d
4 x L(x) .
(6.2.17)
1 Of course, this description holds only for the diffeomorphisms that are connected to the identity.
6 Appendix
L ξ , L ξ
≡ L (L ξ ξ ) ,
(6.2.13)
and obey a Jacobi identity
L ξ ,
L ξ , L ξ
+
L ξ ,
L ξ , L ξ
+
L ξ ,
L ξ , L ξ
≡ 0 .
(6.2.14)
These properties imply that the pull-back transformation is nothing but the exponential map of the generator
1
ϕ ∗ (T ) = e
−L ξ T .
(6.2.15)
In this form we obtain a generalization of the Taylor expansion in the generallycovariant context. For instance, for constant ξ
μ in some coordinate system we retrieve
the translations of field theory in flat space-time, where the generators are L ξ = ξ
μ
∂ μ
f (x − ξ) = e
−ξ
μ ∂ μ f (x) ,
ξ
μ
= constant .
(6.2.16)
In general ξ
μ is an arbitrary vector field and, depending on its shape, it can generate
translations, rotations, etc., so (6.2.6) corresponds to translating T along the integral
lines of ξ
μ .
In conclusion, on the one hand we have coordinate transformations, while on the
other hand we have pull-back transformations of tensor fields. In the former case, the
coordinate-independent expression of the tensor T is invariant, by definition, i.e. we
do not change the configuration, only the way it is parametrized (6.2.5), hence the
name “passive” diffeomorphism. In the latter case, the tensor field T is genuinely
transformed (6.2.7), hence the name “active” diffeomorphism. The latter are therefore
the “true” transformations since they do modify the physical configuration and can
be defined without requiring a coordinate system.
Nevertheless, these two transformations are different manifestations of the same
symmetry, since they are derived from the same relation (6.2.5) through two different interpretations. In particular, if an equation is invariant under passive diffeomorphisms, it is also invariant under the active ones, and vice-versa. Therefore, for questions of covariance, the distinction is irrelevant. In contrast, one of the cases where
the distinction between the two transformations is relevant are the transformation
properties of integrals over M. So let us consider, as an example, a four-dimensional
submanifold U ⊂ M and the integral of some scalar density of weight one L over U
S U :=
U
d
4 x L(x) .
(6.2.17)
1 Of course, this description holds only for the diffeomorphisms that are connected to the identity.